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Chapter 8: Problem 1

In Exercises 1 through \(20,\) find both first-order partial derivatives. Then evaluate each partial derivative at the indicated point. $$ f(x, y)=x^{2}+y^{2},(1,3) $$

Short Answer

Expert verified
Partial derivatives are 2 at x and 6 at y for (1, 3).

Step by step solution

01

Find the Partial Derivative with respect to x

The function given is \( f(x, y) = x^2 + y^2 \). To find the partial derivative with respect to \( x \), differentiate the function treating \( y \) as a constant. Thus, we have:\[ f_x(x, y) = \frac{\partial}{\partial x} (x^2 + y^2) = 2x. \]
02

Evaluate \( \frac{\partial}{\partial x} f(x, y) \) at the point (1, 3)

Now evaluate the partial derivative with respect to \( x \) at the point \((1, 3)\):\[ f_x(1, 3) = 2 \times 1 = 2. \]
03

Find the Partial Derivative with respect to y

Differentiating the function \( f(x, y) = x^2 + y^2 \) with respect to \( y \) while treating \( x \) as a constant:\[ f_y(x, y) = \frac{\partial}{\partial y} (x^2 + y^2) = 2y. \]
04

Evaluate \( \frac{\partial}{\partial y} f(x, y) \) at the point (1, 3)

Now evaluate the partial derivative with respect to \( y \) at the point \((1, 3)\):\[ f_y(1, 3) = 2 \times 3 = 6. \]
05

Conclusion: Presenting the Results

The first-order partial derivatives of the function \( f(x, y) = x^2 + y^2 \) are \( f_x(x, y) = 2x \) and \( f_y(x, y) = 2y \). Evaluated at the point \( (1, 3) \), we have: - \( f_x(1, 3) = 2 \) - \( f_y(1, 3) = 6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics that studies continuous change. It provides tools to analyze how quantities change with respect to one another. There are two main types of calculus: differential calculus and integral calculus. Differential calculus focuses on the concept of derivative, which measures how a function changes as its input changes. In simple terms, it tells us the rate at which one quantity changes with respect to another. This is crucial for understanding dynamic systems.
  • Derivative: The derivative of a function measures how the function value changes as its input changes. It's like finding the slope of a curve at a specific point.
  • Integral: Integral calculus deals with determining the total size or value, such as the area under a curve.
Differentiation and integration are powerful techniques that let us solve real-world problems in fields like physics, engineering, and economics.
Multivariable Functions
Multivariable functions are functions with more than one input variable. Unlike single-variable functions, which depend on one variable, multivariable functions depend on two or more.For example, the function given in the exercise, \( f(x, y) = x^2 + y^2 \), is a multivariable function because it depends on both \( x \) and \( y \). Such functions are often used to model real-world phenomena where several factors change simultaneously.
  • Inputs: Multivariable functions have more than one input, typically denoted as \( x, y, z, \) etc.
  • Outputs: These functions provide output based on the combination of inputs.
Understanding multivariable functions allows you to explore complex systems, such as temperature changes across a landscape, where temperature might change with both latitude and longitude.
Differentiation
Differentiation is the process of finding the derivative, which tells us how a function's output changes concerning its input. When dealing with functions of more than one variable, we use partial derivatives.
  • Partial Derivatives: These derivatives are called 'partial' because they measure how a function changes as one variable changes, keeping others constant.
  • Notation: The partial derivative of \( f(x, y) \) with respect to \( x \) is denoted \( \frac{\partial f}{\partial x} \).
In our case, to find the partial derivative of \( f(x, y) = x^2 + y^2 \) with respect to \( x \), treat \( y \) as constant, yielding \( 2x \). Similarly, treating \( x \) as constant gives us \( 2y \) when differentiating with respect to \( y \).This technique is essential in analyzing how multivariable systems change and is used in optimization, where you seek to maximize or minimize functions based on certain conditions.

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