Calculus Practice Test: Chapter 3

Calculus Practice Test: chapter 3 Name: ______

1. Use implicit differentiation to find dy/dx : 2x2y + 5xy4 – 2x = 5

2a. Use implicit differentiation to find d2y/dx2 : x2 + xy = 5

2b. Use implicit differentiation to find d2y/dx2 : xcosy = y For questions 4-19, find the derivative. Circle your final, simplified answer.

 x 2  2   4. y = ln(4 + x ) 5. y = ln 3   x  5 

6. y = (lnx3)2 7. y = ln4 x5

2 2 8. f(x) = log(tan x) 9. f(x) = log5(cos x)

3 cosx 10. y = x log5x 11. y = e

2e x 12. y = 2xsinx 13. f(x) = 4  e x 5 14. f(x) = ln1 2e x  15. f(x) =sin 1 (e5x )

16. f(x) = cos-1(2sinx) 17. f(x) = e5x cot-1(lnx)

18. f(x) = 2 tan-1( x ) 19. y = sec-1 ( 5x )

2 20. y = csc-1( 4 x ) For 21-22, use logarithmic differentiation to find the derivative.

2 3 x x  4  cos x ln x 21. y = 4 3 22. y = ln x (2  x )  

23. For f(x) = 4 – x2, find f(1), and (f-1)’(3)

f(1) =

(f -1)’(3) =

24. For f(x) = 5 – 2x2, find f(2), and (f-1)’(-3)

f(2) =

(f -1)’(-3) =

25. Find the derivative of the inverse of: f(x) = 2x2 + x – 2

26. A pebble is thrown into a pond forming ripples whose radius increases at the rate of 4 in/sec. How fast is the area of the ripple changing when the radius is 12 inches? (Area of a circle is πr2) 27. A conical paper cup (vertex down) is being filled water at the rate of 3 cm3/sec. If the height of the water is always 1/3 of the base diameter, how fast is the height of the water changing when the water is 2 cm deep? (Volume of a cylinder is (1/3)πr2h)

28. A 20 foot ladder is leaning against a building. The ladder starts sliding down the wall so that the top of the ladder moves down at the rate of 0.5 ft/sec. How fast is the foot of the ladder moving away from the wall when the foot of the ladder is 12 feet from the wall?

29. A ladder 25 feet long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 2 ft/sec. Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 ft from the wall. 30a. Find the local linear approximation of f(x) = 3x2 – 4x at the point where x = 3.

b. Use your approximation to estimate f(2.9), and f(3.1)

f(1.8) ≈ ______f(2.2) ≈ ______

31. Find dy and Δy for f(x) = 3x2 + 4x at x = -2 and dx = Δx = .01.

dy = Δy =

32. The measurement of the radius of a sphere is found to be 8 inches, with a possible error of ±0.03 inch.

a. Use differentials to estimate the error in the calculated volume.

b. Find the percent error in the radius and the volume of the sphere.