7. 7. 6. y= g(x) 4. 3. 4. 1. 12345678 Ol 1 234 567 8 The graphs of the piecewise linear functions f and g are shown above. If the function h is defined by h (1) = f (1) · g (x). then h' (2) i- 14 nonexistent 20
7. 7. 6. y= g(x) 4. 3. 4. 1. 12345678 Ol 1 234 567 8 The graphs of the piecewise linear functions f and g are shown above. If the function h is defined by h (1) = f (1) · g (x). then h' (2) i- 14 nonexistent 20
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![The eight graphs of the piecewise linear functions f and g are shown above. If the function \( h \) is defined by \( h(x) = f(x) \cdot g(x) \), then \( h'(2) \) is:
- 14
- nonexistent
- 20
### Explanation of the Graphs
The first graph represents the function \( f(x) \). Here are the key points:
- For \( x = 0 \), \( f(x) = 1 \)
- For \( x = 1 \), \( f(x) = 5 \)
- For \( x = 2 \), \( f(x) = 7 \)
- For \( x = 5 \), \( f(x) = 1 \)
The second graph depicts the function \( g(x) \). Key points include:
- For \( x = 0 \), \( g(x) = 7 \)
- For \( x = 2 \), \( g(x) = 5 \)
- For \( x = 3 \), \( g(x) = 6 \)
- For \( x = 5 \), \( g(x) = 1 \)
- For \( x = 7 \), \( g(x) = 0.5 \)
### Calculation of \( h'(2) \)
Given \( h(x) = f(x) \cdot g(x) \), \( h'(x) \) can be found using the product rule:
\[ h'(x) = f'(x) g(x) + f(x) g'(x) \]
At \( x = 2 \):
- From the graph of \( f(x) \), \( f(2) = 7 \) and evaluating the slope on the interval \( [1, 2] \), \( f'(2) = 2 \)
- From the graph of \( g(x) \), \( g(2) = 5 \) and evaluating the slope on the interval \( [2, 3] \), \( g'(2) = -2 \))
Using the product rule:
\[ h'(2) = f'(2) \cdot g(2) + f(2) \cdot g'(2) \]
\[ h'(2) = 2 \cdot 5 + 7 \cdot (-2) \]
\[ h'(2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4143f45a-5c96-4449-af2f-abda697f3e91%2F123ffe3d-d190-4056-a6f1-c23ed51a55bc%2F156z27l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The eight graphs of the piecewise linear functions f and g are shown above. If the function \( h \) is defined by \( h(x) = f(x) \cdot g(x) \), then \( h'(2) \) is:
- 14
- nonexistent
- 20
### Explanation of the Graphs
The first graph represents the function \( f(x) \). Here are the key points:
- For \( x = 0 \), \( f(x) = 1 \)
- For \( x = 1 \), \( f(x) = 5 \)
- For \( x = 2 \), \( f(x) = 7 \)
- For \( x = 5 \), \( f(x) = 1 \)
The second graph depicts the function \( g(x) \). Key points include:
- For \( x = 0 \), \( g(x) = 7 \)
- For \( x = 2 \), \( g(x) = 5 \)
- For \( x = 3 \), \( g(x) = 6 \)
- For \( x = 5 \), \( g(x) = 1 \)
- For \( x = 7 \), \( g(x) = 0.5 \)
### Calculation of \( h'(2) \)
Given \( h(x) = f(x) \cdot g(x) \), \( h'(x) \) can be found using the product rule:
\[ h'(x) = f'(x) g(x) + f(x) g'(x) \]
At \( x = 2 \):
- From the graph of \( f(x) \), \( f(2) = 7 \) and evaluating the slope on the interval \( [1, 2] \), \( f'(2) = 2 \)
- From the graph of \( g(x) \), \( g(2) = 5 \) and evaluating the slope on the interval \( [2, 3] \), \( g'(2) = -2 \))
Using the product rule:
\[ h'(2) = f'(2) \cdot g(2) + f(2) \cdot g'(2) \]
\[ h'(2) = 2 \cdot 5 + 7 \cdot (-2) \]
\[ h'(2)
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