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6.3E: Exercises

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Exercise $$\PageIndex{1}$$

For the following exercises, calculate the partial derivative using the limit definitions only.

1) $$\displaystyle \frac{∂z}{∂x}$$ for $$\displaystyle z=x^2−3xy+y^2$$

2) $$\displaystyle \frac{∂z}{∂y}$$ for $$\displaystyle z=x^2−3xy+y^2$$

Solution:$$\displaystyle \frac{∂z}{∂y}=−3x+2y$$

Exercise $$\PageIndex{2}$$

For the following exercises, calculate the sign of the partial derivative using the graph of the surface.

1) $$\displaystyle f_x(1,1)$$

2) $$\displaystyle f_x(−1,1)$$

Solution:The sign is negative.

3) $$\displaystyle f_y(1,1)$$

4) $$\displaystyle f_x(0,0)$$

Solution:The partial derivative is zero at the origin.

Exercise $$\PageIndex{3}$$

For the following exercises, calculate the partial derivatives.

1) $$\displaystyle \frac{∂z}{∂x}$$ for $$\displaystyle z=sin(3x)cos(3y)$$

2) $$\displaystyle \frac{∂z}{∂y}$$ for $$\displaystyle z=sin(3x)cos(3y)$$

Solution:$$\displaystyle \frac{∂z}{∂y}=−3sin(3x)sin(3y)$$

3) $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$ for $$\displaystyle z=x^8e^3y$$

4) $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$ for $$\displaystyle z=ln(x^6+y^4)$$

Solution:$$\displaystyle \frac{∂z}{∂x}=\frac{6x^5}{x^6+y^4};\frac{∂z}{∂y}=\frac{4y^3}{x^6+y^4}$$

5) Find $$\displaystyle f_y(x,y)$$ for $$\displaystyle f(x,y)=e^{xy}cos(x)sin(y).$$

6) Let $$\displaystyle z=e^{xy}.$$ Find $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$.

Solution:$$\displaystyle \frac{∂z}{∂x}=ye^{xy};\frac{∂z}{∂y}=xe^{xy}\ 7) Let \(\displaystyle z=ln(\frac{x}{y})$$. Find $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$.

8) Let $$\displaystyle z=tan(2x−y).$$ Find $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$.

Solution:$$\displaystyle \frac{∂z}{∂x}=2sec^2(2x−y),\frac{∂z}{∂y}=−sec^2(2x−y)$$

9) Let $$\displaystyle z=sinh(2x+3y).$$ Find $$\displaystyle \frac{∂z}{∂x}$$ and $$\displaystyle \frac{∂z}{∂y}$$.

10) Let $$\displaystyle f(x,y)=arctan(\frac{y}{x}).$$ Evaluate $$\displaystyle f_x(2,−2)$$ and $$\displaystyle f_y(2,−2)$$.

Solution:$$\displaystyle f_x(2,−2)=\frac{1}{4}=f_y(2,−2)$$

11) Let $$\displaystyle f(x,y)=\frac{xy}{x−y}.$$ Find $$\displaystyle f_x(2,−2)$$ and $$\displaystyle f_y(2,−2).$$ Evaluate the partial derivatives at point $$\displaystyle P(0,1).$$

Exercise $$\PageIndex{4}$$

1) Find $$\displaystyle \frac{∂z}{∂x}$$ at $$\displaystyle (0,1)$$ for $$\displaystyle z=e^{−x}cos(y)$$.

Solution:$$\displaystyle \frac{∂z}{∂x}=−cos(1)$$

2) Given $$\displaystyle f(x,y,z)=x^3yz^2,$$ find $$\displaystyle \frac{∂^2f}{∂x∂y}$$ and $$\displaystyle f_z(1,1,1).$$

3) Given $$\displaystyle f(x,y,z)=2sin(x+y),$$ find $$\displaystyle f_x(0,\frac{π}{2},−4), f_y(0,\frac{π}{2},−4),$$ and $$\displaystyle f_z(0,\frac{π}{2},−4).$$

Solution:$$\displaystyle f_x=0,f_y=0,f_z=0$$

Exercise $$\PageIndex{5}$$

1) The area of a parallelogram with adjacent side lengths that are $$\displaystyle a$$ and $$\displaystyle b$$, and in which the angle between these two sides is $$\displaystyle θ$$, is given by the function $$\displaystyle A(a,b,θ)=basin(θ).$$Find the rate of change of the area of the parallelogram with respect to the following:

a. Side a

b. Side b

c. $$\displaystyle Angle θ$$

2) Express the volume of a right circular cylinder as a function of two variables:

a. its radius $$\displaystyle r$$ and its height $$\displaystyle h$$.

b. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height.

c. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.

