AP Calculus Chapter 4 Testbank - Part I - Multiple-Choice Questions

Q: How do you determine local extrema for a polynomial like f(x)=x^3+3x^2-45x+4?

Compute f'(x), solve for critical points, then classify each point as a local minimum or maximum based on the sign changes in f'(x) (or the values compared to neighboring intervals).

Q: How is the volume change of a sphere approximated when the radius increases slightly?

Use differentials: express volume V in terms of radius r, compute dV=V'(r)dr, and substitute r=10 with dr=10.02-10 to estimate the change.

Q: What is the role of the Mean Value Theorem in these problems?

It guarantees the existence of a point c in the interval where f'(c) equals the average rate of change. The tasks then solve for c using that equation on the specified interval.

Sep 30, 2026

Upload by MrHarris58

AP Calculus Chapter 4 practice testbank covering local extrema, monotonicity, volume approximation via differentials, concavity and points of inflection, and applications of the Mean Value Theorem. Includes multiple-choice problems on finding minima/maxima using derivatives, solving for coordinates of inflection points, determining intervals where functions are decreasing or concave down, and related word problems such as a ladder sliding down a wall. Part II provides free-response solutions for extreme values using critical points and endpoint comparisons.

LanguageEnglish

CategoryExam

FormatPDF

Pages23

Reading Time58 min

AccessFree to download

Part I. Multiple-Choice Questions

Questions 1-15

Part II. Free-Response Questions

Questions 16-21

AP Calculus Chapter 4 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions 1. Let f(x) = x3 + 3x2 􀀀 45x + 4 . Then the local extrema of f are (A) a local minimum of 􀀀179 at x = 5 and a local maximum of 77 at x = 􀀀3 . (B) a local minimum of 􀀀77 at x = 3 and a local maximum of 179 at x = 􀀀5 . (C) a local minimum of 􀀀179 at x = 􀀀5 and a local maximum of 􀀀77 at x = 3 . (D) a local minimum of 􀀀77 at x = 3 and a local maximum of 77 at x = 5 . (E) a local minimum of 77 at x = 􀀀5 and a local maximum of 179 at x = 3 . 2. Consider the function f(x) = x3 + 3x2 􀀀 45x + 4 . Then I. f is decreasing on (􀀀5 ; 3) and increasing on ( 􀀀1 ; 􀀀5)[(3 ; 1 ) . II. f has a local minimum at x = 3 . III. f has a local maximum at x = 􀀀5 . (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 3. For what value of x does the function f(x) = x3 􀀀 9x2 􀀀 120x + 6 have a local minimum? (A) 10 (B) 4 (C) 3 (D) 􀀀4 (E) 􀀀10 4. Use differentials to approximate the change in the volume of a sphere when the radius is increased from 10 to 10.02 cm. (A) 4213.973 (B) 1261.669 (C) 1256.637 (D) 25.233 (E) 25.133 5. The graph of y = x3 􀀀 5x2 + 4x + 2 has a local minimum at (A) (0:46 ; 2:87) (B) (0:46 ; 0) (C) (2:94 ; 􀀀4:05) (D) (4:06 ; 2:87) (E) (1:66 ; 􀀀0:59) 6. Find a positive value, c, for x, that satisﬁes the conclusion of the Mean Value Theorem for Derivatives for f(x) = 3x2 􀀀 5x + 1 on the interval [2 ; 5] . (A) 1 13 (B) ~~ ~~ 6 11 (C) ~~ ~~ 6 23 (D) ~~ ~~ 6 (E) | 7 | | --- | | 2 | 