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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how do i solve the attached calculus question?
![**Problem Statement:**
Find an equation of the tangent plane at the given point:
\[ F(r, s) = 2r^4 \frac{1}{s^{0.5}} - 2 \frac{1}{s^3} \]
Given point: \((-2, 1)\)
**Solution:**
The equation for the tangent plane at a point \((r_0, s_0)\) for a function \(z = F(r, s)\) is given by:
\[ z = F(r_0, s_0) + \frac{\partial F}{\partial r}(r_0, s_0)(r - r_0) + \frac{\partial F}{\partial s}(r_0, s_0)(s - s_0) \]
**Calculation Steps:**
1. **Evaluate \(F(r_0, s_0)\) at \((-2, 1)\)**:
Calculate \(F(-2, 1)\) by substituting \(r = -2\) and \(s = 1\) into the function.
2. **Partial Derivatives:**
- Compute \(\frac{\partial F}{\partial r}\) and evaluate it at \((-2, 1)\).
- Compute \(\frac{\partial F}{\partial s}\) and evaluate it at \((-2, 1)\).
3. **Substitute Values:**
Substitute \(F(-2, 1)\), \(\frac{\partial F}{\partial r}(-2, 1)\), and \(\frac{\partial F}{\partial s}(-2, 1)\) into the tangent plane equation.
**Final Formula:**
\[ z = \text{[calculated function value]} + \text{[calculated partial derivative w.r.t. \(r\)]}(r + 2) + \text{[calculated partial derivative w.r.t. \(s\)]}(s - 1) \]
Fill in the box for \(z\) with the derived expression once the calculations are complete.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2Ff1342e27-bde9-49b7-8e70-eedc7c295542%2Fi9d3lek_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find an equation of the tangent plane at the given point:
\[ F(r, s) = 2r^4 \frac{1}{s^{0.5}} - 2 \frac{1}{s^3} \]
Given point: \((-2, 1)\)
**Solution:**
The equation for the tangent plane at a point \((r_0, s_0)\) for a function \(z = F(r, s)\) is given by:
\[ z = F(r_0, s_0) + \frac{\partial F}{\partial r}(r_0, s_0)(r - r_0) + \frac{\partial F}{\partial s}(r_0, s_0)(s - s_0) \]
**Calculation Steps:**
1. **Evaluate \(F(r_0, s_0)\) at \((-2, 1)\)**:
Calculate \(F(-2, 1)\) by substituting \(r = -2\) and \(s = 1\) into the function.
2. **Partial Derivatives:**
- Compute \(\frac{\partial F}{\partial r}\) and evaluate it at \((-2, 1)\).
- Compute \(\frac{\partial F}{\partial s}\) and evaluate it at \((-2, 1)\).
3. **Substitute Values:**
Substitute \(F(-2, 1)\), \(\frac{\partial F}{\partial r}(-2, 1)\), and \(\frac{\partial F}{\partial s}(-2, 1)\) into the tangent plane equation.
**Final Formula:**
\[ z = \text{[calculated function value]} + \text{[calculated partial derivative w.r.t. \(r\)]}(r + 2) + \text{[calculated partial derivative w.r.t. \(s\)]}(s - 1) \]
Fill in the box for \(z\) with the derived expression once the calculations are complete.
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