What does \lim_{x \to a} f(x)=L say?

The values of f(x) approach L as x approaches a from both sides. The statement doesn't require f(a)=L or even require f(a) to exist.

How can a table estimate \lim_{x \to a} f(x)?

Use inputs approaching a from below and above, then look for a common output value. Values exactly at x=a don't determine the limit.

When does direct substitution evaluate a limit?

When the function is continuous at the target input. Then \lim_{x \to a} f(x)=f(a).

Three conditions for continuity at x=a?

f(a) exists, \lim_{x \to a}f(x) exists, and \lim_{x \to a}f(x)=f(a).

Intermediate Value Theorem: hypotheses and conclusion?

If f is continuous on [a,b] and N lies between f(a) and f(b), then some c in [a,b] satisfies f(c)=N. If N is strictly between the endpoint values, c lies in (a,b).

When does a two-sided limit equal L?

Exactly when both one-sided limits equal L: \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L. If the one-sided limits differ, the two-sided limit doesn't exist.

How do you read a finite limit from a graph?

Follow the graph toward the target x-value from both sides. The common approached y-value is the limit, regardless of a hole or a differently placed filled point.

Limit law for a sum or difference?

If both component limits exist, \lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).

What makes a discontinuity removable?

The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.

Squeeze Theorem: usable form?

If g(x)\le f(x)\le h(x) near a and \lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L, then \lim_{x\to a}f(x)=L.

What does \lim_{x\to a}f(x)=+\infty mean?

f(x) grows without bound above as x approaches a. It describes unbounded behavior, not a finite limit value.

What must a table show for a left-hand limit?

Inputs less than the target and moving toward it. For \lim_{x\to a^-}f(x), use x<a with x getting closer to a.

Limit law for a product?

If both limits exist, \lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).

Graph signature of a jump discontinuity?

The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.

Which theorem can guarantee a root on [a,b]?

The Intermediate Value Theorem. If f is continuous on [a,b] and 0 lies between f(a) and f(b), then f(c)=0 for some c in the interval.

Horizontal asymptote from a limit at infinity?

If \lim_{x\to\infty}f(x)=L or \lim_{x\to-\infty}f(x)=L, then y=L is a horizontal asymptote in that direction.

What does an open circle say about a graph's limit?

Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.

Limit law for a quotient—and its condition?

If both limits exist and the denominator limit is nonzero, \lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}. The law doesn't apply when the denominator limit is 0.

What does continuity on [a,b] require at the endpoints?

Continuity on (a,b), right-continuity at a, and left-continuity at b: \lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).

When is the Squeeze Theorem a natural choice?

When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.

Vertical asymptote from one-sided behavior?

If at least one one-sided limit at x=a is +\infty or -\infty, then x=a is a vertical asymptote.

Limit at infinity of equal-degree rational functions?

The ratio of the leading coefficients: \lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}. This assumes b_n\ne0.

When can a limit pass through a continuous outer function?

If \lim_{x\to a}g(x)=L and f is continuous at L, then \lim_{x\to a}f(g(x))=f(L).

What makes a discontinuity infinite?

The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.

Left limit =2 and right limit =5: two-sided limit?

It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.

Standard trigonometric limit behind \frac{\sin x}{x}?

With angles in radians, \lim_{x\to0}\frac{\sin x}{x}=1. Equivalent scaled forms follow by substitution.

Continuity of a composition?

If g is continuous at a and f is continuous at g(a), then f\circ g is continuous at a.

Limit at infinity when a rational numerator has lower degree?

0. If the numerator's degree is less than the denominator's, the denominator dominates as x\to\pm\infty.

What does the indeterminate form 0/0 tell you?

Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; 0/0 isn't the limit value.

When do opposite infinite one-sided limits give a two-sided limit?

They don't. For example, +\infty from the left and -\infty from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.

How do you choose a parameter to make a piecewise function continuous?

Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.

Value of \lim_{x\to0}\frac{1-\cos x}{x}?

0. Rationalizing gives a product involving \sin x/x and a factor that approaches 0.

Can \lim_{x\to a}f(x) exist when f(a) doesn't?

Yes. A limit uses nearby values, so a hole at x=a can coexist with a finite two-sided limit.

What graph behavior makes a finite limit fail even without a jump?

Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.

Average rate of change of f on [a,b]?

\frac{f(b)-f(a)}{b-a} It is the slope of the secant line through (a,f(a)) and (b,f(b)).

Derivative at x=a using an increment h?

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h} The derivative exists only if this finite limit exists.

Tangent-line equation to y=f(x) at x=a?

y-f(a)=f'(a)(x-a) This requires f'(a) to exist.

What does differentiability imply about continuity?

If f is differentiable at a, then f is continuous at a. The converse is false: continuity alone doesn't guarantee differentiability.

Power rule for derivatives?

\frac{d}{dx}x^n=nx^{n-1} Apply it where the original real-valued power function and its derivative are defined.

Output units of f per input unit of x. A derivative is a rate of change, so its units are a quotient.

Derivative at x=a using x\to a?

f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a} This is equivalent to the h-form after setting h=x-a.

How does a graph of f show the sign of f'?

f'(x)>0 where f rises as x increases, and f'(x)<0 where f falls. A horizontal tangent gives f'(x)=0 when the derivative exists.

Derivative of a constant?

\frac{d}{dx}C=0 A constant function has zero rate of change.

Derivative of \sin x?

\frac{d}{dx}(\sin x)=\cos x The angle must be measured in radians for the standard formula.

\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x) Differentiating each factor and multiplying the results is not the product rule.

How can nearby table values estimate f'(a)?

Use a difference quotient with inputs close to a. A symmetric estimate is f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}. Smaller h often helps, subject to the data's precision.

What does f''(x) measure?

The rate of change of f'(x) with respect to x. Its units are the units of f per square input unit.

Instantaneous rate of change of f at a?

f'(a). It is the limit of average rates over intervals shrinking to a, and geometrically it is the tangent-line slope.

Derivative of a sum or difference?

\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)

Derivative of \cos x?

\frac{d}{dx}(\cos x)=-\sin x The standard formula assumes radians.

For g(x)\ne0, \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}. The order in the numerator matters.

AP Calculus AB Flashcards: 8-Unit Formulas, Theorems & Concepts

Review all eight AP Calculus AB units with formula conditions, theorem hypotheses, notation, graph interpretation, and concept cards.

Tác giả Open Study Contributor

Về bộ thẻ này

Review the formulas, theorem conditions, notation, and representation links that support all eight AP Calculus AB units. The cards practice short recall in both useful directions: a name or situation can cue a formula, condition, theorem, or interpretation, and selected formulas or hypotheses can cue their meaning or use. Graph, table, symbolic, and word-based cards connect derivatives and integrals to rates, accumulation, extrema, concavity, motion, area, and volume.

The sequence starts with limits and the derivative definition, then builds through derivative rules and applications before moving to integration, differential equations, and applications of integration. Related variants are separated in the final order so one card does not simply reveal the next.

This is an AB-only review deck. It excludes parametric, polar, and vector-valued calculus; infinite sequences and series; integration by parts; partial fractions; improper integrals; logistic differential equations; arc length; and other BC-only material. It also excludes copied exam questions, mark schemes, curriculum prose, full theorem proofs, and long multi-step practice problems. Flashcards support recall; learners should still solve original practice problems and complete timed exam practice.

Every prompt, answer, explanation, metadata field, and ordering choice is independently written from common mathematical knowledge after checking the current official scope. No College Board question, mark scheme, curriculum text, logo, or trade dress is copied. AP® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this product. Official course requirements: AP Calculus AB Course.

