6.5E: Exercises

Exercise $\PageIndex{1}$

For the following exercises, use the information provided to solve the problem.

1) Let $\displaystyle w(x,y,z)=xycosz,$ where $\displaystyle x=t,y=t^2,$ and $\displaystyle z=arcsint.$ Find $\displaystyle \frac{dw}{dt}$.

Answer

$\displaystyle \frac{dw}{dt}=ycosz+xcosz(2t)−\frac{xysinz}{\sqrt{1−t^2}}$

2) Let $\displaystyle w(t,v)=e^{tv}$ where $\displaystyle t=r+s$ and $\displaystyle v=rs$. Find $\displaystyle \frac{∂w}{∂r}$ and $\displaystyle \frac{∂w}{∂s}$.

3) If $\displaystyle w=5x^2+2y^2,x=−3s+t,$ and $\displaystyle y=s−4t,$ find $\displaystyle \frac{∂w}{∂s}$ and $\displaystyle \frac{∂w}{∂t}$.

Answer

$\displaystyle \frac{∂w}{∂s}=−30x+4y, \frac{∂w}{∂t}=10x−16y$

4) If $\displaystyle w=xy^2,x=5cos(2t),$ and $\displaystyle y=5sin(2t)$, find $\displaystyle \frac{∂w}{∂t}$.

5) If $\displaystyle f(x,y)=xy,x=rcosθ,$ and $\displaystyle y=rsinθ$, find ∂f∂r and express the answer in terms of $\displaystyle r$ and $\displaystyle θ$.

Answer

$\displaystyle \frac{∂f}{∂r}=rsin(2θ)$

6) Suppose $\displaystyle f(x,y)=x+y,u=e^xsiny,x=t^2$ and $\displaystyle y=πt$, where $\displaystyle x=rcosθ$ and $\displaystyle y=rsinθ$. Find $\displaystyle \frac{∂f}{∂θ}$.

Exercise $\PageIndex{2}$

For the following exercises, find $\displaystyle \frac{df}{dt}$ using the chain rule and direct substitution.

7) $\displaystyle f(x,y)=x^2+y^2, x=t,y=t^2$

Answer

$\displaystyle \frac{df}{dt}=2t+4t^3$

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

9) $\displaystyle f(x,y)=xy,x=1−\sqrt{t},y=1+\sqrt{t}$

Answer

$\displaystyle \frac{df}{dt}=−1$

10) $\displaystyle f(x,y)=\frac{x}{y},x=e^t,y=2e^t$

11) $\displaystyle f(x,y)=ln(x+y), x=e^t,y=e^t$

Answer

$\displaystyle \frac{df}{dt}=1$

12) $\displaystyle f(x,y)=x^4, x=t,y=t$

Exercise $\PageIndex{3}$

13) Let $\displaystyle w(x,y,z)=x^2+y^2+z^2, x=cost,y=sint,$ and $\displaystyle z=e^t$. Express $\displaystyle w$ as a function of $\displaystyle t$ and find $\displaystyle \frac{dw}{dt}$ directly. Then, find $\displaystyle \frac{dw}{dt}$ using the chain rule.

Answer

$\displaystyle \frac{dw}{dt}=2e^{2t}$ in both cases

14) Let $\displaystyle z=x^2y,$ where $\displaystyle x=t^2$ and $\displaystyle y=t^3$. Find $\displaystyle \frac{dz}{dt}$.

15) Let $\displaystyle u=e^xsiny,$ where $\displaystyle x=t^2$ and $\displaystyle y=πt$. Find $\displaystyle \frac{du}{dt}$ when $\displaystyle x=ln2$ and $\displaystyle y=\frac{π}{4}$.

Answer

$\displaystyle 2\sqrt{2}t+\sqrt{2}π=\frac{du}{dt}$

Exercise $\PageIndex{4}$

For the following exercises, find $\displaystyle \frac{dy}{dx}$using partial derivatives.

16) $\displaystyle sin(6x)+tan(8y)+5=0$

17) $\displaystyle x^3+y^2x−3=0$

Answer

$\displaystyle \frac{dy}{dx}=−\frac{3x^2+y^2}{2xy}$

18) $\displaystyle sin(x+y)+cos(x−y)=4$

19) $\displaystyle x^2−2xy+y^4=4$

Answer

$\displaystyle \frac{dy}{dx}=\frac{y−x}{−x+2y^3}$

20) $\displaystyle xe^y+ye^x−2x^2y=0$

21) $\displaystyle x^{2/3}+y^{2/3}=a^{2/3}$

Answer

$\displaystyle \frac{dy}{dx}=−\sqrt[3]{\frac{y}{x}}$

22) $\displaystyle xcos(xy)+ycosx=2$

23) $\displaystyle e^{xy}+ye^y=1$

Answer

$\displaystyle \frac{dy}{dx}=−\frac{ye^{xy}}{xe^{xy}+e^y(1+y)}$

24) $\displaystyle x^2y^3+cosy=0$

Exercise $\PageIndex{5}$

25) Find $\displaystyle \frac{dz}{dt}$ using the chain rule where $\displaystyle z=3x^2y^3,x=t^4,$ and $\displaystyle y=t^2$.

