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Chapter 13: Problem 1

Suppose \(z=f(x, y),\) where \(x\) and \(y\) are functions of \(t .\) How many dependent, intermediate, and independent variables are there?

Short Answer

Expert verified
Answer: In the given function, there is 1 dependent variable (z), 2 intermediate variables (x and y), and 1 independent variable (t).

Step by step solution

01

Identify the dependent variable

A dependent variable is an output or response variable that relies on other variables. In our case, \(z\) depends on \(x\) and \(y\). So, \(z\) is the dependent variable.
02

Identify the intermediate variables

Intermediate variables are those that depend on the independent variable(s) but also affect the dependent variable. In our case, \(x\) and \(y\) are both functions of \(t\) and affect \(z\). Therefore, \(x\) and \(y\) are intermediate variables.
03

Identify the independent variable

An independent variable is one that can be manipulated without being affected by any other variable. In our case, \(t\) is the only variable that is not affected by any other variable, and both \(x\) and \(y\) depend on it. Thus, \(t\) is the independent variable.
04

Count the number of dependent, intermediate, and independent variables

Based on our analysis, there is: - 1 dependent variable (\(z\)), - 2 intermediate variables (\(x\) and \(y\)), and - 1 independent variable (\(t\)).

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Most popular questions from this chapter

Use the Second Derivative Test to prove that if \((a, b)\) is a critical point of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\).

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=\ln \left(1+x^{2}+y^{2}\right)$$

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)
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