Thẻ trong bộ này

Thẻ 1
Câu hỏi
What does $\lim_{x \to a} f(x)=L$ say?
Câu trả lời
The values of $f(x)$ approach $L$ as $x$ approaches $a$ from both sides. The statement doesn't require $f(a)=L$ or even require $f(a)$ to exist.
Thẻ 2
Câu hỏi
How can a table estimate $\lim_{x \to a} f(x)$ ?
Câu trả lời
Use inputs approaching $a$ from below and above, then look for a common output value. Values exactly at $x=a$ don't determine the limit.
Thẻ 3
Câu hỏi
When does direct substitution evaluate a limit?
Câu trả lời
When the function is continuous at the target input. Then
$\lim_{x \to a} f(x)=f(a).$
Thẻ 4
Câu hỏi
Three conditions for continuity at $x=a$ ?
Câu trả lời
$f(a)$ exists, $\lim_{x \to a}f(x)$ exists, and
$\lim_{x \to a}f(x)=f(a).$
Thẻ 5
Câu hỏi
Intermediate Value Theorem: hypotheses and conclusion?
Câu trả lời
If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$ , then some $c$ in $[a,b]$ satisfies $f(c)=N$ . If $N$ is strictly between the endpoint values, $c$ lies in $(a,b)$ .
Thẻ 6
Câu hỏi
When does a two-sided limit equal $L$ ?
Câu trả lời
Exactly when both one-sided limits equal $L$ :
$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$
If the one-sided limits differ, the two-sided limit doesn't exist.
Thẻ 7
Câu hỏi
How do you read a finite limit from a graph?
Câu trả lời
Follow the graph toward the target $x$ -value from both sides. The common approached $y$ -value is the limit, regardless of a hole or a differently placed filled point.
Thẻ 8
Câu hỏi
Limit law for a sum or difference?
Câu trả lời
If both component limits exist,
$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$
Thẻ 9
Câu hỏi
What makes a discontinuity removable?
Câu trả lời
The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.
Thẻ 10
Câu hỏi
Squeeze Theorem: usable form?
Câu trả lời
If $g(x)\le f(x)\le h(x)$ near $a$ and
$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$
then $\lim_{x\to a}f(x)=L$ .
Thẻ 11
Câu hỏi
What does $\lim_{x\to a}f(x)=+\infty$ mean?
Câu trả lời
$f(x)$ grows without bound above as $x$ approaches $a$ . It describes unbounded behavior, not a finite limit value.
Thẻ 12
Câu hỏi
What must a table show for a left-hand limit?
Câu trả lời
Inputs less than the target and moving toward it. For $\lim_{x\to a^-}f(x)$ , use $x<a$ with $x$ getting closer to $a$ .
Thẻ 13
Câu hỏi
Limit law for a product?
Câu trả lời
If both limits exist,
$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$
Thẻ 14
Câu hỏi
Graph signature of a jump discontinuity?
Câu trả lời
The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.
Thẻ 15
Câu hỏi
Which theorem can guarantee a root on $[a,b]$ ?
Câu trả lời
The Intermediate Value Theorem. If $f$ is continuous on $[a,b]$ and $0$ lies between $f(a)$ and $f(b)$ , then $f(c)=0$ for some $c$ in the interval.
Thẻ 16
Câu hỏi
Horizontal asymptote from a limit at infinity?
Câu trả lời
If $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$ , then $y=L$ is a horizontal asymptote in that direction.
Thẻ 17
Câu hỏi
What does an open circle say about a graph's limit?
Câu trả lời
Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.
Thẻ 18
Câu hỏi
Limit law for a quotient—and its condition?
Câu trả lời
If both limits exist and the denominator limit is nonzero,
$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}.$
The law doesn't apply when the denominator limit is $0$ .
Thẻ 19
Câu hỏi
What does continuity on $[a,b]$ require at the endpoints?
Câu trả lời
Continuity on $(a,b)$ , right-continuity at $a$ , and left-continuity at $b$ :
$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$
Thẻ 20
Câu hỏi
When is the Squeeze Theorem a natural choice?
Câu trả lời
When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.
Thẻ 21
Câu hỏi
Vertical asymptote from one-sided behavior?
Câu trả lời
If at least one one-sided limit at $x=a$ is $+\infty$ or $-\infty$ , then $x=a$ is a vertical asymptote.
Thẻ 22
Câu hỏi
Limit at infinity of equal-degree rational functions?
Câu trả lời
The ratio of the leading coefficients:
$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$
This assumes $b_n\ne0$ .
Thẻ 23
Câu hỏi
When can a limit pass through a continuous outer function?
Câu trả lời
If $\lim_{x\to a}g(x)=L$ and $f$ is continuous at $L$ , then
$\lim_{x\to a}f(g(x))=f(L).$
Thẻ 24
Câu hỏi
What makes a discontinuity infinite?
Câu trả lời
The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.
Thẻ 25
Câu hỏi
Left limit $=2$ and right limit $=5$ : two-sided limit?
Câu trả lời
It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.
Thẻ 26
Câu hỏi
Standard trigonometric limit behind $\frac{\sin x}{x}$ ?
Câu trả lời
With angles in radians,
$\lim_{x\to0}\frac{\sin x}{x}=1.$
Equivalent scaled forms follow by substitution.
Thẻ 27
Câu hỏi
Continuity of a composition?
Câu trả lời
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$ , then $f\circ g$ is continuous at $a$ .
Thẻ 28
Câu hỏi
Limit at infinity when a rational numerator has lower degree?
Câu trả lời
$0$ . If the numerator's degree is less than the denominator's, the denominator dominates as $x\to\pm\infty$ .
Thẻ 29
Câu hỏi
What does the indeterminate form $0/0$ tell you?
Câu trả lời
Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; $0/0$ isn't the limit value.
Thẻ 30
Câu hỏi
When do opposite infinite one-sided limits give a two-sided limit?
Câu trả lời
They don't. For example, $+\infty$ from the left and $-\infty$ from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.
Thẻ 31
Câu hỏi
How do you choose a parameter to make a piecewise function continuous?
Câu trả lời
Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.
Thẻ 32
Câu hỏi
Value of $\lim_{x\to0}\frac{1-\cos x}{x}$ ?
Câu trả lời
$0$ . Rationalizing gives a product involving $\sin x/x$ and a factor that approaches $0$ .
Thẻ 33
Câu hỏi
Can $\lim_{x\to a}f(x)$ exist when $f(a)$ doesn't?
Câu trả lời
Yes. A limit uses nearby values, so a hole at $x=a$ can coexist with a finite two-sided limit.
Thẻ 34
Câu hỏi
What graph behavior makes a finite limit fail even without a jump?
Câu trả lời
Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.
Thẻ 35
Câu hỏi
Average rate of change of $f$ on $[a,b]$ ?
Câu trả lời
$\frac{f(b)-f(a)}{b-a}$
It is the slope of the secant line through $(a,f(a))$ and $(b,f(b))$ .
Thẻ 36
Câu hỏi
Derivative at $x=a$ using an increment $h$ ?
Câu trả lời
$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$
The derivative exists only if this finite limit exists.
Thẻ 37
Câu hỏi
Tangent-line equation to $y=f(x)$ at $x=a$ ?
Câu trả lời
$y-f(a)=f'(a)(x-a)$
This requires $f'(a)$ to exist.
Thẻ 38
Câu hỏi
What does differentiability imply about continuity?
Câu trả lời
If $f$ is differentiable at $a$ , then $f$ is continuous at $a$ . The converse is false: continuity alone doesn't guarantee differentiability.
Thẻ 39
Câu hỏi
Power rule for derivatives?
Câu trả lời
$\frac{d}{dx}x^n=nx^{n-1}$
Apply it where the original real-valued power function and its derivative are defined.
Thẻ 40
Câu hỏi
Units of $f'(x)$ ?
Câu trả lời
Output units of $f$ per input unit of $x$ . A derivative is a rate of change, so its units are a quotient.
Thẻ 41
Câu hỏi
Derivative at $x=a$ using $x\to a$ ?
Câu trả lời
$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$
This is equivalent to the $h$ -form after setting $h=x-a$ .
Thẻ 42
Câu hỏi
How does a graph of $f$ show the sign of $f'$ ?
Câu trả lời
$f'(x)>0$ where $f$ rises as $x$ increases, and $f'(x)<0$ where $f$ falls. A horizontal tangent gives $f'(x)=0$ when the derivative exists.
Thẻ 43
Câu hỏi
Derivative of a constant?
Câu trả lời
$\frac{d}{dx}C=0$
A constant function has zero rate of change.
Thẻ 44
Câu hỏi
Derivative of $\sin x$ ?
Câu trả lời
$\frac{d}{dx}(\sin x)=\cos x$
The angle must be measured in radians for the standard formula.
Thẻ 45
Câu hỏi
Product rule?
Câu trả lời
$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$
