AP Calculus AB · Flashcards
Unit 3 — Applications of Differentiation
Course framework: Units 4–5 (contextual and analytical applications of differentiation) · 70 cards. Answer each card in your head, flip it, then mark it Got it or Review again. Your progress is saved in this browser.
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Derivatives in Context✓ Known #1
\(V(t)\) is the volume of water in a tank, in liters, \(t\) minutes after noon. Interpret \(V'(5)=-2\).
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Answer#1
At 12:05, the volume of water is decreasing at a rate of 2 liters per minute.
Trap: Leaving out the units, the time, or the direction (decreasing).
Derivatives in Context✓ Known #2
What are the units of \(f'(x)\)?
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Answer#2
Units of \(f\) divided by units of \(x\) — for example, meters per second.
Trap: Giving the units of \(f\) itself.
Linearization & Differentials✓ Known #3
Linearization at \(x=a\): write the formula for \(L(x)\).
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Answer#3
\(L(x)=f(a)+f'(a)(x-a)\).
Trap: Using \(f(x)\) instead of \(f(a)\).
Linearization & Differentials✓ Known #4
When is \(L(x)\) a good approximation to \(f(x)\)?
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Answer#4
- When \(x\) is close to \(a\).
- When \(f\) is differentiable (locally linear) near \(a\).
Trap: Using an \(x\)-value far from \(a\).
Linearization & Differentials✓ Known #5
What does \(f'(a)\) represent in the linearization formula?
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Answer#5
The slope of the tangent line at \(x=a\).
Trap: Using a secant slope from two points.
Linearization & Differentials✓ Known #6
Is a tangent-line approximation an over- or underestimate?
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Answer#6
- If \(f\) is concave up near \(a\), the tangent line lies below the graph: underestimate.
- If \(f\) is concave down, the tangent line lies above the graph: overestimate.
Trap: Deciding from the sign of \(f’\) instead of \(f”\).
Linearization & Differentials✓ Known #7
Define the differential \(dy\) in terms of \(dx\).
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Answer#7
\(dy=f'(x)\,dx\).
Trap: Writing \(dy=f(x)\,dx\).
Linearization & Differentials✓ Known #8
If \(dx\) is small, how do \(\Delta y\) and \(dy\) compare?
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Answer#8
\(\Delta y\approx dy\) when \(dx\) is small.
Trap: Claiming they are always equal.
Linearization & Differentials✓ Known #9
Relative error: what is it?
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Answer#9
\(\text{relative error}=\dfrac{|\text{approx}-\text{exact}|}{|\text{exact}|}\).
Trap: Dividing by the approximate value (unless the problem says so).
Linearization & Differentials✓ Known #10
How is percent error related to relative error?
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Answer#10
Percent error = relative error × 100%.
Trap: Reporting a decimal and calling it a percent.
Linearization & Differentials✓ Known #11
To approximate \(\sqrt{1.02}\), why choose \(a=1\)?
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Answer#11
For \(f(x)=\sqrt{x}\), \(a=1\) is close to 1.02 and \(f(1)=1\), \(f'(1)=\frac12\) are easy: \(\sqrt{1.02}\approx1+\frac12(0.02)=1.01\).
Trap: Choosing \(a=0\), where \(f’\) is undefined.
Linearization & Differentials✓ Known #12
Estimate the change in \(y=f(x)\) when \(x\) changes from \(a\) to \(a+dx\).
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Answer#12
\(\Delta y\approx dy=f'(a)\,dx\).
Trap: Evaluating the derivative at \(a+dx\) instead of \(a\).
Linearization & Differentials✓ Known #13
Tangent line and linearization: are they the same object?
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Answer#13
Yes. The linearization is the tangent-line function at \(a\).
Trap: Thinking the linearization is a different line.
Linearization & Differentials✓ Known #14
If \(dx<0\), must \(dy\) be negative?
