AP Unit 5 Problems
Thirty problems for AP Unit 5. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
Choose your next practice
Find the problems you need.
Mix the choices to focus your practice. Problem numbers and worked-solution links stay the same.
30 of 30 problems shown.
Selection: All topics and practice types.
Printing includes the currently visible selection. Reset filters first to print all 30 problems.
Problem 1
Verify MVT for f(x) = x² on [1,3], and find every c in the open interval satisfying its conclusion.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 2
Can MVT be applied to f(x) = |x| on [−1,1]? Does any interior derivative equal the endpoint secant slope?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 3
A temperature function is continuous on [2,6] and differentiable on (2,6). T(2) = 30°C and T(6) = 20°C, with t in minutes. What instantaneous rate is guaranteed, and where? Is the time unique?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 4
Compare f(x) = x on [0,1] and on (0,1). Give their absolute extrema, and explain why the domain changes the answer.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 5
For f(x) = x^(2/3) and g(x) = x³ on the real line, classify the critical point at zero for each. Explain why “critical” does not mean “extremum.”
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 6
A differentiable function has f′(x) = (x + 1)(x − 2). Determine all increasing and decreasing intervals.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 7
A differentiable function has exact derivative values in the table. What can you conclude at each listed input, and can the table prove increasing behavior throughout [0,4]?
| x | 0 | 2 | 4 |
|---|---|---|---|
| f′(x) | 3 | 1 | 2 |
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 8
For a differentiable function with f′(x) = (x − 1)²(x + 2), classify the critical numbers by the first derivative test.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 9
A continuous derivative graph passes linearly through (−1,2), (0,0), (1,−2), (2,0), and (3,2). Classify the relative extrema of f in (−1,3).
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 10
Find all absolute extrema of f(x) = x³ − 3x on [−2,3]. Show a complete candidates comparison.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 11
Find all absolute extrema of f(x) = |x − 1| on [−1,3]. Include the nondifferentiable candidate.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 12
A twice-differentiable function on the real line has f″(x) = x(x − 2). Determine concavity intervals, where f′ increases/decreases, and the x-coordinates of inflection points.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 13
Compare the origin for f(x) = x⁴ and g(x) = x^(1/3). Which has an inflection there? Explain using concavity, including any undefined derivatives.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 14
Use the second derivative test to classify critical points of f(x) = x³ − 3x, including their function values.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 15
Both f(x) = x⁴ and g(x) = x³ have f′(0) = g′(0) = 0 and f″(0) = g″(0) = 0. What does the second derivative test say, and what actually happens?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 16
For f(x) = x + 9/x on (0,∞), find and justify the absolute minimum. State why a local conclusion extends to the entire interval.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 17
For f(x) = x³ − 3x, give the labeled features needed for a calculus sketch: intercepts, extrema, concavity, and end behavior.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 18
The derivative is f′(x) = |x| − 1, with the graph shown. f(0) = 2. Describe extrema, concavity, the inflection, and the anchor for a sketch of f. Is f differentiable at zero?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 19
On an interval, f′ > 0 and f″ < 0. Explain the behavior of f and f′. Can you determine the sign of f?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 20
These are exact instantaneous values, and f′ and f″ are continuous near each listed input. Classify the local direction and concavity near each listed x; then say whether the three rows alone establish behavior on the full intervals between rows.
| x | f′(x) | f″(x) |
|---|---|---|
| −1 | −2 | 3 |
| 0 | 4 | −1 |
| 1 | −3 | −2 |
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 21
A rectangular garden borders a wall, so only three sides need fencing. With 96 m of fencing, let x be a perpendicular side. Build the area model, give its feasible domain, and find the maximizing dimensions and area.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 22
An open box is made by cutting equal squares of side h from the corners of a 20-cm by 12-cm sheet and folding the sides. Write its volume objective, feasible domain, and derivative equation for candidates. You need not solve it.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 23
A closed cylinder must hold 54π cm³. Minimize its surface area S = 2πr² + 2πrh, with positive dimensions. Give radius, height, area, and a global justification.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 24
A rectangle must have area 100 m². Find positive side lengths that minimize its perimeter, with a global justification.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 25
For the box model V(h) = h(20 − 2h)(12 − 2h), 0 < h < 6, find the maximizing h numerically to three decimals and the volume to three decimals. Justify that the chosen root is feasible and globally maximal.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 26
On the upper branch of x² + y² = 25, with −5 < x < 5, find the interior critical point and classify it using implicit derivative evidence.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 27
The implicit relation y² + xy = 6 passes through (1,2). Find y′ and y″ there and describe the local direction and concavity.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 28
Near (0,1), the relation y² = x³ + 1 uses its positive branch. Show that the point is critical, explain why the second derivative test is inconclusive, and classify extrema using first-derivative signs.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 29
A differentiable function has f′(x) = sin x on [0,2π]. Determine increasing/decreasing intervals, interior relative extrema, concavity intervals, and inflection x-coordinates. Can you determine the inflection heights without an f-value?
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.
Problem 30
Find all points on y = x² closest to (0,3). Minimize squared distance and give the actual minimum distance, with a global justification.
Written working workspace
Drawing is temporarily unavailable. Work on paper if needed, then record your reasoning here.
Saving written working requires JavaScript. Keep your work on paper or in your own notes; the source problem above remains available.