Differentiating each factor and multiplying the results is not the product rule.
Thẻ 46
Câu hỏi
How can nearby table values estimate $f'(a)$ ?
Câu trả lời
Use a difference quotient with inputs close to $a$ . A symmetric estimate is
$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}.$
Smaller $h$ often helps, subject to the data's precision.
Thẻ 47
Câu hỏi
What does $f''(x)$ measure?
Câu trả lời
The rate of change of $f'(x)$ with respect to $x$ . Its units are the units of $f$ per square input unit.
Thẻ 48
Câu hỏi
Instantaneous rate of change of $f$ at $a$ ?
Câu trả lời
$f'(a)$ . It is the limit of average rates over intervals shrinking to $a$ , and geometrically it is the tangent-line slope.
Thẻ 49
Câu hỏi
Derivative of a sum or difference?
Câu trả lời
$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$
Thẻ 50
Câu hỏi
Derivative of $\cos x$ ?
Câu trả lời
$\frac{d}{dx}(\cos x)=-\sin x$
The standard formula assumes radians.
Thẻ 51
Câu hỏi
Quotient rule?
Câu trả lời
For $g(x)\ne0$ ,
$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$
The order in the numerator matters.
Thẻ 52
Câu hỏi
Common notations for the first derivative?
Câu trả lời
$f'(x)$ , $y'$ , $\dfrac{dy}{dx}$ , and $\dfrac{d}{dx}f(x)$ . They describe the same derivative in different contexts.
Thẻ 53
Câu hỏi
Derivative of $e^x$ ?
Câu trả lời
$\frac{d}{dx}e^x=e^x$
Thẻ 54
Câu hỏi
What graph features can make $f$ nondifferentiable?
Câu trả lời
A discontinuity, corner, cusp, vertical tangent, or other failure of the difference-quotient limit. Continuity is necessary but not sufficient.
Thẻ 55
Câu hỏi
Derivative of $\tan x$ ?
Câu trả lời
Where $\tan x$ is defined,
$\frac{d}{dx}(\tan x)=\sec^2x.$
Angles are in radians.
Thẻ 56
Câu hỏi
What does the derivative function $f'$ assign to each input?
Câu trả lời
The instantaneous rate of change—or tangent slope—of $f$ at that input, wherever the derivative exists.
Thẻ 57
Câu hỏi
Derivative of $\ln x$ ?
Câu trả lời
For $x>0$ ,
$\frac{d}{dx}\ln x=\frac1x.$
More generally, $d(\ln|x|)/dx=1/x$ for $x\ne0$ .
Thẻ 58
Câu hỏi
How does the power rule handle roots or negative powers?
Câu trả lời
Rewrite them as $x^n$ and apply $nx^{n-1}$ on intervals where the real-valued expression is defined. Domain restrictions still matter.
Thẻ 59
Câu hỏi
Derivative of $\csc x$ ?
Câu trả lời
Where $\csc x$ is defined,
$\frac{d}{dx}(\csc x)=-\csc x\cot x.$
Angles are in radians.
Thẻ 60
Câu hỏi
If $f'(x)>0$ throughout an interval, what does $f$ do there?
Câu trả lời
$f$ is increasing on that interval.
Thẻ 61
Câu hỏi
Derivative of $a^x$ for a constant base?
Câu trả lời
For $a>0$ ,
$\frac{d}{dx}a^x=a^x\ln a.$
When $a=1$ , the derivative is $0$ .
Thẻ 62
Câu hỏi
How can a graph estimate $f'(a)$ ?
Câu trả lời
Estimate the slope of the tangent line at $x=a$ , using two convenient points on a drawn tangent when available. A steeper tangent has a derivative with larger magnitude.
Thẻ 63
Câu hỏi
Derivative of $\sec x$ ?
Câu trả lời
Where $\sec x$ is defined,
$\frac{d}{dx}(\sec x)=\sec x\tan x.$
Angles are in radians.
Thẻ 64
Câu hỏi
If $f''(x)>0$ , how is $f'$ changing?
Câu trả lời
$f'$ is increasing. This is also the derivative condition associated with $f$ being concave up.
Thẻ 65
Câu hỏi
Derivative of $\log_a x$ ?
Câu trả lời
For $x>0$ , $a>0$ , and $a\ne1$ ,
$\frac{d}{dx}\log_a x=\frac{1}{x\ln a}.$
Thẻ 66
Câu hỏi
Product rule from a table at $x=a$ ?
Câu trả lời
For $h=fg$ ,
$h'(a)=f'(a)g(a)+f(a)g'(a).$
Use the four table entries at the same input.
Thẻ 67
Câu hỏi
Derivative of $\cot x$ ?
Câu trả lời
Where $\cot x$ is defined,
$\frac{d}{dx}(\cot x)=-\csc^2x.$
Angles are in radians.
Thẻ 68
Câu hỏi
Why isn't $|x|$ differentiable at $x=0$ ?
Câu trả lời
Its left-hand slope is $-1$ and right-hand slope is $1$ . The one-sided derivative limits disagree, creating a corner.
Thẻ 69
Câu hỏi
Constant-multiple rule?
Câu trả lời
For a constant $c$ ,
$\frac{d}{dx}[c f(x)]=c f'(x).$
Thẻ 70
Câu hỏi
Quotient rule from a table at $x=a$ ?
Câu trả lời
For $h=f/g$ with $g(a)\ne0$ ,
$h'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}.$
Thẻ 71
Câu hỏi
Chain rule for $f(g(x))$ ?
Câu trả lời
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
Differentiate the outer function at the unchanged inner input, then multiply by the inner derivative.
Thẻ 72
Câu hỏi
How do you identify inner and outer functions in a composite?
Câu trả lời
Find the expression evaluated first—that is the inner function. The operation applied to its result is the outer function.
Thẻ 73
Câu hỏi
Core rule when differentiating an implicit equation in $x$ and $y$ ?
Câu trả lời
Treat $y$ as a differentiable function of $x$ . Every derivative of an expression involving $y$ gains a factor of $dy/dx$ by the chain rule.
Thẻ 74
Câu hỏi
Derivative of an inverse function at $x$ ?
Câu trả lời
If $f$ is differentiable and one-to-one near $f^{-1}(x)$ , with $f'(f^{-1}(x))\ne0$ ,
$[f^{-1}]'(x)=\frac{1}{f'(f^{-1}(x))}.$
Thẻ 75
Câu hỏi
Derivative of $\arcsin x$ ?
Câu trả lời
For $|x|<1$ ,
$\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}.$
Thẻ 76
Câu hỏi
Notation for the third derivative of $f$ ?
Câu trả lời
$f'''(x)$ or $\dfrac{d^3f}{dx^3}$ . The exponent on $d$ indicates derivative order; it is not an ordinary power.
Thẻ 77
Câu hỏi
If $h(x)=f(g(x))$ , what table entries give $h'(a)$ ?
Câu trả lời
$h'(a)=f'(g(a))g'(a)$
Use $g(a)$ to find the input needed for the table entry of $f'$ .
Thẻ 78
Câu hỏi
For $x^2+y^2=r^2$ , what is $dy/dx$ ?
Câu trả lời
Where $y\ne0$ ,
$\frac{dy}{dx}=-\frac{x}{y}.$
Differentiate to get $2x+2y(dy/dx)=0$ .
Thẻ 79
Câu hỏi
If $f(a)=b$ , how do you find $[f^{-1}]'(b)$ ?
Câu trả lời
Provided $f'(a)\ne0$ ,
$[f^{-1}]'(b)=\frac{1}{f'(a)}.$
The inverse swaps the input-output pair $(a,b)$ .
Thẻ 80
Câu hỏi
Derivative of $\arctan x$ ?
Câu trả lời
For every real $x$ ,
$\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}.$
Thẻ 81
Câu hỏi
Derivative of $e^{g(x)}$ ?
Câu trả lời
$\frac{d}{dx}e^{g(x)}=e^{g(x)}g'(x)$
The extra factor is the chain rule.
Thẻ 82
Câu hỏi
Slope of a tangent to an implicit curve $F(x,y)=0$ ?
Câu trả lời
Differentiate the relation with respect to $x$ , solve for $dy/dx$ , then substitute the point. Confirm the point lies on the curve and the resulting slope is defined.
Thẻ 83
Câu hỏi
Why must $f'(a)\ne0$ to use $[f^{-1}]'(f(a))=1/f'(a)$ ?
Câu trả lời
Because the reciprocal slope would be undefined when $f'(a)=0$ . The inverse may have a vertical tangent or fail to be differentiable there.
Thẻ 84
Câu hỏi
Derivative of $\arccos x$ ?
Câu trả lời
For $|x|<1$ ,
$\frac{d}{dx}(\arccos x)=-\frac{1}{\sqrt{1-x^2}}.$
Thẻ 85
Câu hỏi
How do you find $d^2y/dx^2$ for an implicit relation?
Câu trả lời
Differentiate the first-derivative equation again with respect to $x$ , include $dy/dx$ factors, then substitute the known expression for $dy/dx$ if needed.
Thẻ 86
Câu hỏi
Derivative of $\ln(g(x))$ ?
Câu trả lời
Where $g(x)>0$ ,
$\frac{d}{dx}\ln(g(x))=\frac{g'(x)}{g(x)}.$
For $\ln|g(x)|$ , the same derivative holds where $g(x)\ne0$ .
Thẻ 87
Câu hỏi
Derivative of $y^n$ when $y=y(x)$ ?
Câu trả lời
$\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}$
The $dy/dx$ factor comes from the chain rule.
Thẻ 88
Câu hỏi
How are tangent slopes of inverse graphs related?
Câu trả lời
At reflected points $(a,b)$ and $(b,a)$ , the slopes are reciprocals when both are defined and nonzero.
Thẻ 89
Câu hỏi
Derivative of $\arcsin(g(x))$ ?
Câu trả lời
$\frac{d}{dx}\arcsin(g(x))=\frac{g'(x)}{\sqrt{1-[g(x)]^2}},\qquad |g(x)|<1.$
Thẻ 90
Câu hỏi
Derivative of $\sin(g(x))$ ?
Câu trả lời
$\frac{d}{dx}\sin(g(x))=\cos(g(x))g'(x)$
Thẻ 91
Câu hỏi
Horizontal tangent on an implicit curve: derivative condition?
Câu trả lời
$dy/dx=0$ at a valid point, with the derivative defined there. In a fraction for $dy/dx$ , the numerator is typically zero while the denominator is nonzero.
Thẻ 92
Câu hỏi
How do you differentiate $f^{-1}(x)$ without solving for the inverse?
