AP CALCULUS AB
CURVE SKETCHING WORKSHEET
NAME_____________________________
NO CALCULATORS ALLOWED
1978-BC2
1. Given the function f defined by f (x) = x x + 2 for all x such that −3 ≤ x ≤ 1 .
a) Find the values of x in the given interval for which f is increasing. Justify your answer.
b) For what values of x is the graph of f concave downward?
c) On the axes provided, sketch the graph of f.
d) Is f ′(x) continuous for all x in the given interval? Justify your answer.
1980-BC2
2. Given the function f defined by f (x) = cos x − cos 2 x for −π ≤ x ≤ π .
a) Find the x-intercepts of the graph of f.
b) Find the x- and y-coordinates of all relative maximum points of f. Justify your answer.
c) Find the intervals on which the graph of f is increasing.
d) Using the information found in parts (a), (b), and (c), sketch the graph of f on the axes
provided.
1981-BC1
2
3
3. Let f be the function defined by f (x) = 12x − 4x.
a) Find the intervals on which f is increasing.
b) Find the x- and y-coordinates of all relative maximum points.
c) Find the x- and y-coordinates of all relative minimum points.
d) Find the intervals on which f is concave downward.
e) Using the information found in parts (a), (b), (c) and (d), sketch the graph of f on the
axes provided.
1984-BC3
4. A function f is continuous on the closed interval [-3,3] such that f(-3) = 4 and f(3) = 1. The
functions f ′ and f ′′ have the properties given in the table below.
x
−3 < x < −1
x = −1
−1 < x < 1
x=1
1< x < 3
f ′(x)
Positive
Fails to exist
Negative
0
Negative
f ′′(x)
Positive
Fails to exist
Positive
0
Negative
a) What are the x-coordinates of all absolute maximum and absolute minimum points of f on
the interval [-3,3]? Justify your answer.
b) What are the x-coordinates of all points of inflection of f on the interval [-3,3]? Justify
your answer.
c). On the axes provided, sketch a graph that satisfies the given properties of f.