Solution:$$\displaystyle a. V(r,h)=πr^2h$$ $$\displaystyle b. \frac{∂V}{∂r}=2πrh$$ $$\displaystyle c. \frac{∂V}{∂h}=πr^2$$

3) Calculate $$\displaystyle \frac{∂w}{∂z}$$ for $$\displaystyle w=zsin(xy^2+2z).$$

Exercise $$\PageIndex{6}$$

Find the indicated higher-order partial derivatives.

1) $$\displaystyle f_{xy}$$ for $$\displaystyle z=ln(x−y)$$

Solution:$$\displaystyle f_{xy}=\frac{1}{(x−y)^2}$$

2) $$\displaystyle f_{yx}$$ for $$\displaystyle z=ln(x−y)$$

3) Let $$\displaystyle z=x^2+3xy+2y^2.$$ Find $$\displaystyle \frac{∂^2z}{∂x^2}$$ and $$\displaystyle \frac{∂^2z}{∂y^2}$$.

Solution:$$\displaystyle \frac{∂^2z}{∂x^2}=2,\frac{∂^2z}{∂y^2}=4$$

4) Given $$\displaystyle z=e^xtany,$$ find $$\displaystyle \frac{∂^2z}{∂x∂y}$$ and $$\displaystyle \frac{∂^2z}{∂y∂x}$$.

5) Given $$\displaystyle f(x,y,z)=xyz,$$ find $$\displaystyle f_{xyy},f_{yxy},$$ and $$\displaystyle f_{yyx}$$.

Solution:$$\displaystyle f_{xyy}=f_{yxy}=f_{yyx}=0$$

6) Given $$\displaystyle f(x,y,z)=e^{−2x}sin(z^2y),$$ show that $$\displaystyle f_{xyy}=f_{yxy}$$.

7) Show that $$\displaystyle z=\frac{1}{2}(e^y−e^{−y})sinx$$ is a solution of the differential equation $$\displaystyle \frac{∂^2z}{∂x^2}+\frac{∂^2z}{∂y^2}=0.$$

Solution:

$$\displaystyle \frac{d^2z}{dx^2}=−\frac{1}{2}(e^y−e^{−y})sinx$$

$$\displaystyle \frac{d^2z}{dy^2}=\frac{1}{2}(e^y−e^{−y})sinx$$

$$\displaystyle \frac{d^2z}{dx^2}+\frac{d^2z}{dy^2}=0$$

8) Find $$\displaystyle f_{xx}(x,y)$$ for $$\displaystyle f(x,y)=\frac{4x^2}{y}+\frac{y^2}{2x}.$$

9) Let $$\displaystyle f(x,y,z)=x^2y^3z−3xy^2z^3+5x^2z−y^3z.$$ Find $$\displaystyle f_{xyz}.$$

Solution:$$\displaystyle f_{xyz}=6y^2x−18yz^2$$

10) Let $$\displaystyle F(x,y,z)=x^3yz^2−2x^2yz+3xz−2y^3z.$$ Find $$\displaystyle F_{xyz}$$.

11) Given $$\displaystyle f(x,y)=x^2+x−3xy+y^3−5,$$ find all points at which $$\displaystyle f_x=f_y=0$$ simultaneously.

Solution:$$\displaystyle (\frac{1}{4},\frac{1}{2}),(1,1)$$

12) Given $$\displaystyle f(x,y)=2x^2+2xy+y^2+2x−3,$$ find all points at which $$\displaystyle \frac{∂f}{∂x}=0$$ and $$\displaystyle \frac{∂f}{∂y}=0$$ simultaneously.

13) Given $$\displaystyle f(x,y)=y^3−3yx^2−3y^2−3x^2+1$$, find all points on $$\displaystyle f$$ at which $$\displaystyle f_x=f_y=0$$ simultaneously.

Solution:$$\displaystyle (0,0),(0,2),(\sqrt{3},−1),(−\sqrt{3},−1)$$

14) Given $$\displaystyle f(x,y)=15x^3−3xy+15y^3,$$ find all points at which $$\displaystyle f_x(x,y)=f_y(x,y)=0$$ simultaneously.

