Graphs of Functions and Derivatives Practice Problems for AP Calculus

Review the following concepts if needed:

• Rolles Theorem for AP Calculus
• Mean Value Theorem for AP Calculus
• Extreme Value Theorem for AP Calculus
• Test for Increasing and Decreasing Functions for AP Calculus
• First Derivative Test and Second Derivative Test for Relative Extremas for AP Calculus
• Test for Concavity and Points of Inflection for AP Calculus
• Graphing without Calculators for AP Calculus
• Graphing with Calculators for AP Calculus
• Graphs of Derivatives for AP Calculus
• Parametric Curves for AP Calculus
• Polar Equations, Polar Graphs, and Symmetry of Polar Graphs for AP Calculus
• Vectors and Vector Arithmetic for AP Calculus

Part A—The use of a calculator is not allowed.

1. If f(x )=x³ – x² – 2x, show that the hypotheses of Rolle's Theorem are satisfied on the interval [–1, 2] and find all values of c that satisfy the conclusion of the theorem.
2. Let f (x )=e^x. Show that the hypotheses of the Mean Value Theorem are satisfied on [0, 1] and find all values of c that satisfy the conclusion of the theorem.
3. Determine the intervals in which the graph of is concave upward or downward.
4. Given f (x)=x + sin x 0 ≤ x ≥ 2π, find all points of inflection of f.
5. Show that the absolute minimum of on [–5, 5] is 0 and the absolute maximum is 5.
6. Given the function f in Figure 7.7-1, identify the points where:
   1. f '< 0 and f'' > 0,
   2. f ' < 0 and f '' < 0,
   3. f ' =0,
   4. f '' does not exist.



7. Given the graph of f ''in Figure 7.7-2, determine the values of x at which the function f has a point of inflection. (See Figure 7.7-2.)



8. If f ''(x)=x²(x +3)(x – 5), find the values of x at which the graph of f has a change of concavity.
9. The graph of f ' on [–3, 3] is shown in Figure 7.7-3. Find the values of x on [–3, 3] such that (a) f is increasing and (b) f is concave downward.



10. The graph of f is shown in Figure 7.7-4 and f is twice differentiable. Which of the following has the largest value:
    1. f (–1)
    2. f '(–1)
    3. f ''(–1)
    4. f (–1) and f '(–1)
    5. f '(–1) and f ''(–1)

Sketch the graphs of the following functions indicating any relative and absolute extrema, points of inflection, intervals on which the function is increasing, decreasing, concave upward or concave downward.



11. f (x)=x⁴ – x²
12. 

Part B—Calculators are allowed.

13. Given the graph of f ' in Figure 7.7-5, determine at which of the four values of x (x₁, x₂, x₃, x₄) f has:
    1. the largest value,
    2. the smallest value,
    3. a point of inflection,
    4. and at which of the four values of x does f '' have the largest value.

    

14. Given the graph of f in Figure 7.7-6, determine at which values of x is



1. f ''(x )=0
2. f ''(x )=0
3. f '' a decreasing function.
15. A function f is continuous on the interval [–2, 5] with f (–2)=10 and f (5)=6 and the following properties:



1. Find the intervals on which f is increasing or decreasing.
2. Find where f has its absolute extrema.
3. Find where f has points of inflection.
4. Find the intervals where f is concave upward or downward.
5. Sketch a possible graph of f.
16. Given the graph of f ' in Figure 7.7-7, find where the function f
    1. Has its relative extrema.
    2. Is increasing or decreasing.
    3. Has its point(s) of inflection.
    4. Is concave upward or downward.
    5. If f (0)=1 and f (6)=5, draw a sketch of f.

    

17. If f (x )=|x² – 6x –7|, which of the following statements about f are true?
    1. f has a relative maximum at x =3.
    2. f is differentiable at x =7.
    3. f has a point of inflection at x = – 1.
18. How many points of inflection does the graph of y = cos(x²) have on the interval [–π, π]?

Sketch the graphs of the following functions indicating any relative extrema, points of inflection, asymptotes, and intervals where the function is increasing, decreasing, concave upward or concave downward.

19. 
20. f (x )= cos x sin² x [0, 2π]
21. Find the Cartesian equation of the curve defined by , y =t² – 4t +1.
22. Find the polar equation of the line with Cartesian equation y =3x – 5.
23. Identify the type of graph defined by the equation r =2 – sin θ and determine its symmetry, if any.
24. Find the value of k so that the vectors 3, –2 and 1, k are orthogonal.
25. Determine whether the vectors 5, –3 and 5, 3 are orthogonal. If not, find the angle between the vectors.

(Calculator) indicates that calculators are permitted.

26. Find
27. Evaluate
28. Find
29. (Calculator) Determine the value of k such that the function



30. A function f is continuous on the interval [–1, 4] with f (–1)=0 and f (4)=2 and the following properties:



1. Find the intervals on which f is increasing or decreasing.
2. Find where f has its absolute extrema.
3. Find where f has points of inflection.
4. Find intervals on which f is concave upward or downward.
5. Sketch a possible graph of f.
31. Evaluate
32. Evaluate
33. Find the polar equation of the ellipse x² +4y² =4.

 Solutions for these practice problems can be found at:  Solutions to Graphs of Functions and Derivatives Practice Problems for AP Calculus