1. 58 – First Derivative Graphs Calculator Required
Derivative Graph Investigations
58 – First Derivative Graphs Calculator Required
First Derivative
Slope of the Tangent Line

2. Given the graph of FUNCTION f(x):
Slope of tangent line positive
This is the graph of f(x)
This is the graph of f(x)

3. Given the graph of FUNCTION f(x):
Slope of tangent line negative
This is the graph of f(x)
This is the graph of f(x)

4. Given the function
If f ' (x) is positive:
Slopes of f(x) are positive
f(x) is increasing
If f ' (x) is negative:
Slopes of f(x) are negative
f(x) is decreasing
This is the graph of

5. This is the graph of f(x)

6. At x = 3…..
The graph of f(x) is increasing.
At x = -1…..
The graph of f(x) is decreasing.
This is the graph of f ' (x)

7. On what intervals is the graph
of f(x) increasing? X
On what intervals is the graph
of f(x) decreasing? X X X X X
For what values of x is f ' (x) = 0? -2
, 0
, 1
, 2 X
BONUS QUESTION:
This is the graph of f ' (x)
For what values of x is f " (x) = 0?
-1.2
, 0.4
, 1.5

8. The First Derivative Test For Maximum/Minimum
The solutions to f ' (x) = 0 are
CRITICAL POINTS.
If f ' (x) changes from
positive to negative, a
RELATIVE MAXIMUM exists.
If f ' (x) changes from
negative to postive, a
RELATIVE MINIMUM exists.

9. For what values of x is f ' (x) = 0?-2
, 0
, 1
, 2
The critical points of f(x) are
-2, 0, 1, 2
The relative maxima of f(x) are at
-2 and 1
because f ' (x) changes from
positive to negative
The relative minima of f(x) are at
0 and 2
because f ' (x) changes from
negative to positive
This is the graph of f ' (x)

10. Copy the graph -> X
If the graph represents f(x),
mark with an x the critical
numbers on f(x). X X X X X
If the graph represents f ' (x),
mark with an x the critical
numbers on f(x). X
If the graph represents f(x), estimate to one decimal place the
value(s) of x at which there is a relative maximum on f(x).
-1.4, 0.4
If the graph represents f ' (x), estimate to one decimal place the
value(s) of x at which there is a relative minimum on f(x).
-1.9, 1.8

11. CALCULATOR REQUIRED
a) For what value(s) of x will there be a horizontal tangent on f(x) ? 1
b) For what value(s) of x will the graph of f(x) be increasing?
c) For what value(s) of x will there be a relative minimum on f(x)? 1
d) For what value(s) of x will there be a relative maximum on f(x)?
none

12. The graph is on the interval
If the graph represented is f(x), for what values of x is the first derivative equal to zero?
If the graph represented is f ' (x), for what values of x would the local max(s) and local min(s) of f(x) be?
If the graph represented is f(x), write using interval notation the interval(s) on which the graph of f(x) is increasing.
If the graph represented is f ' (x), write using interval notation the interval(s) on which the graph of f(x) is decreasing.
-1 and 2
-2, 1, 3
(-3, -1), (2, 4)
[-3, -2) U (1, 3)
The graph is on the interval
[-3, 4]

13. The graph is on the interval
If the graph represented is f(x), for what value(s) of x if f ' (x) = 0?
If the graph represented is f ' (x), for what values of x is there a relative minimum on f(x)?
If the graph represented is f(x), write using interval notation the interval(s) on which f ' (x) is positive.
If the graph represented is f ' (x), at what value(s) of x is there a relative maximum on f(x)?
-1, 1
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
-1.5, -0.5, 0.5, 1.5
The graph is on the interval
[-2, 2]

14. This is the graph of f ' (x)
Where are the critical point(s)
of f(x)?
x = 1
x = 3
x = -5
What is f ' (1)?
Where is the ABSOLUTE
maximum of f(x) on [-5, 3]?
This is the graph of f ' (x)
on the interval [-5, 3]
Where is the ABSOLUTE
minimum of f(x) on [-5, 3]?