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Answer#14
No. \(dy=f'(x)\,dx\), so its sign depends on both \(dx\) and \(f'(x)\).
Trap: Assuming \(dy\) has the same sign as \(dx\).
Linearization & Differentials✓ Known #15
“Use the linearization to estimate …”: what are the 3 steps?
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Answer#15
- Pick a convenient \(a\) near the target \(x\).
- Compute \(f(a)\) and \(f'(a)\).
- Evaluate \(L(x)=f(a)+f'(a)(x-a)\) at the target.
Trap: Using the wrong center \(a\).
L’Hospital’s Rule✓ Known #16
State L’Hospital’s Rule.
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Answer#16
If \(\lim f(x)\) and \(\lim g(x)\) are both \(0\) (or both \(\pm\infty\)) and \(\lim\frac{f'(x)}{g'(x)}\) exists, then \(\lim\dfrac{f(x)}{g(x)}=\lim\dfrac{f'(x)}{g'(x)}\).
Trap: Using the quotient rule on \(\frac{f}{g}\), or applying the rule when the form is not \(\frac00\) or \(\frac{\infty}{\infty}\).
L’Hospital’s Rule✓ Known #17
Evaluate \(\displaystyle\lim_{x\to0}\frac{e^x-1}{x}\).
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Answer#17
- The form is \(\frac00\), so L’Hospital’s Rule applies.
- \(\displaystyle\lim_{x\to0}\frac{e^x}{1}=1\).
Trap: Not checking (or not stating) the \(\frac00\) form first.
Related Rates✓ Known #18
Related rates: what is the 4-step workflow?
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Answer#18
- Draw a picture and define variables.
- Write an equation relating the variables.
- Differentiate with respect to time \(t\).
- Substitute the values at that instant and solve.
Trap: Substituting numbers before differentiating.
Related Rates✓ Known #19
Why is it wrong to substitute numbers before differentiating?
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Answer#19
A quantity that is changing would be frozen as a constant, so its rate would wrongly disappear from the derivative.
Trap: Turning a variable into a constant too early.
Related Rates✓ Known #20
If \(A=\pi r^2\), what is \(\dfrac{dA}{dt}\)?
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Answer#20
\(\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\).
Trap: Forgetting the chain rule: \(\frac{d}{dt}(r^2)=2r\frac{dr}{dt}\).
Related Rates✓ Known #21
If \(V=\frac{4}{3}\pi r^3\), what is \(\dfrac{dV}{dt}\)?
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Answer#21
\(\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}\).
Trap: Missing \(\frac{dr}{dt}\).
Related Rates✓ Known #22
If \(s^2=x^2+y^2\), differentiate with respect to \(t\).
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Answer#22
\(2s\dfrac{ds}{dt}=2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}\).
Trap: Differentiating \(x^2\) as \(2x\) (missing \(\frac{dx}{dt}\)).
Related Rates✓ Known #23
If \(r^2=x^2+y^2\), solve for \(\dfrac{dr}{dt}\).
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Answer#23
\(\dfrac{dr}{dt}=\dfrac{x\frac{dx}{dt}+y\frac{dy}{dt}}{r}\), with \(r=\sqrt{x^2+y^2}\) at that instant.
Trap: Forgetting to compute \(r\) from \(x\) and \(y\) when substituting.
Related Rates✓ Known #24
Sliding ladder: what is the key equation?
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Answer#24
\(x^2+y^2=L^2\), where the ladder length \(L\) is constant.
Trap: Differentiating \(L\) as if it changes.
Related Rates✓ Known #25
Conical tank: how do you relate volume and height?
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Answer#25
- Use similar triangles: \(r=kh\).
- Then \(V=\frac13\pi r^2h=\frac13\pi k^2h^3\), a function of \(h\) only.
Trap: Skipping similar triangles and keeping too many variables.
Related Rates✓ Known #26
If a distance is decreasing, what sign does \(\dfrac{ds}{dt}\) have?