Câu trả lời
Use the reciprocal derivative formula and the matching original input: find $a$ with $f(a)=x$ , then compute $1/f'(a)$ .
Thẻ 93
Câu hỏi
Difference between $f''(x)$ and $[f'(x)]^2$ ?
Câu trả lời
$f''(x)$ is the derivative of $f'(x)$ . The expression $[f'(x)]^2$ is the square of the first derivative; they are generally unrelated.
Thẻ 94
Câu hỏi
Derivative of $[g(x)]^n$ ?
Câu trả lời
$\frac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)$
This combines the power rule with the chain rule.
Thẻ 95
Câu hỏi
Vertical tangent on an implicit curve: derivative clue?
Câu trả lời
$dy/dx$ becomes unbounded or undefined while the curve still has a vertical tangent. In a simplified derivative fraction, the denominator is often zero and the numerator nonzero.
Thẻ 96
Câu hỏi
Table formula for an inverse derivative at $x=b$ ?
Câu trả lời
Find $a$ in the table with $f(a)=b$ . If $f'(a)\ne0$ , then
$[f^{-1}]'(b)=\frac{1}{f'(a)}.$
Thẻ 97
Câu hỏi
Derivative of $\arctan(g(x))$ ?
Câu trả lời
$\frac{d}{dx}\arctan(g(x))=\frac{g'(x)}{1+[g(x)]^2}$
Thẻ 98
Câu hỏi
How do product and chain rules combine in $f(x)g(h(x))$ ?
Câu trả lời
$\frac{d}{dx}[f(x)g(h(x))]=f'(x)g(h(x))+f(x)g'(h(x))h'(x).$
Use the product rule outside and the chain rule on the composite factor.
Thẻ 99
Câu hỏi
Why can $d^2y/dx^2$ depend on both $x$ and $y$ ?
Câu trả lời
An implicit relation may never solve explicitly for $y$ . After differentiating twice and replacing $dy/dx$ , the result can naturally remain a function of both coordinates.
Thẻ 100
Câu hỏi
A quantity $y$ changes through $u$ , which changes with $x$ . How are the rates connected?
Câu trả lời
When the functions are differentiable, the chain rule gives
$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$
Thẻ 101
Câu hỏi
What local property lets a function have an inverse derivative?
Câu trả lời
The function must be one-to-one near the point, differentiable there, and have a nonzero derivative. Then the inverse is locally differentiable with reciprocal slope.
Thẻ 102
Câu hỏi
Derivative of $a^{g(x)}$ ?
Câu trả lời
For $a>0$ ,
$\frac{d}{dx}a^{g(x)}=a^{g(x)}\ln(a)\,g'(x).$
Thẻ 103
Câu hỏi
How should $Q'(t)$ be interpreted in context?
Câu trả lời
At time $t$ , the quantity $Q$ changes at an instantaneous rate of $Q'(t)$ output units per unit of time. Include the quantity, time, direction or sign, and units.
Thẻ 104
Câu hỏi
Position, velocity, and acceleration relationships?
Câu trả lời
For position $s(t)$ ,
$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$
Thẻ 105
Câu hỏi
Central idea of a related-rates problem?
Câu trả lời
Write an equation connecting the changing quantities, differentiate it with respect to time, then use the data for the specified instant.
Thẻ 106
Câu hỏi
Linearization of $f$ near $x=a$ ?
Câu trả lời
$L(x)=f(a)+f'(a)(x-a)$
For $x$ close to $a$ , $f(x)\approx L(x)$ .
Thẻ 107
Câu hỏi
L’Hospital’s Rule: basic conditions?
Câu trả lời
For a quotient with differentiable numerator and denominator near the target, denominator derivative nonzero nearby, and indeterminate form $0/0$ or $\infty/\infty$ , the original limit equals the limit of the derivative quotient when that latter limit exists or is infinite.
Thẻ 108
Câu hỏi
If distance is in meters and time in seconds, units of acceleration?
Câu trả lời
Meters per second squared, $\text{m}/\text{s}^2$ . Acceleration is the rate of change of velocity with respect to time.
Thẻ 109
Câu hỏi
Speed in terms of velocity?
Câu trả lời
$\text{speed}=|v(t)|$
Velocity includes direction; speed is nonnegative magnitude.
Thẻ 110
Câu hỏi
Why do $x$ and $y$ gain $dx/dt$ and $dy/dt$ in related rates?
Câu trả lời
They are functions of time. Differentiating an expression such as $x^2$ with respect to $t$ gives $2x(dx/dt)$ by the chain rule.
Thẻ 111
Câu hỏi
Differential approximation connecting $dx$ and $dy$ ?
Câu trả lời
$dy=f'(x)\,dx$
For a small change $\Delta x$ , the actual change satisfies $\Delta y\approx f'(x)\Delta x$ .
Thẻ 112
Câu hỏi
Which indeterminate forms directly allow L’Hospital’s Rule?
Câu trả lời
$0/0$ and $\infty/\infty$ . Other indeterminate forms must first be rewritten as an appropriate quotient.
Thẻ 113
Câu hỏi
How do you estimate an instantaneous contextual rate from a table?
Câu trả lời
Use a nearby difference quotient, preferably with times on both sides of the target. Report the result with output units per time unit.
Thẻ 114
Câu hỏi
What does positive acceleration say about velocity?
Câu trả lời
Velocity is increasing. It does not by itself say the object moves forward or speeds up; the sign of velocity also matters.
Thẻ 115
Câu hỏi
Related rates: when should numerical values be substituted?
Câu trả lời
After differentiating the general relationship with respect to time. Substituting fixed values too early can erase the rates being sought.
Thẻ 116
Câu hỏi
How does concavity predict linearization error?
Câu trả lời
Near the tangency point, a concave-up graph generally lies above its tangent line, so the linearization underestimates. A concave-down graph generally lies below, so it overestimates.
Thẻ 117
Câu hỏi
Why can't L’Hospital’s Rule be applied directly to a product?
Câu trả lời
The rule applies to quotients with $0/0$ or $\infty/\infty$ form. Rewrite an indeterminate product such as $0\cdot\infty$ as a quotient first.
Thẻ 118
Câu hỏi
When is a particle moving in the positive direction?
Câu trả lời
When $v(t)>0$ . Position then increases as time increases.
Thẻ 119
Câu hỏi
How can velocity show a change of direction?
Câu trả lời
Velocity changes sign. A time with $v(t)=0$ is only a candidate; confirm the sign differs on the two sides.
Thẻ 120
Câu hỏi
First equation to seek in a geometric related-rates problem?
Câu trả lời
A geometric constraint involving the changing quantities, such as a Pythagorean, area, volume, or similar-triangle relation.
Thẻ 121
Câu hỏi
Tangent-line approximation of $f(a+\Delta x)$ ?
Câu trả lời
$f(a+\Delta x)\approx f(a)+f'(a)\Delta x$
It is most reliable for small $|\Delta x|$ where the function is well approximated by its tangent.
Thẻ 122
Câu hỏi
When may L’Hospital’s Rule be applied more than once?
Câu trả lời
When the derivative quotient still has $0/0$ or $\infty/\infty$ form and the rule's conditions continue to hold.
Thẻ 123
Câu hỏi
What must a contextual derivative sentence include?
Câu trả lời
The changing quantity, the instant or input, the rate's sign or direction when relevant, and correct output-per-input units.
Thẻ 124
Câu hỏi
Velocity negative and acceleration positive: what happens?
Câu trả lời
The particle moves in the negative direction while its velocity increases toward zero. Its speed decreases as long as velocity and acceleration have opposite signs.
Thẻ 125
Câu hỏi
How should a negative related rate be interpreted?
Câu trả lời
The measured quantity is decreasing at that instant. Keep the sign and state the units; don't silently report only its magnitude.
Thẻ 126
Câu hỏi
When is local linearity a sound approximation tool?
Câu trả lời
When $f$ is differentiable near the base point and the target input is close enough that curvature has limited effect.
Thẻ 127
Câu hỏi
Can L’Hospital’s Rule handle a one-sided limit?
Câu trả lời
Yes, if the required indeterminate form and differentiability conditions hold on that side and the derivative-quotient limit exists or is infinite.
Thẻ 128
Câu hỏi
When is speed increasing?
Câu trả lời
When velocity and acceleration have the same sign, so $v(t)a(t)>0$ .
Thẻ 129
Câu hỏi
Volume changes with time: notation for its rate?
Câu trả lời
$dV/dt$ . Its units are cubic length units per time unit.
Thẻ 130
Câu hỏi
Why are similar triangles useful in related rates?
Câu trả lời
They can reduce several changing lengths to one constraint before differentiation, keeping the rate equation tied to the geometry.
Thẻ 131
Câu hỏi
