Activity 5.1.2. Suppose that the function y = f(x) is given by the graph shown in Figure 5.1.2, and that the pieces of ƒ are either portions of lines or portions of circles. In addition, let ♬ be an antiderivative of f and say that F(0) = −1. Finally, assume that for x ≤ 0 and x ≥ 7, ƒ(x) = y = f(x) निजी 2 3 4 5 1- -1- 6 7 Figure 5.1.2. At left, the graph of y = f(x). a. On what interval(s) is F an increasing function? On what intervals is F decreasing? b. On what interval(s) is F concave up? concave down? neither? c. At what point(s) does ♬ have a relative minimum? a relative maximum? d. Use the given information to determine the exact value of F(x) for x = 1, 2, ..., 7. In addition, what are the values of F(−1) and F(8)? e. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of y = F(x) on the axes provided, being sure to indicate the behavior of F for x < 0 and x > 7. Clearly indicate the scale on the vertical and horizontal axes of your graph. f. What happens if we change one key piece of information: in particular, say that G is an antiderivative of f and G(0) = 0. How (if at all) would your answers to the preceding questions change? Sketch a graph of G on the same axes as the graph of F you constructed in (e).
Activity 5.1.2. Suppose that the function y = f(x) is given by the graph shown in Figure 5.1.2, and that the pieces of ƒ are either portions of lines or portions of circles. In addition, let ♬ be an antiderivative of f and say that F(0) = −1. Finally, assume that for x ≤ 0 and x ≥ 7, ƒ(x) = y = f(x) निजी 2 3 4 5 1- -1- 6 7 Figure 5.1.2. At left, the graph of y = f(x). a. On what interval(s) is F an increasing function? On what intervals is F decreasing? b. On what interval(s) is F concave up? concave down? neither? c. At what point(s) does ♬ have a relative minimum? a relative maximum? d. Use the given information to determine the exact value of F(x) for x = 1, 2, ..., 7. In addition, what are the values of F(−1) and F(8)? e. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of y = F(x) on the axes provided, being sure to indicate the behavior of F for x < 0 and x > 7. Clearly indicate the scale on the vertical and horizontal axes of your graph. f. What happens if we change one key piece of information: in particular, say that G is an antiderivative of f and G(0) = 0. How (if at all) would your answers to the preceding questions change? Sketch a graph of G on the same axes as the graph of F you constructed in (e).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Only need help with parts e and f please. Thank you!
Transcribed Image Text:Activity 5.1.2. Suppose that the function y = f(x) is given by the graph
shown in Figure 5.1.2, and that the pieces of f are either portions of lines or
portions of circles. In addition, let F be an antiderivative of f and say that
F(0) = -1. Finally, assume that for x ≤ 0 and x ≥ 7, ƒ(x) = 0.
1
-1+
y = f(x)
2
3
4
5
6
7
Figure 5.1.2. At left, the graph of y = f(x).
a. On what interval(s) is F an increasing function? On what intervals is F
decreasing?
b. On what interval(s) is F concave up? concave down? neither?
c. At what point(s) does F have a relative minimum? a relative maximum?
d. Use the given information to determine the exact value of F(x) for
x = 1, 2, ‚ 7. In addition, what are the values of F(−1) and F(8)?
..
"...
e. Based on your responses to all of the preceding questions, sketch a
complete and accurate graph of y = F(x) on the axes provided, being
sure to indicate the behavior of F for x < 0 and x > 7. Clearly indicate
the scale on the vertical and horizontal axes of your graph.
f. What happens if we change one key piece of information: in particular,
say that G is an antiderivative of ƒ and G(0) = 0. How (if at all) would
your answers to the preceding questions change? Sketch a graph of G on
the same axes as the graph of F you constructed in (e).
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