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 Problem: Estimating the Derivative**
**Problem Statement:**
Estimate \( g'(3) \) of the function \( g(t) = \frac{1}{t^2} \).
**Instructions:**
Please enter your answer as a fraction.
---
For an educational website, you can follow this structure to guide students in solving this problem.
1. **Detailed Explanation of the Problem:**
The given problem asks for the derivative of the function \( g(t) = \frac{1}{t^2} \) evaluated at \( t = 3 \).
2. **Solution Steps:**
- **Step 1:** Find the general derivative \( g'(t) \) of the function \( g(t) \).
The function \( g(t) = \frac{1}{t^2} \) can be rewritten as \( g(t) = t^{-2} \). Using the power rule for differentiation, which states \( \frac{d}{dt}[t^n] = n \cdot t^{n-1} \):
\[
g'(t) = \frac{d}{dt}[t^{-2}] = -2t^{-3} = -\frac{2}{t^3}
\]
- **Step 2:** Evaluate the derivative at \( t = 3 \).
Substitute \( t = 3 \) into the derivative \( g'(t) \):
\[
g'(3) = -\frac{2}{3^3} = -\frac{2}{27}
\]
3. **Final Answer:**
The estimated value of \( g'(3) \) is:
\[
g'(3) = -\frac{2}{27}
\]
**Note to Students:**
Remember to always express your final answer in the required form, in this case, as a fraction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05505020-2279-400a-8a18-a0a1ff0eea2a%2F4425c0d0-f3df-4e39-b563-0664a9469a83%2Fvlmik47_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculus Problem: Estimating the Derivative**
**Problem Statement:**
Estimate \( g'(3) \) of the function \( g(t) = \frac{1}{t^2} \).
**Instructions:**
Please enter your answer as a fraction.
---
For an educational website, you can follow this structure to guide students in solving this problem.
1. **Detailed Explanation of the Problem:**
The given problem asks for the derivative of the function \( g(t) = \frac{1}{t^2} \) evaluated at \( t = 3 \).
2. **Solution Steps:**
- **Step 1:** Find the general derivative \( g'(t) \) of the function \( g(t) \).
The function \( g(t) = \frac{1}{t^2} \) can be rewritten as \( g(t) = t^{-2} \). Using the power rule for differentiation, which states \( \frac{d}{dt}[t^n] = n \cdot t^{n-1} \):
\[
g'(t) = \frac{d}{dt}[t^{-2}] = -2t^{-3} = -\frac{2}{t^3}
\]
- **Step 2:** Evaluate the derivative at \( t = 3 \).
Substitute \( t = 3 \) into the derivative \( g'(t) \):
\[
g'(3) = -\frac{2}{3^3} = -\frac{2}{27}
\]
3. **Final Answer:**
The estimated value of \( g'(3) \) is:
\[
g'(3) = -\frac{2}{27}
\]
**Note to Students:**
Remember to always express your final answer in the required form, in this case, as a fraction.
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