Answer

$\displaystyle \frac{dz}{dt}=42t^{13}$

26) Let $\displaystyle z=3cosx−sin(xy),x=\frac{1}{t},$ and $\displaystyle y=3t.$ Find $\displaystyle \frac{dz}{dt}$.

27) Let $\displaystyle z=e^{1−xy},x=t^{1/3},$ and $\displaystyle y=t^3$. Find $\displaystyle \frac{dz}{dt}$.

Answer

$\displaystyle \frac{dz}{dt}=−\frac{10}{3}t^{7/3}×e^{1−t^{10/3}}$

28) Find $\displaystyle \frac{dz}{dt}$ by the chain rule where $\displaystyle z=cosh^2(xy),x=\frac{1}{2}t,$ and $\displaystyle y=e^t$.

29) Let $\displaystyle z=\frac{x}{y},x=2cosu,$ and $\displaystyle y=3sinv.$ Find $\displaystyle \frac{∂z}{∂u}$ and $\displaystyle \frac{∂z}{∂v}$.

Answer

$\displaystyle \frac{∂z}{∂u}=\frac{−2sinu3}{sinv}$ and $\displaystyle \frac{∂z}{∂v}=\frac{−2cosucosv3}{sin^2v}$

30) Let $\displaystyle z=e^{x^2y}$, where $\displaystyle x=\sqrt{uv}$ and $\displaystyle y=\frac{1}{v}$. Find $\displaystyle \frac{∂z}{∂u}$ and $\displaystyle \frac{∂z}{∂v}$.

31) If $\displaystyle z=xye^{x/y}, x=rcosθ,$ and $\displaystyle y=rsinθ$, find $\displaystyle \frac{∂z}{∂r}$ and $\displaystyle \frac{∂z}{∂θ}$ when $\displaystyle r=2$ and $\displaystyle θ=\frac{π}{6}$.

Answer

$\displaystyle \frac{∂z}{∂r}=\sqrt{3}e^{\sqrt{3}}, \frac{∂z}{∂θ}=(2−4\sqrt{3})e^{\sqrt{3}}$

32) Find $\displaystyle \frac{∂w}{∂s}$ if $\displaystyle w=4x+y^2+z^3,x=e^{rs^2},y=ln(\frac{r+s}{t}),$ and $\displaystyle z=rst^2$.

33) If $\displaystyle w=sin(xyz),x=1−3t,y=e^{1−t},$ and $\displaystyle z=4t$, find $\displaystyle \frac{∂w}{∂t}$.

Answer

$\displaystyle \frac{∂w}{∂t}=cos(xyz)×yz×(−3)−cos(xyz)xze^{1−t}+cos(xyz)xy×4$

Exercise $\PageIndex{6}$

For the following exercises, use this information: A function $\displaystyle f(x,y)$ is said to be homogeneous of degree $\displaystyle n$ if $\displaystyle f(tx,ty)=t^nf(x,y)$. For all homogeneous functions of degree $\displaystyle n$, the following equation is true: $\displaystyle x\frac{∂f}{∂x}+y\frac{∂f}{∂y}=nf(x,y)$. Show that the given function is homogeneous and verify that $\displaystyle x\frac{∂f}{∂x}+y\frac{∂f}{∂y}=nf(x,y)$.

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

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

Answer

$\displaystyle f(tx,ty)=\sqrt{t^2x^2+t^2y^2}=t^1f(x,y), \frac{∂f}{∂y}=x\frac{1}{2}(x^2+y^2)^{−1/2}×2x+y\frac{1}{2}(x^2+y^2)^{−1/2}×2y=1f(x,y)$

36) $\displaystyle f(x,y)=x^2y−2y^3$

Exercise $\PageIndex{7}$

37) The volume of a right circular cylinder is given by $\displaystyle V(x,y)=πx^2y,$ where $\displaystyle x$ is the radius of the cylinder and $\displaystyle y$ is the cylinder height. Suppose $\displaystyle x$ and $\displaystyle y$ are functions of $\displaystyle t$ given by $\displaystyle x=\frac{1}{2}t$ and $\displaystyle y=\frac{1}{3}t$ so that $\displaystyle x$ and $\displaystyle y$ are both increasing with time. How fast is the volume increasing when $\displaystyle x=2$ and $\displaystyle y=5$?

