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# 6.1E: Exercises

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## Exercise $$\PageIndex{1}$$ functional value

For the following exercises, evaluate each function at the indicated values.

1) $$\displaystyle W(x,y)=4x^2+y^2.$$ Find $$\displaystyle W(2,−1), W(−3,6).$$

Solution:$$\displaystyle 17,72$$

2) $$\displaystyle W(x,y)=4x^2+y^2$$. Find $$\displaystyle W(2+h,3+h).$$

3) The volume of a right circular cylinder is calculated by a function of two variables, $$\displaystyle V(x,y)=πx^2y,$$ where $$\displaystyle x$$ is the radius of the right circular cylinder and $$\displaystyle y$$ represents the height of the cylinder. Evaluate $$\displaystyle V(2,5)$$ and explain what this means.

Solution:$$\displaystyle 20π.$$ This is the volume when the radius is $$\displaystyle 2$$ and the height is $$\displaystyle 5$$.

4) An oxygen tank is constructed of a right cylinder of height $$\displaystyle y$$ and radius $$\displaystyle x$$ with two hemispheres of radius $$\displaystyle x$$ mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, $$\displaystyle x$$ and $$\displaystyle y$$, find $$\displaystyle V(10,2)$$, and explain what this means.

## Exercise $$\PageIndex{2}$$ Domain

For the following exercises, find the domain of the function.

1) $$\displaystyle V(x,y)=4x^2+y^2$$

Solution:All points in the $$\displaystyle xy-plane$$

2) $$\displaystyle f(x,y)=\sqrt{x^2+y^2−4}$$

3) $$\displaystyle f(x,y)=4ln(y^2−x)$$

Solution:$$\displaystyle x<y^2$$

4) $$\displaystyle g(x,y)=\sqrt{16−4x^2−y^2}$$

5) $$\displaystyle z(x,y)=y^2−x^2$$

Solution:All real ordered pairs in the $$\displaystyle xy-plane$$ of the form $$\displaystyle (a,b)$$

6) $$\displaystyle f(x,y)=\frac{y+2}{x^2}$$

## Exercise $$\PageIndex{3}$$ range

Find the range of the functions.

1) $$\displaystyle g(x,y)=\sqrt{16−4x^2−y^2}$$

Solution:$$\displaystyle \{z|0≤z≤4 \}$$

2) $$\displaystyle V(x,y)=4x^2+y^2$$

3) $$\displaystyle z=y^2−x^2$$

Solution:The set $$\displaystyle R$$.

## Exercise $$\PageIndex{4}$$ Level Curves

For the following exercises, find the level curves of each function at the indicated value of $$\displaystyle c$$ to visualize the given function.

1) $$\displaystyle z(x,y)=y^2−x^2, c=1$$

2) $$\displaystyle z(x,y)=y^2−x^2, c=4$$

Solution:$$\displaystyle y^2−x^2=4,$$ a hyperbola

3) $$\displaystyle g(x,y)=x^2+y^2;c=4,c=9$$

4) $$\displaystyle g(x,y)=4−x−y;c=0,4$$

Solution:$$\displaystyle 4=x+y,$$ a line; $$\displaystyle x+y=0,$$ line through the origin

5) $$\displaystyle h(x,y)=2x−y;c=0,−2,2$$

Solution:$$\displaystyle 2x−y=0,2x−y=−2,2x−y=2;$$ three lines

6) $$\displaystyle f(x,y)=x^2−y;c=1,2$$

7) $$\displaystyle g(x,y)=\frac{x}{x+y};c=−1,0,2$$

Solution:$$\displaystyle \frac{x}{x+y}=−1,\frac{x}{x+y}=0,\frac{x}{x+y}=2$$

8) $$\displaystyle g(x,y)=x^3−y;c=−1,0,2$$

9) $$\displaystyle g(x,y)=e^{xy};c=\frac{1}{2},3$$

Solution:$$\displaystyle e^{xy}=\frac{1}{2},e^{xy}=3$$

10) $$\displaystyle f(x,y)=x^2;c=4,9$$

12) $$\displaystyle f(x,y)=xy−x;c=−2,0,2$$

Solution:$$\displaystyle xy−x=−2,xy−x=0,xy−x=2$$

13) $$\displaystyle h(x,y)=ln(x^2+y^2);c=−1,0,1$$

14) $$\displaystyle g(x,y)=ln(\frac{y}{x^2});c=−2,0,2$$

Solution:$$\displaystyle e^{−2}x^2=y,y=x^2,y=e^2x^2$$

15) $$\displaystyle z=f(x,y)=\sqrt{x^2+y^2}, c=3$$

16) $$\displaystyle f(x,y)=\frac{y+2}{x^2}, c=$$any constant

Solution:The level curves are parabolas of the form $$\displaystyle y=cx^2−2.$$

## Exercise $$\PageIndex{5}$$ Vertical Traces

For the following exercises, find the vertical traces of the functions at the indicated values of $$\displaystyle x$$ and $$\displaystyle y$$, and plot the traces.

