AP Physics C Calculus Practice

1. Draw a graph of the function y = -2x + 5 from x = -2 to x = +2.

2. A child sits at the end of a merry-go-round with a diameter of 6.0m.  What distance do they travel if the merry-go-round turns 2.5 radians?

3. Convert (25m, 40º) to Cartesian coordinates and (14m/s, -12m/s) to polar coordinates.

4. In terms of x, y, and r, what is the definition of the cotangent function?

5. Prove 1 + cot2θ = csc2θ.

6. A right triangle has vertices (0,0), (4,0) and (4,3).  What is the tangent of the left acute angle and what is the cosine of the right acute angle?

7. Graph y = sec x from x = 0 to x = 2π.

8. What are tan 90º, sin 45º, and csc 30º?

9. Simplify (xa)b, log x – log y, and xa times xb divided by xc.

10. What is the limit of [(x + 2)2 – 4] / x as x approaches zero? 11. Given the graph above, what are the instantaneous velocities at 1s and 4.5s?  What is the average velocity from 0s to 5s?

12. Use limits to show that the derivative of y = 10x2 equals 20x.

13. Sketch the graph of velocity versus time for the graph provided in number eleven.

14. Differentiate the functions y = -5x3 + 12x + 10, y = 12x1/3, and y = 12 / x2.

15. Differentiate the function y = (5x2 + 12x – 20)(-4x3 + 10).

16. Differentiate the function y = (2x2 – 10) / x3.

17. Differentiate the function y = (-5x2 + 4x)1/2.

18. Differentiate the function y = 5cos(x) + 10sin(x).

19. Differentiate the function y = -8sin(4x).

20. Differentiate the functions y = ln(5x) + ln x3 and y = 10x + e-2x.

21. Find the second derivative of the function for y = -12x3 + cos(x) + e3x­­.

22. Find the minimum or maximum of the function y = -4x2 + 16x – 12 and state whether it is a minimum or maximum. 23. For the graph above, find the area under the curve from 4s to 12s, including units.

24. Integrate the function f(x) = 4/x + 5ex – sin(x) + 12x2.

25. Integrate the function ∫2sin(6x)dx.

26. Integrate the function ∫θcos(θ)dθ.

27. Find the area under the curve y = -4x2 + 10x + 20 between x = 0 and x = 5.

28. Find the area under the curve of y = 4ex + x from zero to x.

29. If a car begins with a position of 50m, a velocity of 10m/s, and accelerates at a rate a = 4t2, what are the functions for velocity and position?

30. Use integral notation to show that the volume of a hemisphere is (2/3)πR3.