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Chapter 13: Problem 18

These exercises are concerned with functions of four or more variables. $$ \begin{array}{l}{\text { (a) Let } f(u, v, \lambda, \phi)=e^{u+v} \cos \lambda \tan \phi} \\ {\text { Find } f(-2,2,0, \pi / 4)} \\ {\text { (b) Let } f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}} \\ {\quad \text { Find } f(1,2, \ldots, n)}\end{array} $$

Short Answer

Expert verified
(a) 1; (b) \(\frac{n(n+1)(2n+1)}{6}\).

Step by step solution

01

Understand the Function for Part (a)

The function given is \( f(u, v, \lambda, \phi) = e^{u+v} \cos \lambda \tan \phi \). We need to find the value of this function at the specific point \((-2, 2, 0, \pi/4)\).
02

Substitute Values for Part (a)

Substitute \( u = -2 \), \( v = 2 \), \( \lambda = 0 \), and \( \phi = \pi/4 \) into the function: \( f(-2, 2, 0, \pi/4) = e^{-2+2} \cos(0) \tan(\pi/4) \).
03

Simplify the Expression for Part (a)

Simplify each component of the expression: - \( e^{-2+2} = e^0 = 1 \)- \( \cos(0) = 1 \)- \( \tan(\pi/4) = 1 \)Thus, the expression becomes \( 1 \times 1 \times 1 = 1 \). So, \( f(-2, 2, 0, \pi/4) = 1 \).
04

Understand the Function for Part (b)

The function given is \( f(x_1, x_2, \ldots, x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 \). We need to find the value of this function for the sequence \( (1, 2, \ldots, n) \).
05

Substitute Values for Part (b)

Substitute the sequence \( (1, 2, \ldots, n) \) into the function. The function becomes \( 1^2 + 2^2 + \cdots + n^2 \).
06

Simplify Using the Sum of Squares Formula

The sum of the squares of the first \( n \) natural numbers is calculated using the formula \( 1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} \). Substitute to get the result for any \( n \).
07

Write Results for Part (a) and Part (b)

For part (a), the value is \( f(-2, 2, 0, \pi/4) = 1 \). For part (b), the expression simplifies to \( f(1, 2, \ldots, n) = \frac{n(n+1)(2n+1)}{6} \).

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Key Concepts

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

Functions of Multiple Variables
In multivariable calculus, functions can depend on more than one variable. Unlike single-variable functions which contain an input and an output, multivariable functions can take several inputs.
For instance, the function in Part (a) of our exercise is \( f(u, v, \lambda, \phi) = e^{u+v} \cos \lambda \tan \phi \), which depends on four variables: \( u, v, \lambda, \phi \). This allows us to model more complex systems where outcomes depend on multiple factors.
  • Each variable can be manipulated independently and can represent different dimensions or units.
  • Functions are evaluated by substituting specific values for each variable and simplifying.
This multiplicity provides the flexibility needed in fields such as physics, engineering, and economics to describe models that involve multiple interacting quantities.
Mathematical Substitution
Mathematical substitution is a technique used to simplify the evaluation of functions. It involves replacing variable names with given values.
In Part (a), for the function \( f(u, v, \lambda, \phi) \), substitution occurs when you insert the given values to find \( f(-2, 2, 0, \pi/4) \). This means replacing:
  • \( u = -2 \)
  • \( v = 2 \)
  • \( \lambda = 0 \)
  • \( \phi = \pi/4 \)
After substituting, the task is to compute these expressions:
  • \( e^{0} = 1 \) since \( e^{-2+2} = e^0 \)
  • \( \cos(0) = 1 \)
  • \( \tan(\pi/4) = 1 \)
The end result of this substitution and simplification is a concrete numerical output for a specific point.
Sum of Squares Formula
The sum of squares formula is an elegant mathematical tool that simplifies the summation of squared numbers. It is especially useful in situations involving sequences.
For instance, Part (b) of our exercise covers a function \( f(x_1, x_2, \ldots, x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 \). Instead of squaring each number individually, we use the formula:
\[ 1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} \]Where:
  • \( n \) is the number of terms in the sequence.
  • The formula itself provides a shortcut that saves time and reduces calculation errors.
By substituting any integer \( n \) into this formula, you get the sum of squares quickly. This technique is common in statistics and analysis, where managing large data sets is necessary.

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Most popular questions from this chapter

Let \(f\) denote a differentiable function of two variables. Although we have defined what it means to say that \(f\) is differentiable, we have not defined the "derivative" of \(f\) Write a short paragraph that discusses the merits of defining the derivative of \(f\) to be the gradient \(\nabla f\).

Let \(\alpha, \beta,\) and \(\gamma\) be the angles of a triangle. (a) Use Lagrange multipliers to find the maximum value of \(f(\alpha, \beta, \gamma)=\cos \alpha \cos \beta \cos \gamma,\) and determine the angles for which the maximum occurs. (b) Express \(f(\alpha, \beta, \gamma)\) as a function of \(\alpha\) and \(\beta\) alone, and use a CAS to graph this function of two variables. Confirm that the result obtained in part (a) is consistent with the graph.

The area \(A\) of a triangle is given by \(A=\frac{1}{2} a b \sin \theta,\) where \(a\) and \(b\) are the lengths of two sides and \(\theta\) is the angle between these sides. Suppose that \(a=5, b=10,\) and \(\theta=\pi / 3\). (a) Find the rate at which \(A\) changes with respect to \(a\) if \(b\) and \(\theta\) are held constant. (b) Find the rate at which \(A\) changes with respect to \(\theta\) if \(a\) and \(b\) are held constant. (c) Find the rate at which \(b\) changes with respect to \(a\) if \(A\) and \(\theta\) are held constant.

Let \(w=(4 x-3 y+2 z)^{5} .\) Find $$ \begin{array}{lll}{\text { (a) } \frac{\partial^{2} w}{\partial x \partial z}} & {\text { (b) } \frac{\partial^{3} w}{\partial x \partial y \partial z}} & {\text { (c) } \frac{\partial^{4} w}{\partial z^{2} \partial y \partial x}}\end{array} $$

A company manager wants to establish a relationship between the sales of a certain product and the price. The company research department provides the following data: $$ \begin{array}{|l|c|c|c|c|c|}\hline \text { PRICE (x) IN DOLLARS } & {\$ 35.00} & {\$ 40.00} & {\$ 45.00} & {\$ 48.00} & {\$ 50.00} \\ \hline \text { DAILY sALES voLUME (y) } & {80} & {75} & {68} & {66} & {63} \\ \hline\end{array} $$ (a) Use a calculating utility to find the regression line of \(y\) as a function of \(x\) (b) Use a graphing utility to make a graph that shows the data points and the regression line. (c) Use the regression line to make a conjecture about the number of units that would be sold at a price of \(\$ 60.00\).
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