Meaning of $dy=f'(x)dx$ in approximation?
Câu trả lời
$dy$ is the tangent-line estimate of the actual output change $\Delta y$ caused by an input change $dx$ .
Thẻ 132
Câu hỏi
What conclusion does L’Hospital’s Rule permit?
Câu trả lời
Under its conditions,
$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.$
It does not say the two quotients are equal as functions.
Thẻ 133
Câu hỏi
When is speed decreasing?
Câu trả lời
When velocity and acceleration have opposite signs, so $v(t)a(t)<0$ .
Thẻ 134
Câu hỏi
What does a tangent slope read from a contextual graph represent?
Câu trả lời
The instantaneous rate of the vertical quantity with respect to the horizontal quantity, with vertical units per horizontal unit.
Thẻ 135
Câu hỏi
Does $v(t)=0$ guarantee a particle changes direction?
Câu trả lời
No. It is momentarily at rest, but direction changes only if velocity changes sign across that time.
Thẻ 136
Câu hỏi
How do you translate “ $Q$ increases by 3 units per minute” into derivative notation?
Câu trả lời
$dQ/dt=3$ in the stated time interval or at the stated instant. “Decreases by 3” would give $dQ/dt=-3$ .
Thẻ 137
Câu hỏi
Extreme Value Theorem: hypothesis and conclusion?
Câu trả lời
If $f$ is continuous on the closed interval $[a,b]$ , then $f$ has at least one absolute minimum value and at least one absolute maximum value on $[a,b]$ .
Thẻ 138
Câu hỏi
What is a critical number of $f$ ?
Câu trả lời
A number $c$ in the domain of $f$ where $f'(c)=0$ or $f'(c)$ doesn't exist.
Thẻ 139
Câu hỏi
First derivative test for a local maximum?
Câu trả lời
$f'$ changes from positive to negative at the critical point, so $f$ changes from increasing to decreasing.
Thẻ 140
Câu hỏi
Second-derivative sign for concave up?
Câu trả lời
If $f''(x)>0$ on an interval, then $f$ is concave up there and $f'$ is increasing.
Thẻ 141
Câu hỏi
If the graph of $f'$ is above the $x$ -axis, what does $f$ do?
Câu trả lời
$f$ is increasing because $f'(x)>0$ .
Thẻ 142
Câu hỏi
First step in an optimization model?
Câu trả lời
Define the objective quantity and the constraint, including the feasible domain. Rewrite the objective as a function of one variable before differentiating.
Thẻ 143
Câu hỏi
Mean Value Theorem: hypotheses and conclusion?
Câu trả lời
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then some $c$ in $(a,b)$ satisfies
$f'(c)=\frac{f(b)-f(a)}{b-a}.$
Thẻ 144
Câu hỏi
Candidates test for absolute extrema on $[a,b]$ ?
Câu trả lời
Assuming $f$ is continuous on $[a,b]$ , evaluate $f$ at every critical number in $(a,b)$ and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.
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Thẻ 145
Câu hỏi
First derivative test for a local minimum?
Câu trả lời
$f'$ changes from negative to positive at the critical point, so $f$ changes from decreasing to increasing.
Thẻ 146
Câu hỏi
What must happen at an inflection point?
Câu trả lời
The graph's concavity changes. A zero or undefined value of $f''$ is only a candidate; verify a concavity change.
Thẻ 147
Câu hỏi
If $f'$ has a local maximum, what can that say about $f$ ?
Câu trả lời
$f''$ may change from positive to negative there, so $f$ may change from concave up to concave down. Confirm the sign change rather than relying only on the point.
Thẻ 148
Câu hỏi
How do you confirm an optimization answer is absolute?
Câu trả lời
Compare objective values at all feasible critical points and relevant endpoints, or use a justified monotonicity argument over the feasible domain.
Thẻ 149
Câu hỏi
Rolle’s Theorem: hypotheses and conclusion?
Câu trả lời
If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ , then some $c$ in $(a,b)$ satisfies $f'(c)=0$ .
Thẻ 150
Câu hỏi
Difference between absolute and relative extrema?
Câu trả lời
An absolute maximum is at least every other value on the stated domain, and an absolute minimum is at most every other value. A relative extremum makes the corresponding comparison only with nearby values.
Thẻ 151
Câu hỏi
If $f$ is continuous at a critical number $c$ and $f'$ is positive on both sides, is there a local extremum?
Câu trả lời
No. The function is increasing through $c$ , so it has no local extremum there.
Thẻ 152
Câu hỏi
Second derivative test for a local minimum?
Câu trả lời
If $f'(c)=0$ and $f''(c)>0$ , then $f$ has a local minimum at $c$ .
Thẻ 153
Câu hỏi
Zeros of $f'$ correspond to what features of $f$ ?
Câu trả lời
Horizontal tangents where $f'$ exists. They are critical numbers and possible extrema, but each needs further sign or value analysis.
Thẻ 154
Câu hỏi
Implicit relation: how can $dy/dx$ reveal local behavior?
Câu trả lời
Its sign shows whether the relation's local branch rises or falls as $x$ increases; zeros and undefined values mark possible horizontal or vertical tangents.
Thẻ 155
Câu hỏi
Which theorem links an average slope to an instantaneous slope?
Câu trả lời
The Mean Value Theorem, provided the function is continuous on the closed interval and differentiable on its interior.
Thẻ 156
Câu hỏi
How can an implicit derivative locate a horizontal tangent?
Câu trả lời
At a valid point on the relation, find where the simplified numerator of $dy/dx$ is zero while its denominator is nonzero, then confirm the local branch has the expected behavior.
Thẻ 157
Câu hỏi
Derivative-sign chart: where is $f$ decreasing?
Câu trả lời
On intervals where $f'(x)<0$ .
Thẻ 158
Câu hỏi
Second derivative test for a local maximum?
Câu trả lời
If $f'(c)=0$ and $f''(c)<0$ , then $f$ has a local maximum at $c$ .
Thẻ 159
Câu hỏi
If $f'$ is increasing, what is the concavity of $f$ ?
Câu trả lời
$f$ is concave up on that interval, assuming the relevant derivatives exist.
Thẻ 160
Câu hỏi
Why must an optimization domain be stated?
Câu trả lời
The context can exclude algebraic candidates and add boundary cases. Absolute extrema depend on the feasible inputs, not just on where the derivative is zero.
Thẻ 161
Câu hỏi
Which theorem guarantees absolute extrema, not where they occur?
Câu trả lời
The Extreme Value Theorem. Continuity on a closed interval guarantees at least one maximum value and one minimum value, but finding them requires analysis.
Thẻ 162
Câu hỏi
Can $f'(c)$ fail to exist at a local extremum?
Câu trả lời
Yes. Corners, cusps, or other nondifferentiable domain points can be local extrema, which is why critical numbers include points where $f'$ is undefined.
Thẻ 163
Câu hỏi
For a function continuous at $c$ , what same-sign pattern in $f'$ rules out a local extremum there?
Câu trả lời
If $f'$ is positive on both sides of $c$ , or negative on both sides, then $f$ keeps the same monotonic direction through $c$ and has no local extremum there.
Thẻ 164
Câu hỏi
If $f'(c)=0$ and $f''(c)=0$ , what does the second derivative test conclude?
Câu trả lời
Nothing. The test is inconclusive; use a first-derivative sign change, higher analysis, or direct comparison.
Thẻ 165
Câu hỏi
If the graph of $f'$ crosses from negative to positive, what feature does $f$ have?
Câu trả lời
A local minimum at the crossing input, provided the input is in the domain of $f$ .
Thẻ 166
Câu hỏi
How can an implicit derivative locate a vertical tangent?
Câu trả lời
Find valid points where the simplified $dy/dx$ denominator is zero and numerator is nonzero, then confirm the curve has the expected local behavior.
Thẻ 167
Câu hỏi
Which extra condition turns the Mean Value Theorem into Rolle’s Theorem?
Câu trả lời
$f(a)=f(b)$ . The average slope becomes zero, so the guaranteed instantaneous slope is also zero.
Thẻ 168
Câu hỏi
Why are endpoints included in the candidates test?
Câu trả lời
An absolute maximum or minimum on a closed interval can occur at an endpoint even though no two-sided derivative test applies there.
Thẻ 169
Câu hỏi
If $f'(x)=0$ throughout an interval, what is $f$ there?
Câu trả lời
$f$ is constant on that interval, assuming the usual continuity and differentiability conditions that let the Mean Value Theorem apply.