7. The graph of y = x3 􀀀 2x2 􀀀 5x + 2 has a local maximum at (A) (2:120 ; 0) (B) (2:120 ; 􀀀8:061) (C) (􀀀0:786 ; 0) (D) (􀀀0:786 ; 4:209) (E) (0:666 ; 􀀀1:926) 8. A 20-foot ladder slides down a wall at 5 ft/sec. At what speed is the bottom sliding out (in ft/sec) when the top is 10 feet from the ﬂoor? (A) 0:346 (B) 2:887 (C) 0:224 (D) 5:774 (E) 4:472 9. What are the coordinates of the point of inﬂection on the graph of y = x3 􀀀 15x2 + 33x + 100? (A) (9 ; 0) (B) (5 ; 􀀀48) (C) (1 ; 119) (D) (9 ; 􀀀89) (E) (5 ; 15) 10. The graph of y = x4 + 8x3 􀀀 72x2 + 4 is concave down for (A) 􀀀6 < x < 2 (B) x > 2 (C) x < 􀀀6 (D) x < 􀀀3 􀀀 3p5 or x > 􀀀3 + 3p5 (E) 􀀀3 􀀀 3p5 < x < 􀀀3 + 3p5 11. The function f is given by f (x) = x4 + 4x3. On which of the following intervals is f decreasing? (A) (􀀀3 ; 0) (B) (0 ; 1 ) (C) (􀀀3 ; 1 ) (D) ( 􀀀1 ; 􀀀3) (E) ( 􀀀1 ; 0) 12. The value of c that satisﬁes the Mean Value Theorem on the interval [0 ; 5] for the function f(x) = x3 􀀀 6x is 5 (A) 􀀀 ~~ ~~ p 3 (B) 0 (C) 1 5 (D) ~~ ~~ 3 (E) 5 p3 13. The graph of the function y = x3 + 12x2 + 15x + 3 has a relative maximum at x = (A) 􀀀10:613 (B) 􀀀:248 (C) 􀀀7:317 (D) 􀀀1:138 (E) 􀀀:683 14. The side of a square is increasing at a constant rate of 0.4 cm/sec. In terms of the perimeter, P, what is the rate of change of the area of the square, in cm2 /sec? (A) 0:05P (B) 0:2P (C) 0:4P (D) 6:4P (E) 51:2P 15. The second derivative of a function is given by f00 (x) = xsin x 􀀀 2. How many points of inﬂection does f have on the interval (􀀀10 ; 10)? (A) Zero (B) Two (C) Four (D) Six (E) Eight Part II. Free-Response Questions 16. Find the extreme values of the function f(x) = x3 +3x2 􀀀 36x +4 on the interval [0 ; 4] . The critical x-values are obtained by solving f 0 (x) = 0 . This leads to 3x2 + 6x 􀀀 36 = 0 ) x = 􀀀1 +p 13 . The extreme values are obtained by comparing: f (0) = 4 f (􀀀1 +p13 ) = 􀀀51:74 f (4) = 􀀀28 Therefore the maximum of f is 4 and the minimum is 􀀀51:74 . 17. Find the extreme values of the function f (x) = x~~2 ~~ on the interval [􀀀3 ; 3] . From f 0 (x) = 11~~ ~~~~ ~~xx2~~2~~, we see that the critical x-values on the given interval are x = 􀀆1 . We compare: f (􀀀3) = 30 1 f (􀀀1) = 􀀀 2 f (1) = 12 3 f (3) = 10 Therefore f has a maximum of 12 (at x = 1) and a minimum of 􀀀 12 (at x = 􀀀1) . 18. Find the extreme values of the

FAQs

How do you determine local extrema for a polynomial like f(x)=x^3+3x^2-45x+4?

Compute f'(x), solve for critical points, then classify each point as a local minimum or maximum based on the sign changes in f'(x) (or the values compared to neighboring intervals).

How is the volume change of a sphere approximated when the radius increases slightly?

Use differentials: express volume V in terms of radius r, compute dV=V'(r)dr, and substitute r=10 with dr=10.02-10 to estimate the change.

What is the role of the Mean Value Theorem in these problems?

It guarantees the existence of a point c in the interval where f'(c) equals the average rate of change. The tasks then solve for c using that equation on the specified interval.

Sep 30, 2026

Upload by MrHarris58