AP Calculus AB
Unit 4 Test Review
Name________________________
Use: f(x) = x3 − 3 x2 + 1 for #1-5
1. State the intervals of increase/decrease. Justify your answer.
2. Find the coordinates of the points of any relative max or relative min. Justify your answer.
3. State the intervals of concavity. Justify your answer.
4. Find the coordinates of any POI. Justify your answer.
5. On what intervals is f(x) decreasing AND concave down? Justify your answer.
Use the derivative: f '(x) = x3 ( x − 6 )
2
( x − 3 ) for #6 & 7.
6. For what values of x does f(x) have a relative minimum, if any. Justify your answer.
7. For what values of x does f(x) have a relative maximum, if any. Justify your answer.
Use the second derivative: f "(x) = − ( x − 3 )
2
( x + 5)
5
For #8-10.
8. On what intervals is f(x) concave down? Justify your answer.
9. On what intervals is f(x) concave up? Justify your answer.
10. For what values of x does f(x) have a POI? Justify your answer.
11. Given that f(2)=3, f’(2) = 0, and f’’(2) = -3, classify x=2 as a maximum, minimum or neither.
12. Given that f(2)=3, f’(2) = 5, and f’’(2) = -3, classify x=2 as a maximum, minimum or neither.
13. Given that f(2)=3, f’(2) = 0, and f’’(2) = 13, classify x=2 as a maximum, minimum or neither.
2
14. Find the absolute minimum value of f(x) = 2 x 3 , [−8,1]
15. Find the absolute maximum value of f(x) = − x2 + 4 x, [0,3]
Use for #16-22: x(t) = t3 − 9t2 + 24t − 3, v(t) = 3t2 − 18t + 24, a(t) = 6t − 18 where t  0 describes the
motion of the particle (in feet) moving along a horizontal line where t is measured in minutes.
16. Over what time interval(s) is the particle moving to the right? Justify your answer.
17. What direction is the particle moving when t=3? Justify your answer.
18. At what times is the particle at rest? Justify your answer.
19. Over what time interval(s) is the particle speeding up? Justify your answer.
20. Find the displacement of the particle over the first 2 seconds. Indicate units of measure.
21. Find the speed of the particle at t=3. Indicate units of measure.
22. Find the average velocity of the particle over the first 3 seconds? Indicate units of measure.