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Answer#26
Negative.
Trap: Using a speed (positive) instead of a signed rate.
Related Rates✓ Known #27
Two objects move along the same line. When is \(\dfrac{ds}{dt}\) the sum of their speeds?
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Answer#27
When they move directly away from each other.
Trap: Adding speeds when the geometry needs the Pythagorean theorem.
Related Rates✓ Known #28
When do you expect a Pythagorean setup?
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Answer#28
When the distances form a right triangle — perpendicular motions or a fixed offset.
Trap: Writing \(s=x+y\) when the picture is a triangle.
Related Rates✓ Known #29
What must be consistent in a related-rates problem?
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Answer#29
Units: all lengths in the same unit and all rates in the same length-per-time unit.
Trap: Mixing meters and centimeters.
Related Rates✓ Known #30
“The area is increasing at 5 cm²/s.” Which rate is this?
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Answer#30
\(\dfrac{dA}{dt}=5\) cm²/s.
Trap: Confusing it with \(\frac{dr}{dt}\).
Motion (s–v–a)✓ Known #31
If \(s(t)\) is position, what are \(v(t)\) and \(a(t)\)?
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Answer#31
- \(v(t)=s'(t)\)
- \(a(t)=v'(t)=s”(t)\)
Trap: Swapping velocity and acceleration.
Motion (s–v–a)✓ Known #32
Average velocity on \([a,b]\)?
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Answer#32
\(\dfrac{s(b)-s(a)}{b-a}\).
Trap: Confusing it with average speed.
Motion (s–v–a)✓ Known #33
Average speed on \([a,b]\)?
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Answer#33
Total distance traveled divided by \(b-a\) — usually \(\dfrac{1}{b-a}\int_a^b|v(t)|\,dt\).
Trap: Using displacement instead of distance.
Motion (s–v–a)✓ Known #34
When is a particle moving to the right (or up)?
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Answer#34
When \(v(t)>0\).
Trap: Using \(a(t)>0\) instead.
Motion (s–v–a)✓ Known #35
When is a particle speeding up?
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Answer#35
- When velocity and acceleration have the same sign.
- \(v>0,\ a>0\) or \(v<0,\ a<0\).
Trap: Thinking \(a>0\) alone means speeding up.
Motion (s–v–a)✓ Known #36
When is a particle slowing down?
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Answer#36
When velocity and acceleration have opposite signs.
Trap: Ignoring the sign of the velocity.
Motion (s–v–a)✓ Known #37
Displacement from \(t=a\) to \(t=b\)?
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Answer#37
\(\displaystyle\int_a^b v(t)\,dt=s(b)-s(a)\).
Trap: Using \(\int|v|\,dt\), which is distance.
Motion (s–v–a)✓ Known #38
How do you find the total distance traveled from \(a\) to \(b\)?
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Answer#38
\(\displaystyle\int_a^b|v(t)|\,dt\): split at the times where \(v\) changes sign and add the absolute displacements.
Trap: Computing \(\int v\,dt\) and taking the absolute value at the end.
Motion (s–v–a)✓ Known #39
If \(v(t)\) changes sign, what does that indicate?
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Answer#39
The particle changes direction.
Trap: Confusing it with speeding up or slowing down.
Motion (s–v–a)✓ Known #40
If \(a(t)=0\) at some time, does the particle stop?
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Answer#40
No. Only the acceleration is zero at that instant; the velocity can be nonzero.
Trap: Claiming the particle is at rest.
Extrema / EVT / Derivative Tests✓ Known #41
What are critical numbers?
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Answer#41
Values \(x=c\) in the domain where \(f'(c)=0\) or \(f'(c)\) does not exist.
Trap: Including points that are not in the domain.
Extrema / EVT / Derivative Tests✓ Known #42
If \(f\) has a local extremum at \(c\) and is differentiable there, what must be true?