Thẻ 170
Câu hỏi
Second-derivative sign for concave down?
Câu trả lời
If $f''(x)<0$ on an interval, then $f$ is concave down there and $f'$ is decreasing.
Thẻ 171
Câu hỏi
Graph of $f'$ has a local minimum: possible effect on $f$ ?
Câu trả lời
$f''$ may change from negative to positive, so $f$ may change from concave down to concave up. Verify the sign change.
Thẻ 172
Câu hỏi
What should the final line of an optimization solution state?
Câu trả lời
The requested maximum or minimum quantity in context, with units and the feasible input that produces it when relevant.
Thẻ 173
Câu hỏi
Can Rolle’s Theorem be used if $f$ has a corner inside $(a,b)$ ?
Câu trả lời
No. A corner breaks differentiability on the open interval, so the theorem's hypotheses are not satisfied.
Thẻ 174
Câu hỏi
How do $f''$ zeros help analyze a graph?
Câu trả lời
They are candidates for changes in concavity. Test the sign of $f''$ on both sides; a zero alone doesn't guarantee an inflection point.
Thẻ 175
Câu hỏi
What does the accumulation function $F(x)=\int_a^x f(t)\,dt$ measure?
Câu trả lời
The signed net accumulation of $f$ from $a$ to $x$ . Contributions above the axis are positive; contributions below it are negative.
Thẻ 176
Câu hỏi
Left Riemann sum on equal subintervals?
Câu trả lời
If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$ , then
$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$
Thẻ 177
Câu hỏi
What does $\int_a^b f(x)\,dx$ represent geometrically?
Câu trả lời
Signed area between the graph and the $x$ -axis from $a$ to $b$ , when $f$ is integrable. Regions below the axis subtract from regions above it.
Thẻ 178
Câu hỏi
Fundamental Theorem of Calculus: evaluate a definite integral?
Câu trả lời
If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ , then
$\int_a^b f(x)\,dx=F(b)-F(a).$
Thẻ 179
Câu hỏi
Derivative of $F(x)=\int_a^x f(t)\,dt$ ?
Câu trả lời
If $f$ is continuous, then
$F'(x)=f(x).$
This connects accumulation with instantaneous rate.
Thẻ 180
Câu hỏi
Why do all antiderivatives of the same function differ by a constant?
Câu trả lời
If $F'=f$ and $G'=f$ on an interval, then $(F-G)'=0$ , so $F-G=C$ on that interval.
Thẻ 181
Câu hỏi
Right Riemann sum on equal subintervals?
Câu trả lời
If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$ , then
$R_n=\sum_{i=1}^{n} f(x_i)\Delta x.$
Thẻ 182
Câu hỏi
How does reversing integral bounds change the value?
Câu trả lời
It changes the sign:
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx.$
Thẻ 183
Câu hỏi
Net Change Theorem?
Câu trả lời
If $Q'(t)$ is the rate of change of a quantity, then
$Q(b)-Q(a)=\int_a^b Q'(t)\,dt.$
Thẻ 184
Câu hỏi
Derivative of $\int_a^{g(x)} f(t)\,dt$ ?
Câu trả lời
If $f$ is continuous on an interval containing $a$ and the range of $g$ , and $g$ is differentiable, then
$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x).$
Thẻ 185
Câu hỏi
Power rule for antiderivatives?
Câu trả lời
For $n\ne-1$ ,
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.$
Thẻ 186
Câu hỏi
Midpoint Riemann sum on equal subintervals?
Câu trả lời
With midpoint $m_i=(x_{i-1}+x_i)/2$ ,
$M_n=\sum_{i=1}^{n} f(m_i)\Delta x.$
Thẻ 187
Câu hỏi
How can an integral be split at an interior point $c$ ?
Câu trả lời
For $a\le c\le b$ ,
$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$
Thẻ 188
Câu hỏi
Derivative of $G(x)=\int_x^b f(t)\,dt$ ?
Câu trả lời
If $f$ is continuous, then
$G'(x)=-f(x).$
The variable lower bound produces the negative sign.
Thẻ 189
Câu hỏi
Antiderivative of $1/x$ ?
Câu trả lời
On any interval not crossing zero,
$\int \frac1x\,dx=\ln|x|+C.$
Thẻ 190
Câu hỏi
Trapezoidal approximation on equal subintervals?
Câu trả lời
$T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].$
Thẻ 191
Câu hỏi
How do geometric regions help evaluate a definite integral?
Câu trả lời
Replace each familiar region with its geometric area, then attach a positive sign above the axis and a negative sign below it.
Thẻ 192
Câu hỏi
Basic antiderivatives of sine and cosine?
Câu trả lời
$\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C.$
Thẻ 193
Câu hỏi
Definite integral as a limit of Riemann sums?
Câu trả lời
For an integrable function and sample points $x_i^*$ ,
$\int_a^b f(x)\,dx=\lim_{\max\Delta x_i\to0}\sum_i f(x_i^*)\Delta x_i.$
Thẻ 194
Câu hỏi
Constant-multiple rule for integrals?
Câu trả lời
For a constant $k$ ,
$\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx.$
The analogous rule holds for indefinite integrals.
Thẻ 195
Câu hỏi
What pattern suggests $u$ -substitution?
Câu trả lời
A composite expression paired with its derivative, such as $f(g(x))g'(x)$ . Set $u=g(x)$ so $du=g'(x)\,dx$ .
Thẻ 196
Câu hỏi
How should bounds change in a definite $u$ -substitution?
Câu trả lời
If $u=g(x)$ , replace the $x$ -bounds $a,b$ with $u$ -bounds $g(a),g(b)$ . Then finish entirely in $u$ , or return to $x$ before applying the original bounds.
Thẻ 197
Câu hỏi
What condition makes $F(x)=\int_a^x f(t)\,dt$ differentiable with $F'(x)=f(x)$ ?
Câu trả lời
Continuity of $f$ on an interval containing $a$ and $x$ is the standard AP Calculus condition.
Thẻ 198
Câu hỏi
Sum-and-difference rule for definite integrals?
Câu trả lời
For integrable $f$ and $g$ ,
$\int_a^b [f(x)\pm g(x)]\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx.$
Thẻ 199
Câu hỏi
Basic antiderivative of $e^x$ ?
Câu trả lời
$\int e^x\,dx=e^x+C.$
Thẻ 200
Câu hỏi
Basic antiderivatives of $\sec^2x$ and $\csc^2x$ ?
Câu trả lời
$\int\sec^2x\,dx=\tan x+C,\qquad \int\csc^2x\,dx=-\cot x+C.$
Thẻ 201
Câu hỏi
For an increasing integrable function, how do left and right sums compare with the integral?
Câu trả lời
On the same partition, the left sum is an underestimate and the right sum is an overestimate. Reverse those conclusions when the function is decreasing.
Thẻ 202
Câu hỏi
How does concavity predict trapezoidal and midpoint error?
Câu trả lời
For a concave-up function, trapezoidal sums overestimate and midpoint sums underestimate. For a concave-down function, the directions reverse.
Thẻ 203
Câu hỏi
Why might polynomial long division help before integrating a rational function?
Câu trả lời
When the numerator's degree is at least the denominator's, division rewrites the expression as a polynomial plus a simpler proper rational function.
Thẻ 204
Câu hỏi
What denominator pattern suggests an arctangent antiderivative?
Câu trả lời
After completing the square and scaling, a form like
$\int\frac{1}{1+u^2}\,du=\arctan u+C.$
Thẻ 205
Câu hỏi
Basic antiderivatives of $\sec x\tan x$ and $\csc x\cot x$ ?
Câu trả lời
$\int\sec x\tan x\,dx=\sec x+C,\qquad \int\csc x\cot x\,dx=-\csc x+C.$
Thẻ 206
Câu hỏi
How does an initial condition determine an antiderivative?
Câu trả lời
First find the family $F(x)+C$ . Substitute the given point, such as $y(a)=b$ , and solve for $C$ .
Thẻ 207
Câu hỏi
Should a definite-integral answer include $+C$ ?
Câu trả lời
No. A definite integral is one number or quantity. The constant of integration belongs to an indefinite integral or general antiderivative.
Thẻ 208
Câu hỏi
Why does an indefinite integral include $+C$ ?
Câu trả lời
Differentiation loses additive constants. The $+C$ represents every function with the stated derivative.
Thẻ 209
Câu hỏi
When is $F(x)=\int_a^x f(t)\,dt$ increasing?
Câu trả lời
Where $F'(x)=f(x)>0$ . It is decreasing where $f(x)<0$ .
Thẻ 210
Câu hỏi
How is the concavity of $F(x)=\int_a^x f(t)\,dt$ determined?
Câu trả lời
Since $F'(x)=f(x)$ , $F$ is concave up where $f$ is increasing and concave down where $f$ is decreasing, assuming the needed derivatives exist.
Thẻ 211
Câu hỏi
How is $\int_a^b |f(x)|\,dx$ interpreted?
Câu trả lời
As total geometric area between $f$ and the $x$ -axis. Split at zeros of $f$ and make every regional contribution nonnegative.
Thẻ 212
Câu hỏi
What constant-factor check completes many $u$ -substitutions?
Câu trả lời
