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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![**Problem: Finding the Derivative**
Determine \( f'(t) \) if \( f(t) = -2t^2 - 8t + 7 \).
---
**Solution:**
To find the derivative \( f'(t) \), apply the power rule to each term of \( f(t) \).
1. **Derivative of \(-2t^2\):**
- The derivative of \( t^n \) is \( n \cdot t^{n-1} \).
- Apply this to \(-2t^2\):
\[
-2 \cdot 2t^{2-1} = -4t
\]
2. **Derivative of \(-8t\):**
- The derivative of \( t \) is 1.
- Apply to \(-8t\):
\[
-8 \cdot 1 = -8
\]
3. **Derivative of \(7\):**
- The derivative of a constant is 0.
**Result:**
Combine the derivatives:
\[
f'(t) = -4t - 8
\]
The derivative \( f'(t) = -4t - 8 \) represents the slope of the tangent line to the curve \( f(t) \) at any point \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfe1180e-b38d-4ac2-b4f3-29ed5eba8d3b%2Fb3301a93-1449-4794-a870-8afe14171aa7%2F8u8zhjp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Finding the Derivative**
Determine \( f'(t) \) if \( f(t) = -2t^2 - 8t + 7 \).
---
**Solution:**
To find the derivative \( f'(t) \), apply the power rule to each term of \( f(t) \).
1. **Derivative of \(-2t^2\):**
- The derivative of \( t^n \) is \( n \cdot t^{n-1} \).
- Apply this to \(-2t^2\):
\[
-2 \cdot 2t^{2-1} = -4t
\]
2. **Derivative of \(-8t\):**
- The derivative of \( t \) is 1.
- Apply to \(-8t\):
\[
-8 \cdot 1 = -8
\]
3. **Derivative of \(7\):**
- The derivative of a constant is 0.
**Result:**
Combine the derivatives:
\[
f'(t) = -4t - 8
\]
The derivative \( f'(t) = -4t - 8 \) represents the slope of the tangent line to the curve \( f(t) \) at any point \( t \).
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