Name:
Date:
Block:
Interpreting the Graph of a Derivative
y  f ' ( x)
y
x
-3
-1
1
3
NOTE: This is the graph of the derivative of f, NOT the graph of f.
The figure above 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  4  x  4 .
Use the graph of the derivative, f ' ( x ) , above to give the values requested below for the
graph of the function, f (x) .
Relative maximums:
Relative minimums:
Increasing intervals:
Decreasing intervals:
Inflection points:
Concave upward intervals:
Concave downward intervals:
Sketch a possible graph for f (x) .
Sketch a possible graph of f "( x) .
y
y
x
-4
-2
2
4
x
-4
-2
2
4
1
Fill in the blank to correctly complete each statement.
1. Let f be defined at c if _____________ or ____________, then c is a critical number of f.
2. Rolle’s Theorem: Let f be continuous on the closed interval [a,b] and differentiable on the open interval
(a,b). If f(a) = f(b) then their is at least one number c in (a,b) such that __________.
3. The Mean Value Theorem: If f is continuous on the closed interval [a,b] and differentiable on the open
interval (a,b), then there exists a number c in (a,b) such that .....
4. If _____________ for all x in (a,b), then f is increasing on [a,b].
5. If _____________ for all x in (a,b), then f is decreasing on [a,b]
6. Let f be differentiable on an open interval I. The graph of f is _____________ on I if f’ is positive and
______________ if f’ is negative.
7. If _____________ for all x in an open interval I, then the graph of f is concave upward in I.
8. If ____________ for all x in an open interval I, then the graph of f is concave downward in I.
9. If (c,f(c)) is a point of inflection of the graph of f, then either ___________ or__________________.
10. Second Derivative Test: Let f be a function such that f’(c) = 0 and the second derivative of f exists on an
open interval containing c.
If f”(c) > 0 , then f(c) is a _______________.
If f”(c) < 0 , then f(c) is a _______________.
11. If the graph of f is concave upward at (c,f(c)), the graph of f lies __________ the tangent
line at (c, f(c)) on some open interval containing c.
2
12. Determine if the Mean Value Theorem applies for the function f ( x) 
2 x 2  3x  1
over the interval [2,3] .
x 1
If it does not apply, state why not. If it does apply, find the value of c that satisfies the theorem.
13. Find all open intervals on which the function f ( x ) 
x2
is increasing and the intervals on which it is
x3
decreasing. Are there any points of inflection? If so, state the points of inflection.
14. Find the equation of the line tangent to the function f ( x)  x3  x  5 at its point of inflection.
15. Find any relative extrema for the function f ( x)  x3 
x2
 2 x . Justify your response using the second
2
derivative test.
16. Use calculus to graph f ( x)  2x  sin x . Use a calculator only when absolutely necessary.
17. Use calculus to graph g ( x) 
x2  2 x  4
.
x2
18. Let f ( x)  4x3  2x and let f ( x) have critical numbers –1, 0, and 1. Use the Second Derivative Test
to determine which critical numbers, if any, give a relative maximum.
3t 3  2t 2
19. A particle moves along the x-axis in such a way that its position at time t is given by x (t )  2
.
t

1
What is the acceleration of the particle at time t  0 ?
3
20. The graph of the derivative of f is shown below. Use the graph of f ' to answer the following questions
about the graph of f . Justify each answer using f '
f ( x )
a. Where are the critical numbers f ( x) ?
b. Give the intervals where f ( x) is increasing.
c. Give the intervals where f ( x) is decreasing.
d. Where does f ( x) have a relative maximum?
e. Where does f ( x) have a relative minimum?
f. Give the intervals where f ( x) is concave up.
g. Give the intervals where f ( x) is concave down.
h. Where are the inflection points for f ( x) ?
i. Based on your answers above, draw a possible graph on the same grid for f ( x) .
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