AP calculus BC 2019

138 STEP 4. Review the Knowledge You Need to Score High

10. The graph of f is shown in Figure 7.7-4 y
and f is twice differentiable. Which of the f′
following has the largest value:
0 x1 x2 x3 x
(A) f (−1) x4
(B) f (−1)
(C) f (−1)
(D) f (−1) and f (−1)

y Figure 7.7-5
f
14. Given the graph of f in Figure 7.7-6,
determine at which values of x is

–1 0 x

Figure 7.7-4

Sketch the graphs of the following functions Figure 7.7-6
indicating any relative and absolute
extrema, points of inflection, intervals on (a) f (x ) = 0
which the function is increasing, decreasing, (b) f (x ) = 0
concave upward or concave downward. (c) f a decreasing function.

11. f (x ) = x 4 − x 2 15. A function f is continuous on the interval
[−2, 5] with f (−2) = 10 and f (5) = 6 and
12. f (x ) = x + 4 the following properties:
x − 4
INTERVALS (−2, 1) x = 1 (1, 3) x = 3 (3, 5)
Part B Calculators are allowed.
f + 0 − undefined +
13. Given the graph of f in Figure 7.7-5,
determine at which of the four values of x f − 0 − undefined +
(x1, x2, x3, x4) f has:
(a) Find the intervals on which f is
(a) the largest value, increasing or decreasing.
(b) the smallest value,
(c) a point of inflection, (b) Find where f has its absolute extrema.
(d) and at which of the four values of x (c) Find where f has points of inflection.
(d) Find the intervals where f is concave
does f have the largest value.
upward or downward.
(e) Sketch a possible graph of f .

Graphs of Functions and Derivatives 139

16. Given the graph of f in Figure 7.7-7, find 18. How many points of inflection does the
where the function f graph of y = cos(x 2) have on the interval
(a) has its relative extrema. [−π, π ]?
(b) is increasing or decreasing.
(c) has its point(s) of inflection. Sketch the graphs of the following functions
(d) is concave upward or downward. indicating any relative extrema, points of
(e) if f (0) = 1 and f (6) = 5, draw a sketch inflection, asymptotes, and intervals where
of f . the function is increasing, decreasing,
concave upward or concave downward.
Figure 7.7-7
19. f (x ) = 3e −x2/2
17. If f (x ) = |x 2 − 6x − 7|, which of the
following statements about f are true? 20. f (x ) = cos x sin2 x [0, 2π ]
I. f has a relative maximum at x = 3.
II. f is differentiable at x = 7. 21. Find the Cartesian equation of the curve
III. f has a point of inflection at x = −1. t
defined by x = 2 , 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.

7.8 Cumulative Review Problems

(Calculator) indicates that calculators are 28. Find d 2y if y = cos(2x ) + 3x 2 − 1.
permitted. dx2

26. Find dy if (x 2 + y 2)2 = 10x y . 29. (Calculator) Determine the value of k such
dx that the function

f (x) = x 2 − 1, x ≤ 1 is continuous

27. Evaluate lim x + 9 − 3 . 2x + k, x > 1
x →0 x
for all real numbers.