AP Unit 4 Problems
Thirty problems for AP Unit 4. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
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Problem 1
W(t) is water in liters, t is minutes, and W′(6) = −2.5. Interpret this derivative. Can you determine W(6)?
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Problem 2
Use the table to estimate T′(3) with a centered secant. Interpret the result. T is °C and t is minutes.
| t | 2.8 | 3.0 | 3.2 |
|---|---|---|---|
| T(t) | 19.1 | 19.8 | 20.5 |
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Problem 3
B(T) is a battery’s operating time in hours at temperature T in °C. B′(20) = −0.08. Explain its units and estimate the change when temperature rises from 20°C to 20.5°C.
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Problem 4
The graph of a differentiable quantity P has a horizontal tangent at t = 4 and lies above the horizontal axis there. What do P(4) > 0 and P′(4) = 0 each say? Must P have a local maximum?
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Problem 5
For s(t) = t³ − 6t² + 9t meters, find velocity, acceleration, direction, and speed at t = 2 seconds.
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Problem 6
A particle has v(t) = (t − 1)(t − 3) m/s for 0 ≤ t ≤ 4. On which open intervals does it move right, and when does it reverse direction?
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Problem 7
The entries are exact instantaneous velocity and acceleration values. Classify direction and whether speed increases or decreases at each time.
| t | v(t) | a(t) |
|---|---|---|
| 1 | −3 | 2 |
| 2 | 4 | −1 |
| 3 | −2 | −5 |
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Problem 8
A velocity graph is the straight line through (0,−4), (2,0), and (4,4), with seconds and m/s. Determine acceleration and describe motion on (0,2) and (2,4).
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Problem 9
For v(t) = (t − 2)², t ≥ 0, find the rest time. Does the particle reverse direction? Where is it slowing down?
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Problem 10
A cost model is C(q) = 100 + 8q + 0.02q² dollars. Find marginal cost at q = 50 and the exact model cost of increasing from 50 to 51 items. Compare them.
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Problem 11
A concentration model is A(t) = 5e^(−0.3t) + 0.2t mg/L, t in hours. Use a numerical derivative or the derivative formula to find and interpret A′(2). Round to three decimals.
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Problem 12
A population N is measured in people and t in years. At t = 2, N′(2) = 40 and N″(2) = −3. Interpret both values and explain how the population can grow while its growth rate falls.
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Problem 13
x(t) and y(t) satisfy x² + xy + y² = 12. Differentiate with respect to t, keeping both variables symbolic.
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Problem 14
Let R(t) = x(t)/y(t). At an instant x = 6, y = 3, x′ = 2, y′ = −1, find R′.
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Problem 15
A student substitutes x = 3 and y = 4 into x² + y² = 25, then differentiates 25 = 25 to get 0 = 0. Explain the mistake and write the useful rate equation.
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Problem 16
A rectangle has x = 8 cm, y = 5 cm, x′ = −0.5 cm/s, y′ = 0.8 cm/s. Find and interpret the area rate.
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Problem 17
A 10-ft ladder leans against a vertical wall. Its foot moves away at 2 ft/s. When its foot is 6 ft from the wall, how fast is the top moving?
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Problem 18
Air expands a spherical balloon at dV/dt = 24π cm³/s. Find dr/dt when r = 2 cm, using V = (4/3)πr³.
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Problem 19
A conical tank is 12 ft high with top radius 4 ft. Water enters at 2π ft³/min. Find the rate of depth increase when h = 3 ft.
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Problem 20
A 6-ft person walks away from a 15-ft lamp at 4 ft/s. Let x be the lamp-to-person distance and s the shadow length. Find s′ and the speed of the shadow tip.
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Problem 21
At an instant, one car is 3 miles east of an intersection and travels east at 40 mph. Another is 4 miles north and travels north at 30 mph. How fast is their separation changing?
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Problem 22
Use linearization at a = 16 to estimate √16.32 and justify overestimate or underestimate.
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Problem 23
A differentiable function has f(2) = 5 and f′(2) = −3. It is concave up throughout [2,2.1]. Approximate f(2.1) and state the error direction.
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Problem 24
A circle has measured radius 10 cm with a small increase of 0.02 cm. Use a differential to estimate the area increase, then find the exact increase.
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Problem 25
Use the tangent to f(x) = eˣ at 0 to estimate e^0.08. Use a calculator to find the actual value and absolute error to six decimals. Explain the direction.
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Problem 26
Find lim as x → 0 of (e^(2x) − 1)/x and justify the method.
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Problem 27
Find lim as x → ∞ of ln x/x², stating its initial form.
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Problem 28
Find lim as x → 0 of (1 − cos(3x))/x² using justified applications of L’Hospital’s Rule.
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Problem 29
A student applies L’Hospital to lim as x → 0⁺ of (1 + x)/x and answers 1. Explain the invalid step and find the limit.
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Problem 30
Evaluate lim as x → 1 of (x² − 1)/(x − 1) two ways. Explain why L’Hospital is valid but unnecessary.
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