Use the information about values of the functions and their derivatives to calculate G'(2), where G(x) = x - g(x). f(x). f(2) 8 f'(2) -12 g(2) 14 g' (2) -5 (Use symbolic notation and fractions where needed.) G' (2) = -5 Incorrect
Use the information about values of the functions and their derivatives to calculate G'(2), where G(x) = x - g(x). f(x). f(2) 8 f'(2) -12 g(2) 14 g' (2) -5 (Use symbolic notation and fractions where needed.) G' (2) = -5 Incorrect
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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![Use the information about values of the functions and their derivatives to calculate \( G'(2) \), where \( G(x) = x \cdot g(x) \cdot f(x) \).
\[
\begin{array}{|c|c|c|}
\hline
f(2) & f'(2) & g(2) & g'(2) \\ \hline
8 & -12 & 14 & -5 \\ \hline
\end{array}
\]
(Use symbolic notation and fractions where needed.)
\[
G'(2) = -5 \quad\text{(In the red box)}
\]
Incorrect
### Explanation of the Problem
The table provided gives values for the functions \( f(x) \) and \( g(x) \) and their derivatives at \( x = 2 \). Use these values to find \( G'(2) \) for the function \( G(x) = x \cdot g(x) \cdot f(x) \).
### Solution Steps
1. **Define the Function \( G(x) \):**
\( G(x) = x \cdot g(x) \cdot f(x) \)
2. **Apply the Product Rule:**
To find \( G'(x) \), apply the product rule: \( (u \cdot v \cdot w)' = u' \cdot v \cdot w + u \cdot v' \cdot w + u \cdot v \cdot w' \), where \( u = x \), \( v = g(x) \), and \( w = f(x) \).
3. **Differentiate Middle Functions:**
- \( u = x \rightarrow u' = 1 \)
- \( v = g(x) \rightarrow v' = g'(x) \)
- \( w = f(x) \rightarrow w' = f'(x) \)
So,
\[
G'(x) = (x \cdot g(x) \cdot f(x))' = 1 \cdot g(x) \cdot f(x) + x \cdot g'(x) \cdot f(x) + x \cdot g(x) \cdot f'(x)
\]
4. **Substitute Given Values:**
- \( f(2) = 8](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F575d721b-26eb-4db6-af93-50c207cd3fab%2F714d76cc-4a98-4326-9e7b-3c003e995dd5%2Frtya75_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use the information about values of the functions and their derivatives to calculate \( G'(2) \), where \( G(x) = x \cdot g(x) \cdot f(x) \).
\[
\begin{array}{|c|c|c|}
\hline
f(2) & f'(2) & g(2) & g'(2) \\ \hline
8 & -12 & 14 & -5 \\ \hline
\end{array}
\]
(Use symbolic notation and fractions where needed.)
\[
G'(2) = -5 \quad\text{(In the red box)}
\]
Incorrect
### Explanation of the Problem
The table provided gives values for the functions \( f(x) \) and \( g(x) \) and their derivatives at \( x = 2 \). Use these values to find \( G'(2) \) for the function \( G(x) = x \cdot g(x) \cdot f(x) \).
### Solution Steps
1. **Define the Function \( G(x) \):**
\( G(x) = x \cdot g(x) \cdot f(x) \)
2. **Apply the Product Rule:**
To find \( G'(x) \), apply the product rule: \( (u \cdot v \cdot w)' = u' \cdot v \cdot w + u \cdot v' \cdot w + u \cdot v \cdot w' \), where \( u = x \), \( v = g(x) \), and \( w = f(x) \).
3. **Differentiate Middle Functions:**
- \( u = x \rightarrow u' = 1 \)
- \( v = g(x) \rightarrow v' = g'(x) \)
- \( w = f(x) \rightarrow w' = f'(x) \)
So,
\[
G'(x) = (x \cdot g(x) \cdot f(x))' = 1 \cdot g(x) \cdot f(x) + x \cdot g'(x) \cdot f(x) + x \cdot g(x) \cdot f'(x)
\]
4. **Substitute Given Values:**
- \( f(2) = 8
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