-5 10 ↑y 1 f(x) g(x) 1. Given f (x) and g(x) are both piecewise functions as shown in the graph above, if h (x) = f (x) · g(x) find h' (1). 710
-5 10 ↑y 1 f(x) g(x) 1. Given f (x) and g(x) are both piecewise functions as shown in the graph above, if h (x) = f (x) · g(x) find h' (1). 710
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I need an explanation. On how they found the points on the graph, and also why they found h'(-1) even though h'(1) was asked to be found.
![### Problem:
Given \( f(x) \) and \( g(x) \) are both piecewise functions as shown in the graph above, if \( h(x) = f(x) \cdot g(x) \), find \( h'(1) \).
### Solution:
1. Evaluate \( f(-1) \):
\[
f(-1) = -8
\]
2. Find \( f'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)):
\[
f'(-1) = 3
\]
3. Evaluate \( g(-1) \):
\[
g(-1) = 4
\]
4. Find \( g'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)):
\[
g'(-1) = -1
\]
5. Use the product rule to find \( h'(-1) \):
\[
h'(-1) = f(-1) \cdot g'(-1) + g(-1) \cdot f'(-1)
\]
6. Substitute the known values:
\[
h'(-1) = (-8) \cdot (-1) + (4) \cdot (3)
\]
7. Calculate \( h'(-1) \):
\[
h'(-1) = 20
\]
### Graph Explanation:
The graph displays two piecewise functions, \( f(x) \) in red and \( g(x) \) in blue, plotted on a Cartesian plane. The x-axis ranges from -7 to 10, and the y-axis ranges from -10 to 10. Points of intersection and changes in slopes are critical for evaluating values and derivatives at specific points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7353de8a-3834-465a-af3b-558129de7e04%2Fd88c8a05-d104-41ef-944f-09c1bbb48e29%2F5ywzar_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem:
Given \( f(x) \) and \( g(x) \) are both piecewise functions as shown in the graph above, if \( h(x) = f(x) \cdot g(x) \), find \( h'(1) \).
### Solution:
1. Evaluate \( f(-1) \):
\[
f(-1) = -8
\]
2. Find \( f'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)):
\[
f'(-1) = 3
\]
3. Evaluate \( g(-1) \):
\[
g(-1) = 4
\]
4. Find \( g'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)):
\[
g'(-1) = -1
\]
5. Use the product rule to find \( h'(-1) \):
\[
h'(-1) = f(-1) \cdot g'(-1) + g(-1) \cdot f'(-1)
\]
6. Substitute the known values:
\[
h'(-1) = (-8) \cdot (-1) + (4) \cdot (3)
\]
7. Calculate \( h'(-1) \):
\[
h'(-1) = 20
\]
### Graph Explanation:
The graph displays two piecewise functions, \( f(x) \) in red and \( g(x) \) in blue, plotted on a Cartesian plane. The x-axis ranges from -7 to 10, and the y-axis ranges from -10 to 10. Points of intersection and changes in slopes are critical for evaluating values and derivatives at specific points.
Expert Solution

Step 1: Given the information
The graph of the functions are given.
The aim is to find the derivative value where
.
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