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Answer#42
\(f'(c)=0\) (Fermat’s theorem): local extrema occur at critical numbers.
Trap: Thinking \(f'(c)=0\) guarantees a local extremum.
Extrema / EVT / Derivative Tests✓ Known #43
State the Extreme Value Theorem.
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Answer#43
If \(f\) is continuous on a closed interval \([a,b]\), then \(f\) attains an absolute maximum and an absolute minimum on \([a,b]\).
Trap: Forgetting “continuous” or “closed interval.”
Extrema / EVT / Derivative Tests✓ Known #44
How do you find the absolute max and min on \([a,b]\)?
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Answer#44
- Find the critical numbers in \((a,b)\).
- Evaluate \(f\) at the critical numbers and at the endpoints.
- The largest value is the absolute max; the smallest is the absolute min.
Trap: Ignoring the endpoints.
Extrema / EVT / Derivative Tests✓ Known #45
First derivative test: how do you classify a critical point?
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Answer#45
- \(f’\) changes from + to −: local max.
- \(f’\) changes from − to +: local min.
- No sign change: neither.
Trap: Using only \(f'(c)=0\) without checking a sign change.
Extrema / EVT / Derivative Tests✓ Known #46
Second derivative test at \(c\) with \(f'(c)=0\): what are the outcomes?
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Answer#46
- \(f”(c)>0\): local min.
- \(f”(c)<0\): local max.
- \(f”(c)=0\): inconclusive — use the first derivative test.
Trap: Declaring a max or min when \(f”(c)=0\).
Extrema / EVT / Derivative Tests✓ Known #47
What do the signs of \(f’\) tell you?
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Answer#47
- \(f’>0\) on an interval ⇒ \(f\) increasing there.
- \(f'<0\) on an interval ⇒ \(f\) decreasing there.
Trap: Confusing this with concavity, which comes from \(f”\).
Extrema / EVT / Derivative Tests✓ Known #48
Candidates test: what are the candidates for absolute extrema on \([a,b]\)?
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Answer#48
The endpoints and the critical numbers inside \((a,b)\).
Trap: Including points where \(f\) is undefined.
Extrema / EVT / Derivative Tests✓ Known #49
If the graph has a corner at \(x=c\), can \(c\) be a critical number?
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Answer#49
Yes — if \(c\) is in the domain and \(f'(c)\) does not exist, \(c\) is a critical number.
Trap: Counting only points where \(f'(c)=0\).
Extrema / EVT / Derivative Tests✓ Known #50
Does \(f'(c)=0\) imply an absolute max or min on \([a,b]\)?
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Answer#50
No. It is only a candidate; compare the values with the endpoints.
Trap: Forgetting the endpoints and the comparison.
MVT & Rolle✓ Known #51
State the Mean Value Theorem.
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Answer#51
If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there is \(c\in(a,b)\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\).
Trap: Forgetting the hypotheses.
MVT & Rolle✓ Known #52
State Rolle’s Theorem.
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Answer#52
If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a)=f(b)\), then there is \(c\in(a,b)\) with \(f'(c)=0\).
Trap: Not checking \(f(a)=f(b)\).
MVT & Rolle✓ Known #53
What does the MVT mean geometrically?
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Answer#53
Somewhere in \((a,b)\), a tangent line is parallel to the secant line from \(a\) to \(b\).
Trap: Saying the tangent equals the secant at the endpoints.
MVT & Rolle✓ Known #54
A problem asks you to justify that some \(c\) exists. What must you write?
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Answer#54
- \(f\) is continuous on \([a,b]\).
- \(f\) is differentiable on \((a,b)\).
- Therefore, by the MVT (or Rolle’s Theorem), such a \(c\) exists.
Trap: Stating the conclusion without the hypotheses.
MVT & Rolle✓ Known #55
If \(f\) is not continuous on \([a,b]\), can you apply the MVT?
