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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![### Calculus Practice Problems
1. **Finding the Derivative:**
Given the function \( f'(x) = -2x \):
\[
\text{Find } f(x)
\]
(Solution noted: \( \frac{-x^2}{2} \))
2. **Differentiation Problem:**
Given:
\[
y = 2e^{2x}
\]
\[
\text{Find } y'
\]
(Solution noted: \( y' = 4e^{2x} \))
3. **Evaluation with Given Conditions:**
Given:
\[
f'(9) = 3 \text{ and } g'(9) = 3
\]
Evaluate:
\[
g'(a) = 3
\text{ and }
f'(a) = 3
\]
4. **True or False Statement:**
Determine the truth value of the following statement:
\[
F(x) = f(g(x)), \quad g(a)=3, \quad f(g(a))=3, \quad \text{and} \, f'(2)=-1
\]
5. **Area Calculation Problem:**
Find the area of the region bounded by the curve \( y = \sin{(x)} \) and the x-axis between \( x = 0 \) and \( x = 4 \).
(Solution noted: 4 square units)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2fc8d61-50af-4e4e-bb29-e3320480694b%2F4e4e9cb7-6eec-41af-a7ca-6bfb7e8f41db%2Fxjr40al_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Practice Problems
1. **Finding the Derivative:**
Given the function \( f'(x) = -2x \):
\[
\text{Find } f(x)
\]
(Solution noted: \( \frac{-x^2}{2} \))
2. **Differentiation Problem:**
Given:
\[
y = 2e^{2x}
\]
\[
\text{Find } y'
\]
(Solution noted: \( y' = 4e^{2x} \))
3. **Evaluation with Given Conditions:**
Given:
\[
f'(9) = 3 \text{ and } g'(9) = 3
\]
Evaluate:
\[
g'(a) = 3
\text{ and }
f'(a) = 3
\]
4. **True or False Statement:**
Determine the truth value of the following statement:
\[
F(x) = f(g(x)), \quad g(a)=3, \quad f(g(a))=3, \quad \text{and} \, f'(2)=-1
\]
5. **Area Calculation Problem:**
Find the area of the region bounded by the curve \( y = \sin{(x)} \) and the x-axis between \( x = 0 \) and \( x = 4 \).
(Solution noted: 4 square units)
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