Compare $du=g'(x)\,dx$ with the remaining factor. Multiply or divide by a constant so the integrand contains exactly the needed differential.
Thẻ 213
Câu hỏi
How do you recover $\Delta x$ from a sigma-form Riemann sum on $[a,b]$ ?
Câu trả lời
Identify the factor multiplying each function value. For $n$ equal subintervals, it should be
$\Delta x=\frac{b-a}{n}.$
Thẻ 214
Câu hỏi
Riemann sum for unequal subinterval widths?
Câu trả lời
If $x_{i-1}$ to $x_i$ has width $\Delta x_i$ and sample point $x_i^*$ , use
$\sum_i f(x_i^*)\Delta x_i.$
Thẻ 215
Câu hỏi
Does continuity guarantee integrability on a closed interval?
Câu trả lời
Yes. A function continuous on $[a,b]$ is integrable there. Some discontinuous functions are also integrable, so continuity is sufficient, not necessary.
Thẻ 216
Câu hỏi
Antiderivative pattern for $g'(x)/g(x)$ ?
Câu trả lời
Where $g(x)\ne0$ ,
$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C.$
Thẻ 217
Câu hỏi
What algebraic rewrites often reveal a basic antiderivative?
Câu trả lời
Split sums, factor out constants, simplify rational powers, and rewrite radicals as exponents. Then apply known antiderivative rules term by term.
Thẻ 218
Câu hỏi
What units does $\int_a^b r(t)\,dt$ have?
Câu trả lời
Rate units multiplied by input units. For example, liters per minute integrated over minutes gives liters.
Thẻ 219
Câu hỏi
What is a differential equation?
Câu trả lời
An equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true on an interval.
Thẻ 220
Câu hỏi
How does a verbal rate statement become a differential equation?
Câu trả lời
Name the changing quantity and its independent variable, translate its rate as a derivative, and express that derivative using the stated relationship. For example, “rate proportional to $y$ ” becomes $dy/dt=ky$ .
Thẻ 221
Câu hỏi
How do you verify that $y=f(x)$ solves a differential equation?
Câu trả lời
Differentiate $f$ as needed, substitute $y$ and its derivatives into the equation, and confirm both sides agree on the claimed interval.
Thẻ 222
Câu hỏi
General solution versus particular solution?
Câu trả lời
A general solution contains an arbitrary constant and represents a family of solution curves. A particular solution uses an initial condition to determine that constant.
Thẻ 223
Câu hỏi
What does one segment in a slope field show?
Câu trả lời
At $(x,y)$ , its slope equals the value of $dy/dx$ given by the differential equation at that point.
Thẻ 224
Câu hỏi
What units does the constant $k$ have in $dy/dt=ky$ ?
Câu trả lời
Inverse time units, such as per hour. That makes the exponent $kt$ dimensionless.
Thẻ 225
Câu hỏi
How do you verify a proposed solution to an initial value problem?
Câu trả lời
Check both requirements: the function must satisfy the differential equation throughout the stated interval, and it must satisfy the given initial condition.
Thẻ 226
Câu hỏi
What makes a first-order differential equation separable?
Câu trả lời
It can be rearranged so all $y$ factors accompany $dy$ and all $x$ factors accompany $dx$ , such as
$g(y)\,dy=f(x)\,dx.$
Thẻ 227
Câu hỏi
What is an initial value problem?
Câu trả lời
A differential equation paired with a value such as $y(x_0)=y_0$ . It asks for a solution through $(x_0,y_0)$ ; depending on the equation, there may be zero, one, or multiple such solutions.
Thẻ 228
Câu hỏi
What is an isocline in a slope field?
Câu trả lời
A curve along which the differential equation gives the same slope. For $dy/dx=F(x,y)$ , an isocline satisfies $F(x,y)=k$ for a constant $k$ .
Thẻ 229
Câu hỏi
How do you draw a slope-field segment at $(x_0,y_0)$ ?
Câu trả lời
Evaluate the differential equation at $(x_0,y_0)$ to find the slope, then draw a short segment through that point with the resulting slope.
Thẻ 230
Câu hỏi
General solution of $dy/dt=ky$ ?
Câu trả lời
$y=Ce^{kt}$
for a constant $C$ . The zero solution is included by $C=0$ .
Thẻ 231
Câu hỏi
Core method for solving a separable differential equation?
Câu trả lời
Separate the variables, integrate both sides, include a constant of integration, and solve for $y$ when practical. Then apply any initial condition.
Thẻ 232
Câu hỏi
How should a solution curve follow a slope field?
Câu trả lời
It must pass through its initial point and remain tangent to the short field segments it crosses. It should not be drawn by connecting segment endpoints.
Thẻ 233
Câu hỏi
If $dy/dx=f(x)$ , what pattern appears in its slope field?
Câu trả lời
Every point with the same $x$ -coordinate has the same slope, so the segments repeat vertically in columns.
Thẻ 234
Câu hỏi
Why is one integration constant enough after integrating both sides?
Câu trả lời
Two constants can be combined: $C_2-C_1$ is still an arbitrary constant. Write a single $C$ .
Thẻ 235
Câu hỏi
Solution of $dy/dt=ky$ with $y(0)=y_0$ ?
Câu trả lời
$y(t)=y_0e^{kt}.$
Thẻ 236
Câu hỏi
Can one differential equation have infinitely many solutions?
Câu trả lời
Yes. A differential equation commonly describes a family of solution functions. An initial condition may select one particular solution when the relevant conditions support uniqueness.
Thẻ 237
Câu hỏi
What is an equilibrium solution of $dy/dx=F(y)$ ?
Câu trả lời
A constant solution $y=c$ where $F(c)=0$ . In the slope field, the segments along that horizontal line have zero slope.
Thẻ 238
Câu hỏi
For continuous $f$ , particular solution of $dy/dx=f(x)$ with $y(a)=b$ ?
Câu trả lời
$y(x)=b+\int_a^x f(t)\,dt.$
The Fundamental Theorem of Calculus gives $y'=f(x)$ , and $y(a)=b$ .
Thẻ 239
Câu hỏi
What can be lost when dividing to separate variables?
Câu trả lời
Equilibrium solutions that make the divided factor zero. Check those constant solutions directly in the original differential equation.
Thẻ 240
Câu hỏi
In $dy/dt=ky$ , what do the signs of $k$ mean?
Câu trả lời
For a positive quantity, $k>0$ produces exponential growth, $k<0$ produces exponential decay, and $k=0$ keeps the quantity constant.
Thẻ 241
Câu hỏi
How can a table of slopes identify the matching differential equation?
Câu trả lời
Test representative $(x,y)$ entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.
Thẻ 242
Câu hỏi
How does the sign of $dy/dx$ describe a solution?
Câu trả lời
The solution is increasing where $dy/dx>0$ and decreasing where $dy/dx<0$ . At one point, $dy/dx=0$ gives a horizontal tangent; a constant $y=c$ is an equilibrium only when the derivative equation gives zero all along that level.
Thẻ 243
Câu hỏi
How can a differential equation determine a solution's concavity?
Câu trả lời
Differentiate the equation with respect to the independent variable to obtain $y''$ , using the chain rule for any $y$ -dependence. Then use the sign of $y''$ along the solution.
Thẻ 244
Câu hỏi
Why must a differential-equation solution include an interval or domain?
Câu trả lời
Algebraic steps may introduce restrictions, and the formula must remain differentiable and satisfy the original equation throughout the interval containing the initial point.
Thẻ 245
Câu hỏi
Doubling time for exponential growth $y=y_0e^{kt}$ ?
Câu trả lời
For $k>0$ ,
$T_d=\frac{\ln2}{k}.$
It is independent of the initial amount.
Thẻ 246
Câu hỏi
How can a slope field reveal whether $dy/dx$ depends only on $y$ ?
Câu trả lời
Slopes repeat horizontally: every point at the same height has the same segment slope.
Thẻ 247
Câu hỏi
How is an initial condition used after separation?
Câu trả lời
Substitute the given $x$ and $y$ values into the integrated relationship to solve for the arbitrary constant, then state the resulting particular solution.
Thẻ 248
Câu hỏi
How do units check a model $dy/dt=F(t,y)$ ?
Câu trả lời
The right side must have the same units as $dy/dt$ : units of $y$ per unit of $t$ . A mismatch signals an incorrect translation or parameter unit.
Thẻ 249
Câu hỏi
Why should a separated solution be checked in the original equation?
Câu trả lời