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Answer#55
No — the conclusion may fail.
Trap: Applying the MVT across a hole, jump or asymptote.
MVT & Rolle✓ Known #56
If \(f'(x)\ge 0\) on an interval, what can you conclude?
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Answer#56
\(f\) is nondecreasing on that interval.
Trap: Claiming strictly increasing without more information.
MVT & Rolle✓ Known #57
When should you think of Rolle’s Theorem specifically?
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Answer#57
When \(f(a)=f(b)\) and the question asks about a point where \(f’=0\).
Trap: Using the MVT slope formula when Rolle’s is more direct.
MVT & Rolle✓ Known #58
If \(f'(x)=0\) for every \(x\) in an interval, what is \(f\)?
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Answer#58
Constant on that interval.
Trap: Thinking \(f\) could still vary.
Concavity & Inflection✓ Known #59
What do the signs of \(f”\) tell you?
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Answer#59
- \(f”>0\): concave up.
- \(f”<0\): concave down.
Trap: Using the signs of \(f’\) for concavity.
Concavity & Inflection✓ Known #60
Define a point of inflection.
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Answer#60
A point where \(f\) is continuous and the concavity changes (\(f”\) changes sign).
Trap: Claiming \(f”=0\) automatically gives an inflection point.
Concavity & Inflection✓ Known #61
What must you check to confirm an inflection point at \(x=c\)?
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Answer#61
That \(f”\) changes sign at \(c\) (with \(f\) continuous there).
Trap: Checking only \(f”(c)=0\) and stopping.
Concavity & Inflection✓ Known #62
If \(f’\) is increasing on an interval, what can you say about \(f\)?
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Answer#62
\(f\) is concave up there — that is the definition of concave up.
Trap: Mixing up “\(f\) increasing” with “\(f’\) increasing.”
Concavity & Inflection✓ Known #63
What does the second derivative test require at \(x=c\)?
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Answer#63
\(f'(c)=0\) and \(f”(c)\) exists. (If \(f'(c)\) does not exist, use the first derivative test.)
Trap: Using the second derivative test where \(f'(c)\ne0\).
Concavity & Inflection✓ Known #64
If \(f'(x)>0\) and \(f”(x)<0\), how is \(f\) changing?
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Answer#64
Increasing and concave down: increasing at a decreasing rate.
Trap: Calling \(f\) decreasing because \(f”<0\).
Optimization✓ Known #65
Optimization: what is the 5-step workflow?
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Answer#65
- Draw a picture and define variables.
- Write the objective function to maximize or minimize.
- Use the constraint to reduce to one variable.
- Find the critical numbers.
- Check candidates (critical numbers and endpoints) and conclude with units.
Trap: Forgetting to use the constraint to eliminate a variable.
Optimization✓ Known #66
Why do you reduce to one variable before differentiating?
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Answer#66
AP Calculus optimizes functions of one variable; critical points come from that single derivative.
Trap: Trying to differentiate with two independent variables.
Optimization✓ Known #67
After finding the critical value, what must you do to answer?
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Answer#67
Compute the quantity the question asks for (the maximum area, minimum cost, …) and state units.
Trap: Reporting only the \(x\)-value when the maximum value was asked.
Optimization✓ Known #68
What domain must you always identify?
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Answer#68
The physically meaningful interval for the variable (lengths \(\ge 0\), limited material, …).
Trap: Accepting negative lengths.
Optimization✓ Known #69
Is a local minimum automatically the absolute minimum?
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Answer#69
Not necessarily. Compare all candidates — or, if it is the only critical number on the interval, justify with the first derivative test.
Trap: Claiming “local = absolute” without justification.
Optimization✓ Known #70
In word problems, what does the candidates test check?
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Answer#70
Critical values and endpoints (or boundary behavior) within the feasible domain.
Trap: Ignoring endpoints because a derivative test “found something.”
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