15) Show that $$\displaystyle z=e^xsiny$$ satisfies the equation $$\displaystyle \frac{∂^2z}{∂x^2}+\frac{∂^2z}{∂y^2}=0.$$

Solution:$$\displaystyle \frac{∂^2z}{∂x^2}+\frac{∂^2z}{∂y^2}=e^xsin(y)−e^xsiny=0$$

16) Show that $$\displaystyle f(x,y)=ln(x^2+y^2)$$ solves Laplace’s equation $$\displaystyle \frac{∂^2z}{∂x^2}+\frac{∂^2z}{∂y^2}=0.$$

17) Show that $$\displaystyle z=e^{−t}cos(\frac{x}{c})$$ satisfies the heat equation $$\displaystyle \frac{∂z}{∂t}=−e^{−t}cos(\frac{x}{c}).$$

Solution:$$\displaystyle c^2\frac{∂^2z}{∂x^2}=e^{−t}cos(\frac{x}{c})$$

18) Find $$\displaystyle \lim_{Δx→0}\frac{f(x+Δx)−f(x,y)}{Δx}$$ for $$\displaystyle f(x,y)=−7x−2xy+7y.$$

19) Find $$\displaystyle \lim_{Δy→0}\frac{f(x,y+Δy)−f(x,y)}{Δy}$$ for $$\displaystyle f(x,y)=−7x−2xy+7y.$$

Solution:$$\displaystyle \frac{∂f}{∂y}=−2x+7$$

20) Find $$\displaystyle \lim_{Δx→0}\frac{Δf}{Δx}=\lim_{Δx→0}\frac{f(x+Δx,y)−f(x,y)}{Δx}$$ for $$\displaystyle f(x,y)=x^2y^2+xy+y.$$

21) Find $$\displaystyle \lim_{Δx→0}\frac{Δf}{Δx}=\lim_{Δx→0}\frac{f(x+Δx,y)−f(x,y)}{Δx}$$ for $$\displaystyle f(x,y)=sin(xy).$$

Solution:$$\displaystyle \frac{∂f}{∂x}=ycosxy$$

Exercise $$\PageIndex{7}$$

1) The function $$\displaystyle P(T,V)=\frac{nRT}{V}$$ gives the pressure at a point in a gas as a function of temperature $$\displaystyle T$$ and volume $$\displaystyle V$$. The letters $$\displaystyle n$$ and $$\displaystyle R$$ are constants. Find $$\displaystyle \frac{∂P}{∂V}$$ and $$\displaystyle \frac{∂P}{∂T}$$, and explain what these quantities represent.

2) The equation for heat flow in the $$\displaystyle xy-plane$$ is $$\displaystyle \frac{∂f}{∂t}=\frac{∂^2f}{∂x^2}+\frac{∂^2f}{∂y^2}$$. Show that $$\displaystyle f(x,y,t)=e^{−2t}sinxsiny$$ is a solution.

3) The basic wave equation is $$\displaystyle f_{tt}=f_{xx}.$$ Verify that $$\displaystyle f(x,t)=sin(x+t)$$ and $$\displaystyle f(x,t)=sin(x−t)$$ are solutions.

4) The law of cosines can be thought of as a function of three variables. Let $$\displaystyle x,y,$$ and $$\displaystyle θ$$ be two sides of any triangle where the angle $$\displaystyle θ$$ is the included angle between the two sides. Then, $$\displaystyle F(x,y,θ)=x^2+y^2−2xycosθ$$ gives the square of the third side of the triangle. Find $$\displaystyle \frac{∂F}{∂θ}$$ and $$\displaystyle \frac{∂F}{∂x}$$ when $$\displaystyle x=2,y=3,$$ and $$\displaystyle θ=\frac{π}{6}.$$

Solution:$$\displaystyle \frac{∂F}{∂θ}=6,\frac{∂F}{∂x}=4−3\sqrt{3}\ 5) Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of \(\displaystyle 2$$in./sec whereas the second side is changing at the rate of $$\displaystyle 4$$ in/sec. How fast is the diagonal of the rectangle changing when the first side measures $$\displaystyle 16$$ in. and the second side measures $$\displaystyle 20$$ in.? (Round answer to three decimal places.)
6) A Cobb-Douglas production function is $$\displaystyle f(x,y)=200x^{0.7}y^{0.3},$$ where $$\displaystyle x$$ and $$\displaystyle y$$ represent the amount of labor and capital available. Let $$\displaystyle x=500$$ and $$\displaystyle y=1000.$$ Find $$\displaystyle \frac{δf}{δx}$$ and $$\displaystyle \frac{δf}{δy}$$ at these values, which represent the marginal productivity of labor and capital, respectively.
Solution:$$\displaystyle \frac{δf}{δx}$$ at $$\displaystyle (500,1000)=172.36, \frac{δf}{δy}$$ at $$\displaystyle (500,1000)=36.93$$
7) The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: $$\displaystyle h$$, which is relative humidity, and $$\displaystyle t$$, which is the air temperature. $$\displaystyle A=0.885t−22.4h+1.20th−0.544.$$ Find $$\displaystyle \frac{∂A}{∂t}$$ and $$\displaystyle \frac{∂A}{∂h}$$ when $$\displaystyle t=20°F$$ and $$\displaystyle h=0.90.$$