Division, logarithms, square roots, or algebraic rearrangement can lose or introduce branches. Direct substitution confirms the formula and its stated domain.
Thẻ 250
Câu hỏi
Half-life for exponential decay $y=y_0e^{kt}$ ?
Câu trả lời
For $k<0$ ,
$T_{1/2}=\frac{\ln(1/2)}{k}=\frac{\ln2}{|k|}.$
Thẻ 251
Câu hỏi
Average value of $f$ on $[a,b]$ ?
Câu trả lời
For integrable $f$ and $a<b$ ,
$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$
Thẻ 252
Câu hỏi
Displacement from velocity $v(t)$ on $[a,b]$ ?
Câu trả lời
$s(b)-s(a)=\int_a^b v(t)\,dt.$
Velocity below zero contributes negative displacement.
Thẻ 253
Câu hỏi
Area between vertical curves $y=f(x)$ and $y=g(x)$ ?
Câu trả lời
On intervals where $f(x)\ge g(x)$ ,
$A=\int_a^b [f(x)-g(x)]\,dx.$
Think top minus bottom.
Thẻ 254
Câu hỏi
Volume from known cross-sectional area $A(x)$ ?
Câu trả lời
If slices are perpendicular to the $x$ -axis,
$V=\int_a^b A(x)\,dx.$
Thẻ 255
Câu hỏi
Mean Value Theorem for Integrals: hypotheses and conclusion?
Câu trả lời
If $f$ is continuous on $[a,b]$ , then some $c\in[a,b]$ satisfies
$f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx.$
If $a<b$ , a point can also be chosen in $(a,b)$ .
Thẻ 256
Câu hỏi
Velocity and acceleration from position $s(t)$ ?
Câu trả lời
$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$
Thẻ 257
Câu hỏi
Cross-sectional area when each slice is a square?
Câu trả lời
If the base segment has length $s(x)$ , then
$A(x)=[s(x)]^2.$
Thẻ 258
Câu hỏi
How do you find accumulation from an inflow rate and an outflow rate?
Câu trả lời
Integrate the net rate:
$Q(b)-Q(a)=\int_a^b [r_{\text{in}}(t)-r_{\text{out}}(t)]\,dt.$
Thẻ 259
Câu hỏi
Area between horizontal curves written as $x=R(y)$ and $x=L(y)$ ?
Câu trả lời
On intervals where $R(y)\ge L(y)$ ,
$A=\int_c^d [R(y)-L(y)]\,dy.$
Think right minus left.
Thẻ 260
Câu hỏi
Disc-method volume formula?
Câu trả lời
For radius $R(x)$ and slices perpendicular to the $x$ -axis,
$V=\pi\int_a^b [R(x)]^2\,dx.$
Thẻ 261
Câu hỏi
What units does average value have?
Câu trả lời
The same units as the original function. Integration adds an input unit, and division by interval length removes it.
Thẻ 262
Câu hỏi
Total distance traveled from velocity $v(t)$ ?
Câu trả lời
$\text{distance}=\int_a^b |v(t)|\,dt.$
Split the interval wherever $v(t)=0$ and its sign changes.
Thẻ 263
Câu hỏi
Cross-sectional area when each slice is a rectangle?
Câu trả lời
If the slice has base $b(x)$ and height $h(x)$ , then
$A(x)=b(x)h(x).$
Use the stated relationship to express both in the integration variable.
Thẻ 264
Câu hỏi
How do you determine bounds for area between curves?
Câu trả lời
Use the stated region boundaries or solve the curve-intersection equations. Check whether additional intersections inside the interval require splitting.
Thẻ 265
Câu hỏi
How is a rotation radius measured from a horizontal axis $y=k$ ?
Câu trả lời
As vertical distance: $|y-k|$ . For washers, identify which boundary stays farther from the axis over the interval.
Thẻ 266
Câu hỏi
When is a particle moving to the right or left?
Câu trả lời
It moves right where $v(t)>0$ and left where $v(t)<0$ . Position alone does not determine direction.
Thẻ 267
Câu hỏi
Cross-sectional area when the diameter of a semicircle is $d(x)$ ?
Câu trả lời
The radius is $d(x)/2$ , so
$A(x)=\frac12\pi\left(\frac{d(x)}2\right)^2=\frac{\pi}{8}[d(x)]^2.$
Thẻ 268
Câu hỏi
Why must an area integral be split where curves intersect?
Câu trả lời
The top/bottom or right/left relationship may switch there. Splitting keeps each integrand nonnegative and prevents signed cancellation.
Thẻ 269
Câu hỏi
How can a velocity table approximate displacement?
Câu trả lời
Use a left, right, midpoint, or trapezoidal sum for $\int v(t)\,dt$ . Each term is velocity times a time width.
Thẻ 270
Câu hỏi
Washer-method volume formula?
Câu trả lời
For outer radius $R(x)$ and inner radius $r(x)$ ,
$V=\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx.$
Thẻ 271
Câu hỏi
How do you recover position from velocity and an initial position?
Câu trả lời
If $s(a)$ is known,
$s(t)=s(a)+\int_a^t v(u)\,du.$
Thẻ 272
Câu hỏi
How do you choose between vertical and horizontal area slices?
Câu trả lời
Choose the direction that describes the region with fewer pieces. Vertical slices use top minus bottom; horizontal slices use right minus left.
Thẻ 273
Câu hỏi
Cross-sectional area of an equilateral triangle with side $s(x)$ ?
Câu trả lời
$A(x)=\frac{\sqrt3}{4}[s(x)]^2.$
Thẻ 274
Câu hỏi
How can a table approximate the average value of $f$ on $[a,b]$ ?
Câu trả lời
First approximate $\int_a^b f(x)\,dx$ with an appropriate Riemann or trapezoidal sum, then divide by $b-a$ .
Thẻ 275
Câu hỏi
Single expression for area between two curves?
Câu trả lời
When the functions are integrable,
$A=\int_a^b |f(x)-g(x)|\,dx.$
For hand evaluation, split where their order changes.
Thẻ 276
Câu hỏi
How is a rotation radius measured from a vertical axis $x=k$ ?
Câu trả lời
As horizontal distance: $|x-k|$ . With $dy$ slices, write the relevant boundaries as $x$ -functions of $y$ .
Thẻ 277
Câu hỏi
How can a rate table approximate total change with unequal time gaps?
Câu trả lời
Multiply a representative rate on each interval by that interval's actual width, then sum. Do not assume a common $\Delta t$ when the table is uneven.
Thẻ 278
Câu hỏi
When should a volume integral use $dy$ ?
Câu trả lời
When slices perpendicular to the $y$ -axis make the cross-sectional area easiest to express as $A(y)$ . Then use $V=\int_c^d A(y)\,dy$ .
Thẻ 279
Câu hỏi
What signals that a washer, not a disc, is needed?
Câu trả lời
The rotated region leaves a hole around the axis. Subtract the inner circular area from the outer circular area.
Thẻ 280
Câu hỏi
What base length is used for cross sections over a planar region?
Câu trả lời
Use the distance across the base region within each slice: top minus bottom for vertical slices, or right minus left for horizontal slices.
Thẻ 281
Câu hỏi
Why must total distance split at velocity sign changes?
Câu trả lời
Distance accumulates speed $|v|$ , not signed velocity. A single integral of $v$ would cancel motion in opposite directions.
Thẻ 282
Câu hỏi
When does an accumulated quantity reach a local maximum?
Câu trả lời
When its net rate changes from positive to negative. A zero rate alone is only a candidate; check the sign change.
Thẻ 283
Câu hỏi
What distinguishes area from a definite integral?
Câu trả lời
Area is nonnegative. A definite integral is signed and may be zero or negative because regions below the axis subtract.
Thẻ 284
Câu hỏi
How do position, velocity, and acceleration graphs correspond?
Câu trả lời
Velocity is the slope of position, and acceleration is the slope of velocity. Conversely, signed area under velocity gives position change, and signed area under acceleration gives velocity change.
Thẻ 285
Câu hỏi
How do you interpret $\int_a^b [r(t)-c(t)]\,dt$ in context?
Câu trả lời
As net change: total amount added by $r$ minus total amount removed by $c$ over the interval. State the resulting quantity and units.
Thẻ 286
Câu hỏi
What units does a volume integral $\int A(x)\,dx$ have?
Câu trả lời
Cubic units. Cross-sectional area contributes square units and slice thickness contributes one more length unit.
Thẻ 287
Câu hỏi
How can a graph of a rate reveal the largest accumulated value?
Câu trả lời
Compare accumulation at endpoints and at times where the rate is zero or undefined. Compute signed areas to track the quantity's changes from its initial value.
Thẻ 288
Câu hỏi
Why should a contextual integral answer include a sentence?
Câu trả lời
The number alone does not say whether it represents displacement, distance, area, volume, or net change. Name the quantity, interval, and units.

Abstract calculus graph with a glowing orange curve, tangent line, shaded area under the curve, and layered contour lines on a dark grid.

288 thẻ