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 )=ex. 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:
a. f '< 0 and f'' > 0,
b. f ' < 0 and f '' < 0,
c. f ' =0,
d. 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:
a. f (–1)
b. f '(–1)
c. f ''(–1)
d. f (–1) and f '(–1)
e. 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:
a. the largest value,
b. the smallest value,
c. a point of inflection,
d. 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
a. f ''(x )=0
b. f ''(x )=0
c. 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:
a. Find the intervals on which f is increasing or decreasing.
b. Find where f has its absolute extrema.
c. Find where f has points of inflection.
d. Find the intervals where f is concave upward or downward.
e. Sketch a possible graph of f.
16. Given the graph of f ' in Figure 7.7-7, find where the function f
a. Has its relative extrema.
b. Is increasing or decreasing.
c. Has its point(s) of inflection.
d. Is concave upward or downward.
e. 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?
I.
f has a relative maximum at x =3.
II.
f is differentiable at x =7.
III.
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:
a.
b.
c.
d.
e.
Find the intervals on which f is increasing or decreasing.
Find where f has its absolute extrema.
Find where f has points of inflection.
Find intervals on which f is concave upward or downward.
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
These are from a series I didn’t expect to like (but did!): Five Steps to a Five, from McGraw Hill.