AP Calculus AB
WS – Analysis of Graphs using the 1st and 2nd derivatives
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1. Let H’(x), the derivative of a continuous functions H(x) be given by the graph shown above in the interval [-5, 5].
a. Determine the intervals for which H(x) is increasing. Justify your answer.
b. Determine the intervals for which H(x) is decreasing. Justify your answer.
c. Determine the approximate intervals for which H(x) is concave up. Justify your answer.
d. Determine the approximate intervals for which H(x) is concave down. Justify your answer.
e. Determine the x – coordinates of all relative minimum(s) of H(x). Justify your answer.
f. Determine the approximate x – coordinates of all inflection points of H(x). Justify your answer.
g. Determine the x – coordinates of all the points on the curve of H(x) for which there is a horizontal tangent.
2. Let H’(x), the derivative of a continuous function H(x) be given by the graph shown above.
a. Determine the intervals for which H(x) is increasing. Justify your answer.
b. Determine the intervals for which H(x) is decreasing. Justify your answer.
c. Determine the approximate intervals for which H(x) is concave up. Justify your answer.
d. Determine the approximate intervals for which H(x) is concave down. Justify your answer.
e. Determine the x – coordinates of all relative maximum of H(x). Justify your answer.
f. Determine the approximate x – coordinates of all inflection points of H(x). Justify your answer.
g. Determine the x – coordinates of all the points on the curve of H(x) for which there is a horizontal tangent.
3. The figure to the right shows the graph f’, the derivative of a function f.
The domain of the function f is the set of all x such that 3  x  3 .
a. For what values of x, -3 < x < 3, does f have a relative maximum?
A relative minimum? Justify your answers.
b. For what values of x is the graph of f concave up? Justify your answer.
c. Use the information found in parts (a) and (b) and the fact that f(-3) = 0 to sketch a possible graph of f on the axes
provided below.
4. The figure to the right shows the graph of f’, the
derivative of a function f. The domain of f is the set
of all real numbers x such that 10  x  10 .
a. For what values of x does the graph of f have a
horizontal tangent?
b. For what values of x in the interval (-10, 10) does
f have a relative minimum? Justify your answer.
c. For what values of x is the graph of f concave downward?
5. Let f be a function that has domain the closed interval [-1, 4] and range the
closed interval [-1, 2]. Let f(-1) = -1, f(0) = 0, and f(4) = 1. Also let f have the
derivative function f’ that is continuous and that has the graph shown in the figure
to the right.
a. Find all values of x for which f assumes a relative maximum. Justify your answer.
b. Find all values of x for which f assumes its absolute maximum. Justify your answer.
c. Find all intervals on which f is concave downward.
d. Give all values of x for which f has a point of inflection.
e. On the axes provided, sketch the graph of f.
6. The figure to the right shows the graph of f’, the derivative of a
function f. The domain of f is the set of all real numbers such that
-3 < x < 5.
a. For what values of x does f have a relative maximum? Why?
b. For what values of x does f have a relative minimum? Why?
c. On what intervals is the graph of f concave upward? Use f’ to justify your answer.
d. Suppose that f(1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function
f on the open interval 0 < x < 2.