Answer

$\displaystyle \frac{34π}{3}$

38) The pressure $\displaystyle P$ of a gas is related to the volume and temperature by the formula $\displaystyle PV=kT$, where temperature is expressed in kelvins. Express the pressure of the gas as a function of both $\displaystyle V$ and $\displaystyle T$. Find $\displaystyle \frac{dP}{dt}$ when $\displaystyle k=1, \frac{dV}{dt}=2$ cm³/min, $\displaystyle \frac{dT}{dt}=12$ K/min, $\displaystyle V=20 cm^3$, and $\displaystyle T=20°F$.

39) The radius of a right circular cone is increasing at $\displaystyle 3$ cm/min whereas the height of the cone is decreasing at $\displaystyle 2$ cm/min. Find the rate of change of the volume of the cone when the radius is $\displaystyle 13$ cm and the height is $\displaystyle 18$ cm.

Answer

$\displaystyle \frac{dV}{dt}=\frac{1066π}{3}cm^3/min$

40) The volume of a frustum of a cone is given by the formula $\displaystyle V=\frac{1}{3}πz(x^2+y^2+xy),$ where $\displaystyle x$ is the radius of the smaller circle, $\displaystyle y$ is the radius of the larger circle, and $\displaystyle z$ is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when $\displaystyle x=10in.,y=12in.,$ and $\displaystyle z=18in.$

41) A closed box is in the shape of a rectangular solid with dimensions $\displaystyle x,y,$ and $\displaystyle z$. (Dimensions are in inches.) Suppose each dimension is changing at the rate of $\displaystyle 0.5$ in./min. Find the rate of change of the total surface area of the box when $\displaystyle x=2in.,y=3in.,$ and $\displaystyle z=1in.$

Answer

$\displaystyle \frac{dA}{dt}=12in.^2/min$

42) The total resistance in a circuit that has three individual resistances represented by $\displaystyle x,y,$ and $\displaystyle z$ is given by the formula $\displaystyle R(x,y,z)=\frac{xyz}{yz+xz+xy}$. Suppose at a given time the $\displaystyle x$ resistance is $\displaystyle 100Ω$, the $\displaystyle y$ resistance is $\displaystyle 200Ω,$ and the $\displaystyle z$ resistance is $\displaystyle 300Ω.$ Also, suppose the $\displaystyle x$ resistance is changing at a rate of $\displaystyle 2Ω/min,$ the $\displaystyle y$ resistance is changing at the rate of $\displaystyle 1Ω/min$, and the $\displaystyle z$ resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

43) The temperature $\displaystyle T$ at a point $\displaystyle (x,y)$ is $\displaystyle T(x,y)$ and is measured using the Celsius scale. A fly crawls so that its position after $\displaystyle t$ seconds is given by $\displaystyle x=\sqrt{1+t}$ and $\displaystyle y=2+\frac{1}{3}t$, where $\displaystyle x$ and $\displaystyle y$ are measured in centimeters. The temperature function satisfies $\displaystyle T_x(2,3)=4$ and $\displaystyle T_y(2,3)=3$. How fast is the temperature increasing on the fly’s path after $\displaystyle 3$ sec?

Answer

$\displaystyle 2°C/sec$

44) The $\displaystyle x$ and $\displaystyle y$ components of a fluid moving in two dimensions are given by the following functions: $\displaystyle u(x,y)=2y$ and $\displaystyle v(x,y)=−2x; x≥0;y≥0.$ The speed of the fluid at the point $\displaystyle (x,y)$ is $\displaystyle s(x,y)=\sqrt{u(x,y)^2+v(x,y)^2}$. Find $\displaystyle \frac{∂s}{∂x}$ and $\displaystyle \frac{∂s}{∂y}$ using the chain rule.

45) Let $\displaystyle u=u(x,y,z),$ where $\displaystyle x=x(w,t),y=y(w,t),z=z(w,t),w=w(r,s),$ and $\displaystyle t=t(r,s).$ Use a tree diagram and the chain rule to find an expression for $\displaystyle \frac{∂u}{∂r}$.

Answer

$\displaystyle \frac{∂u}{∂r}=\frac{∂u}{∂x}(\frac{∂x}{∂w}\frac{∂w}{∂r}+\frac{∂x}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂y}(\frac{∂y}{∂w}\frac{∂w}{∂r}+\frac{∂y}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂z}(\frac{∂z}{∂w}\frac{∂w}{∂r}+\frac{∂z}{∂t}\frac{∂t}{∂r})$