Section 7.5
Antiderivatives - Graphical/Numerical
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
What does the graph of the derivative tell us about the function?
Recall: Section 2.2
To the right is a graph of f ‘(x).
• Where is f (x) increasing?
• 0<𝑥<4
• Where is f (x) decreasing?
• 𝑥 < 0 and 4 < 𝑥
• Suppose f (0) = 0.
Sketch a graph of f (x).
Antiderivatives
The function F(x) is an antiderivative of f(x) if F ′(x) = f(x).
Example: if F’(x)=3, find 17 different antiderivatives, F(x).
But, if I tell you that F(0)=3….?
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Exercise 1
𝑥
𝑓
0
Suppose g x =
𝑡 𝑑𝑡 where 𝑓(𝑡)
is given by the graph.
• At what values of x do the local maximum
and minimum values of 𝑔(𝑥) occur?
• Max at 1.6, 4.7, & 7.9
• Min at 0, 3.2, 6.2, & 8
• At what value of x does 𝑔(𝑥) attain its
absolute maximum value?
• 1.6
• On what intervals is 𝑔(𝑥) concave
downward?
• Where 𝑔′′ 𝑥 = 𝑓 ′ 𝑥 < 0
• (0, 2.2) ∪ (3.8, 5.4) ∪ (7, 8)
• Is 𝑔(𝑥) positive or negative? Explain.
• positive
• On the same axes as above, sketch a
reasonable looking graph of 𝑔(𝑥).
Exercise 2
𝑥
𝑓
0
Suppose g x =
𝑡 𝑑𝑡 where 𝑓(𝑡)
is given by the graph.
• Complete the table.
0
1
2
3
4
5
6
7
0
2
5
57
6.5
5
3
2
• On what interval is g(𝑥) increasing?
• (0, 3)
• Where does g(𝑥) have a maximum value?
• 𝑥=3
Matching a function with its antiderivative
Which of the following graphs (a)-(d) could represent an antiderivative of the function shown in Figure 7.2?
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Matching a function with its antiderivative
Which of the following graphs (a)-(d) could represent an antiderivative of the function shown in Figure 7.3?
What does the graph of the derivative tell us about the function?
The figure below shows a graph of y = f(x) with some areas labeled. Assume Fʹ(x) = f (x) and
F(0) = 10. Then F(5) =
F(5) = 10 + 7 – 6 = 11
What does the graph of the derivative tell us about the function?
The figure below shows a graph of y = Fʹ(x). Where does F(x) have a local maximum? A local
minimum?
F has local maxima at 2 and 8. F has local minima at 0 and 6.