Chapter 15: Chapter 15

Problem 1

Calculate $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}$, and $\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}$ when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=10,000-40 x+20 y $$

Short Answer

Expert verified

The partial derivatives of the function $f(x, y) = 10000 - 40x + 20y$ are $\frac{\partial f}{\partial x} = -40$ and $\frac{\partial f}{\partial y} = 20$. Since they are constants and do not depend on the point, their values at $(1, -1)$ are $\left.\frac{\partial f}{\partial x}\right|_{(1,-1)} = -40$ and $\left.\frac{\partial f}{\partial y}\right|_{(1,-1)} = 20$.

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Finding the partial derivative with respect to x

To find the partial derivative of $f(x, y)$ with respect to $x$, we treat $y$ as a constant and differentiate with respect to $x$. So, \[\frac{\partial f}{\partial x} = \frac{d}{dx}(10000 - 40x + 20y).\]

Step 2. Calculating the partial derivative with respect to x

Differentiating with respect to $x$, we get \[ \frac{\partial f}{\partial x} = -40.\]

Step 3. Finding the partial derivative with respect to y

Similarly, to find the partial derivative of $f(x, y)$ with respect to $y$, we treat $x$ as a constant and differentiate with respect to $y$. So, \[\frac{\partial f}{\partial y} = \frac{d}{dy}(10000 - 40x + 20y).\]

Step 4. Calculating the partial derivative with respect to y

Differentiating with respect to $y$, we get \[ \frac{\partial f}{\partial y} = 20.\]

Step 5. Evaluate the partial derivatives at (1, -1)

Now we need to find the values of the partial derivatives at the given point $(1, -1)$. However, since the partial derivatives we calculated in previous steps are constants and do not depend on $x$ and $y$, their values remain unchanged at the point $(1, -1)$. Thus, \[\left.\frac{\partial f}{\partial x}\right|_{(1,-1)} = \frac{\partial f}{\partial x} = - 40\] and \[\left.\frac{\partial f}{\partial y}\right|_{(1,-1)} = \frac{\partial f}{\partial y} = 20.\]

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Find Study Materials for

Create Study Materials

Select your language

Chapter 15: Chapter 15

Calculate $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}$, and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=10,000-40 x+20 y $$

Short Answer

Step by step solution

Unlock all solutions

Step 1. Finding the partial derivative with respect to x

Step 2. Calculating the partial derivative with respect to x

Step 3. Finding the partial derivative with respect to y

Step 4. Calculating the partial derivative with respect to y

Step 5. Evaluate the partial derivatives at (1, -1)

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Finite Mathematics and Applied Calculus’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 16

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Decision Maths

Probability and Statistics

Pure Maths

Calculus