Review Chapter 4

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 1. Find the value or values of c that satisfy the equation f(b) − f(a) b−a Assignment: Review: Chapter 4 = f′(c) in the conclusion of the mean value theorem for the given function and interval. f(x) = 10x + c= 10 x 1 ​, 1 10 ,10 ​(Use a comma to separate answers as​needed.) 2. You are planning to make an open rectangular box from a 23​-in.-by-45​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this​way, and what is its​volume? The dimensions of box of maximum volume are 4.84,13.32,35.32 in. ​(Round to the nearest hundredth as needed. Use a comma to separate answers as​needed.) 3 2277.04 in . The maximum volume is ​(Round to the nearest hundredth as​needed.) 3. Find the location of the indicated absolute extremum for the function. B. x = 1 ​Extremum: Maximum 5 C. x = 4 h(x) D. No maximum 4 3 2 1 -5 -4 -3 -2 -1 -1 x 1 2 3 4 5 -2 -3 -4 -5 4. Find the extreme values of the function and where they occur. 3 A. x = − 4 2 y = x − 3x + 1 A. Local minimum at ​(2, − ​3) B. Local maximum at​(0,1) C. There are no local extremes D. Local maximum at​(0,1), local minimum at ​(2, − 3) 5. Find the absolute extreme values of the function on the interval. 2 ​g(x) = 10 − 7x ​, − 1 ≤ x ≤ 2 A. absolute maximum is 20 at x = ​0; absolute minimum is 3 at x = 2 B. absolute maximum is 10 at x = ​0; absolute minimum is − 18 at x = 2 C. absolute maximum is 70 at x = ​0; absolute minimum is 3 at x = − 1 D. absolute maximum is 7 at x = ​0; absolute minimum is − 38 at x = 2 6. Find the absolute extreme values of the function on the interval. x ​f(x) = e − ​x, − 5 ≤ x ≤ 2 A. Minimum value is 1 at x = ​0; no maximum value B. Minimum value is 1 at x = ​0; maximum value is e− 5 + 5 at x = − 5 C. Minimum value is e− 5 + 5 at x = − 5​; maximum value is e2 − 2 at x = 2 D. Minimum value is 1 at x = ​0; maximum value is e2 − 2 at x = 2 7. Determine all critical points for the function. 3 2 ​f(x) = x − 9x + 8 A. x = 0 and x = 3 B. x = 0 and x = 6 C. x = − 3 and x = 3 D. x = 0 8. Determine all critical points for the function. 3 5 ​f(x) = 5x − 3x A. x = − 1 and x = 1 B. x = − 1 C. x = 1 D. x = ​0, x = − 1​, and x = 1 9. Determine all critical points for the function. 2 ​f(x) = 6x x+7 A. x = − 14 and x = 0 B. x = − 7 C. x = 0 and x = − 7 D. x = 7 10. Determine all critical points for the function. 2 y = 3x − 96 x A. x = 0 B. x = ​0, x = ​4, and x = − 4 C. x = 0 and x = 4 D. x = 4 11. Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the function and interval. 2 ​f(x) = x + 2x + 1​, ​[ − 3​,2​] A. ​0, − B. − 1 2 1 1 ​, 2 2 C. − 3​, 2 D. − 12. 1 2 Find the value or values of c that satisfy the equation f(b) − f(a) b−a = f′​(c) in the conclusion of the Mean Value Theorem for the function and interval. Round to the nearest thousandth. ​f(x) = ln (x − 1)​, ​[2​,4​] A. 3.885 B. 3.820 C. ± 2.820 D. 2.820 13. Using the derivative of​f(x) given​below, determine the critical points of​f(x). f′​(x) = ​(x + 6​)(x + 9​) A. 6​, 9 B. − 15 C. − 9​, − 6 D. ​0, − 9​, − 6 14. Using the derivative of​f(x) given​below, determine the critical points of​f(x). f′​(x) = ​(x + 8​)(x + 9​) A. 8​, 9 B. − 17 C. ​0, − 9​, − 8 D. − 9​, − 8 15. Using the derivative of​f(x) given​below, determine the critical points of​f(x). 2 f′​(x) = (x − 2) ​(x + 8​) A. − 2​, 8 B. − 8​, 2 C. − 2​, ​0, 8 D. − 8​, − 2​, 2 16. Using the derivative of​f(x) given​below, determine the critical points of​f(x). 2 −x f′​(x) = (x + 2) e A. 2 B. − 2 C. − 2​, 0 D. 0 17. Using the derivative of​f(x) given​below, determine the intervals on which​f(x) is increasing or decreasing. f′​(x) = ​(3 − ​x)(5 − ​x) A. Decreasing on ​( − ∞​, 3​) ∪ ​(5​, ∞​); increasing on ​(3​, 5​) B. Decreasing on ​( − ∞​, − 3​) ∪ ​( − 5​, ∞​); increasing on ​( − 3​, − 5​) C. Decreasing on ​(3​, 5​); increasing on ​( − ∞​, 3​) ∪ ​(5​, ∞​) D. Decreasing on ​( − ∞​, 3​); increasing on ​(5​, ∞) 18. Find the largest open interval where the function is changing as requested. 2 Where ​f(x) = x − 2x + 1 is increasing A. ​(0,∞​) B. ​( − ∞​,0) C. ​(1,∞​) D. ​( − ∞​,1) 19. Identify the​function's local and absolute extreme​values, if​any, saying where they occur. 3 2 ​f(x) = x + 7x + 8x − 2 A. local maximum at x = − 4​; local minimum at x = − B. local maximum at x = − 4 3 2 3 3 ​; local minimum at x = − 2 C. local maximum at x = 2​; local minimum at x = D. local maximum at x = 2 4 3 ​; local minimum at x = 4 20. Find the extrema of the function on the given​interval, and say where they occur. sin 4x​, 0 ≤ x ≤ A. π 2 local​maxima: 1 at x = π 8 and 0 at x = local​minima: 0 at x = 0 and − 1 at x = B. local​maxima: 1 at x = π 8 and 0 at x = local​minimum: − 1 at x = C. local​maxima: 1 at x = π 8 local​minimum: 0 at x = 0 D. local​maxima: 1 at x = π 4 and 0 at x = and 0 at x = 2 x − 49 lim x+7 x→ − 7 A. 7 B. − 14 C. − 7 D. 14 22. Use​l'Hôpital's Rule to evaluate the limit. lim 2 x →1 x−1 A. − 9 B. 12 C. 15 D. 9 8 π 2 8 21. Use​l'Hôpital's Rule to evaluate the limit. 3 2 3π 3π local​minima: 0 at x = 0 and − 1 at x = x − 6x + 5 π π 4 π 2 3π 8 23. Use​l'Hôpital's Rule to evaluate the limit. 2 16x − 4x − 3 lim 2 x→∞ 15x − 3x + 6 A. 1 B. 15 16 C. − D. 16 15 16 15 24. Use​l'Hôpital's Rule to find the limit. lim θ →0 2 − 2 cos θ sin 4θ A. 0 B. 1 2 C. 1 D. ∞ 25. A company is constructing an​open-top, square-based, rectangular metal tank that will have a volume of 53 cubic feet. What dimensions yield the minimum surface​area? Round to the nearest tenth. A. 3.8 feet × 3.8 feet × 3.8 feet B. 5.4 feet × 5.4 feet × 1.8 feet C. 10.3 feet × 10.3 feet × 0.5 feet D. 4.7 feet × 4.7 feet × 2.4 feet