1) $$\displaystyle z=4−x−y;x=2$$

2) $$\displaystyle f(x,y)=3x+y^3,x=1$$

Solution:$$\displaystyle z=3+y^3,$$ a curve in the zy-plane with rulings parallel to the $$\displaystyle x-axis$$ 3) $$\displaystyle z=cos\sqrt{x^2+y^2} x=1$$

## Exercise $$\PageIndex{6}$$ Domain

Find the domain of the following functions.

1) $$\displaystyle z=\sqrt{100−4x^2−25y^2}$$

Solution:$$\displaystyle \frac{x^2}{25}+\frac{y^2}{4}≤1$$

2) $$\displaystyle z=ln(x−y^2)$$

3) $$\displaystyle f(x,y,z)=\frac{1}{\sqrt{36−4x^2−9y^2−z^2}}$$

Solution:$$\displaystyle \frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}<1$$

4) $$\displaystyle f(x,y,z)=\sqrt{49−x^2−y^2−z^2}$$

5) $$\displaystyle f(x,y,z)=\sqrt{16−x^2−y^2−z^2}$$

Solution:All points in $$\displaystyle xyz-space$$

6) $$\displaystyle f(x,y)=cos\sqrt{x^2+y^2}$$

## Exercise $$\PageIndex{7}$$ Graph

For the following exercises, plot a graph of the function.

1) $$\displaystyle z=f(x,y)=\sqrt{x^2+y^2}$$

Solution: 2) $$\displaystyle z=x^2+y^2$$

3) Use technology to graph $$\displaystyle z=x^2y.$$

Solution: ## Exercise $$\PageIndex{8}$$ Level curves

Sketch the following by finding the level curves. Verify the graph using technology.

1) $$\displaystyle f(x,y)=\sqrt{4−x^2−y^2}$$

2) $$\displaystyle f(x,y)=2−\sqrt{x^2+y^2}$$ 3) $$\displaystyle z=1+e^{−x^2−y^2}$$

4) $$\displaystyle z=cos\sqrt{x^2+y^2}$$

Solution: 5) $$\displaystyle z=y^2−x^2$$

## Exercise $$\PageIndex{9}$$ Contour lines

1) Describe the contour lines for several values of $$\displaystyle c$$ for $$\displaystyle z=x^2+y^2−2x−2y.$$

Solution:The contour lines are circles.

## Exercise $$\PageIndex{10}$$ level surface

Find the level surface for the functions of three variables and describe it.

1) $$\displaystyle w(x,y,z)=x−2y+z,c=4$$

2) $$\displaystyle w(x,y,z)=x^2+y^2+z^2,c=9$$

Solution:$$\displaystyle x^2+y^2+z^2=9$$, a sphere of radius $$\displaystyle 3$$

3) $$\displaystyle w(x,y,z)=x^2+y^2−z^2,c=−4$$

4) $$\displaystyle w(x,y,z)=x^2+y^2−z^2,c=4$$

Solution:$$\displaystyle x^2+y^2−z^2=4,$$ a hyperboloid of one sheet

5) $$\displaystyle w(x,y,z)=9x^2−4y^2+36z^2,c=0$$

## Exercise $$\PageIndex{11}$$ level curve at a given point

For the following exercises, find an equation of the level curve of $$\displaystyle f$$ that contains the point $$\displaystyle P$$.

1) $$\displaystyle f(x,y)=1−4x^2−y^2,P(0,1)$$

Solution:$$\displaystyle 4x^2+y^2=1,$$

2) $$\displaystyle g(x,y)=y^2arctanx,P(1,2)$$

3) $$\displaystyle g(x,y)=e^{xy}(x^2+y^2),P(1,0)$$

Solution:$$\displaystyle 1=e^{xy}(x^2+y^2)$$

## Exercise $$\PageIndex{12}$$ Applications

1) The strength $$\displaystyle E$$ of an electric field at point $$\displaystyle (x,y,z)$$ resulting from an infinitely long charged wire lying along the $$\displaystyle y-axis$$ is given by $$\displaystyle E(x,y,z)=k/\sqrt{x^2+y^2}$$, where $$\displaystyle k$$ is a positive constant. For simplicity, let $$\displaystyle k=1$$ and find the equations of the level surfaces for $$\displaystyle E=10$$ and $$\displaystyle E=100.$$

2) A thin plate made of iron is located in the $$\displaystyle xy-plane.$$ The temperature $$\displaystyle T$$ in degrees Celsius at a point $$\displaystyle P(x,y)$$ is inversely proportional to the square of its distance from the origin. Express $$\displaystyle T$$ as a function of $$\displaystyle x$$ and $$\displaystyle y$$.

Solution:$$\displaystyle T(x,y)=\frac{k}{x^2+y^2}$$
3) Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point $$\displaystyle P(1,2)$$ is $$\displaystyle 50°C.$$ Use this constant to determine the temperature at point $$\displaystyle Q(3,4).$$
4) Refer to the preceding problem. Find the level curves for $$\displaystyle T=40°C$$ and $$\displaystyle T=100°C,$$ and describe what the level curves represent.
Solution:$$\displaystyle x^2+y^2=\frac{k}{40}, x^2+y^2=\frac{k}{100}$$. The level curves represent circles of radii $$\displaystyle \sqrt{10k}/20$$ and $$\displaystyle \sqrt{k}/10$$