**Calculus Homework 4** 1. **Problem on Combined Resistance:** \[ \frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2} \] Find the rate at which the combined resistance \( R \) changes with respect to changes in \( r_1 \) given that \( r_2 \) is constant and \( r_1 \) is the variable. *(Hint: Solve for \( R \) and then take the derivative with respect to \( r_1 \).)* 2. **Graph Sketching:** Sketch the graph of a function \( y = f(x) \) that satisfies all the following properties: - \( f(3) = 6 \), \( f'(3) = 0 \), \( f'(8) \) is undefined. - \( \lim_{{x \to -\infty}} f(x) = 0 \), \( \lim_{{x \to \infty}} f(x) = +\infty \). - \( f''(x) > 0 \) for \( x < 1 \) and for \( x > 8 \). - \( f(x) \) is continuous and defined everywhere. 3. **Trigonometric Derivative Using Quotient Rule:** Show the following trigonometric derivative by using the quotient rule. *(Hint: Rewrite cosecant in terms of sine and then use the quotient rule on that.)* \[ \frac{d}{dx} \csc x = -\csc x \cot x \]
**Calculus Homework 4** 1. **Problem on Combined Resistance:** \[ \frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2} \] Find the rate at which the combined resistance \( R \) changes with respect to changes in \( r_1 \) given that \( r_2 \) is constant and \( r_1 \) is the variable. *(Hint: Solve for \( R \) and then take the derivative with respect to \( r_1 \).)* 2. **Graph Sketching:** Sketch the graph of a function \( y = f(x) \) that satisfies all the following properties: - \( f(3) = 6 \), \( f'(3) = 0 \), \( f'(8) \) is undefined. - \( \lim_{{x \to -\infty}} f(x) = 0 \), \( \lim_{{x \to \infty}} f(x) = +\infty \). - \( f''(x) > 0 \) for \( x < 1 \) and for \( x > 8 \). - \( f(x) \) is continuous and defined everywhere. 3. **Trigonometric Derivative Using Quotient Rule:** Show the following trigonometric derivative by using the quotient rule. *(Hint: Rewrite cosecant in terms of sine and then use the quotient rule on that.)* \[ \frac{d}{dx} \csc x = -\csc x \cot x \]
Calculus: Early Transcendentals
8th Edition
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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 Homework 4**
1. **Problem on Combined Resistance:**
\[
\frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2}
\]
Find the rate at which the combined resistance \( R \) changes with respect to changes in \( r_1 \) given that \( r_2 \) is constant and \( r_1 \) is the variable. *(Hint: Solve for \( R \) and then take the derivative with respect to \( r_1 \).)*
2. **Graph Sketching:**
Sketch the graph of a function \( y = f(x) \) that satisfies all the following properties:
- \( f(3) = 6 \), \( f'(3) = 0 \), \( f'(8) \) is undefined.
- \( \lim_{{x \to -\infty}} f(x) = 0 \), \( \lim_{{x \to \infty}} f(x) = +\infty \).
- \( f''(x) > 0 \) for \( x < 1 \) and for \( x > 8 \).
- \( f(x) \) is continuous and defined everywhere.
3. **Trigonometric Derivative Using Quotient Rule:**
Show the following trigonometric derivative by using the quotient rule. *(Hint: Rewrite cosecant in terms of sine and then use the quotient rule on that.)*
\[
\frac{d}{dx} \csc x = -\csc x \cot x
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffcd4407d-2981-4e99-9688-7bc12792f13f%2F9956bd51-581c-4bf9-b411-832621336e3c%2Fd7k1a3.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculus Homework 4**
1. **Problem on Combined Resistance:**
\[
\frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2}
\]
Find the rate at which the combined resistance \( R \) changes with respect to changes in \( r_1 \) given that \( r_2 \) is constant and \( r_1 \) is the variable. *(Hint: Solve for \( R \) and then take the derivative with respect to \( r_1 \).)*
2. **Graph Sketching:**
Sketch the graph of a function \( y = f(x) \) that satisfies all the following properties:
- \( f(3) = 6 \), \( f'(3) = 0 \), \( f'(8) \) is undefined.
- \( \lim_{{x \to -\infty}} f(x) = 0 \), \( \lim_{{x \to \infty}} f(x) = +\infty \).
- \( f''(x) > 0 \) for \( x < 1 \) and for \( x > 8 \).
- \( f(x) \) is continuous and defined everywhere.
3. **Trigonometric Derivative Using Quotient Rule:**
Show the following trigonometric derivative by using the quotient rule. *(Hint: Rewrite cosecant in terms of sine and then use the quotient rule on that.)*
\[
\frac{d}{dx} \csc x = -\csc x \cot x
\]
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