Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

14: Partial Differentiation (Exercises)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here

14.1: Functions of Several Variables

Ex 14.1.1 Let $f(x,y)=(x-y)^2$. Determine the equations and shapes of the cross-sections when $x=0$, $y=0$, $x=y$, and describe the level curves. Use a three-dimensional graphing tool to graph the surface. (answer)

Ex 14.1.2 Let $f(x,y)=|x|+|y|$. Determine the equations and shapes of the cross-sections when $x=0$, $y=0$, $x=y$, and describe the level curves. Use a three-dimensional graphing tool to graph the surface. (answer)

Ex 14.1.3 Let $f(x,y)=e^{-(x^2+y^2)}\sin(x^2+y^2)$. Determine the equations and shapes of the cross-sections when $x=0$, $y=0$, $x=y$, and describe the level curves. Use a three-dimensional graphing tool to graph the surface. (answer)

Ex 14.1.4 Let $f(x,y)=\sin(x-y)$. Determine the equations and shapes of the cross-sections when $x=0$, $y=0$, $x=y$, and describe the level curves. Use a three-dimensional graphing tool to graph the surface. (answer)

Ex 14.1.5 Let $f(x,y)=(x^2-y^2)^2$. Determine the equations and shapes of the cross-sections when $x=0$, $y=0$, $x=y$, and describe the level curves. Use a three-dimensional graphing tool to graph the surface. (answer)

Ex 14.1.6 Find the domain of each of the following functions of two variables:

  • a. $\ds\sqrt{9-x^2}+\sqrt{y^2-4}$ b. $\arcsin(x^2+y^2-2)$ c. $\ds\sqrt{16-x^2-4y^2}$ (answer)

Ex 14.1.7 Below are two sets of level curves. One is for a cone, one is for a paraboloid. Which is which? Explain.

14.2: Limits and Continuity

Determine whether each limit exists. If it does, find the limit and prove that it is the limit; if it does not, explain how you know.

Ex 14.2.1 \( \lim_{(x,y)\to(0,0)}{x^2\over x^2+y^2}\) (answer)

Ex 14.2.2 \( \lim_{(x,y)\to(0,0)}{xy\over x^2+y^2}\) (answer)

Ex 14.2.3 \( \lim_{(x,y)\to(0,0)}{xy\over 2x^2+y^2}\) (answer)

Ex 14.2.4 \( \lim_{(x,y)\to(0,0)}{x^4-y^4\over x^2+y^2}\) (answer)

Ex 14.2.5 \( \lim_{(x,y)\to(0,0)}{\sin(x^2+y^2)\over x^2+y^2}\) (answer)

Ex 14.2.6 \( \lim_{(x,y)\to(0,0)}{xy\over \sqrt{2x^2+y^2}}\) (answer)

Ex 14.2.7 \( \lim_{(x,y)\to(0,0)} {e^{-x^2-y^2}-1\over x^2+y^2}\) (answer)

Ex 14.2.8 \( \lim_{(x,y)\to(0,0)}{x^3+y^3\over x^2+y^2}\) (answer)

Ex 14.2.9 \( \lim_{(x,y)\to(0,0)}{x^2 + \sin^2 y\over 2x^2+y^2}\) (answer)

Ex 14.2.10 \( \lim_{(x,y)\to(1,0)}{(x-1)^2\ln x\over(x-1)^2+y^2}\) (answer)

Ex 14.2.11 \( \lim_{(x,y)\to(1,-1)}{3x+4y}\) (answer)

Ex 14.2.12 \( \lim_{(x,y)\to(0,0)}{4x^2y\over x^2+y^2}\) (answer)

Ex 14.2.13 Does the function \( f(x,y)={x-y\over 1+x+y}\) have any discontinuities? What about \( f(x,y)={x-y\over 1+x^2+y^2}\)? Explain.

14.3: Partial Differentiation

Ex 14.3.1 Find \(f_x\) and \(f_y\) where \( f(x,y)=\cos(x^2y)+y^3\). (answer)

Ex 14.3.2 Find \(f_x\) and \(f_y\) where \( f(x,y)={xy\over x^2+y}\). (answer)

Ex 14.3.3 Find \(f_x\) and \(f_y\) where \( f(x,y)=e^{x^2+y^2}\). (answer)

Ex 14.3.4 Find \(f_x\) and \(f_y\) where \( f(x,y)=xy\ln(xy)\). (answer)

Ex 14.3.5 Find \(f_x\) and \(f_y\) where \( f(x,y)=\sqrt{1-x^2-y^2}\). (answer)

Ex 14.3.6 Find \(f_x\) and \(f_y\) where \( f(x,y)=x\tan(y)\). (answer)

Ex 14.3.7 Find \(f_x\) and \(f_y\) where \( f(x,y)={1\over xy}\). (answer)

Ex 14.3.8 Find an equation for the plane tangent to \( 2x^2+3y^2-z^2=4\) at \((1,1,-1)\). (answer)

Ex 14.3.9 Find an equation for the plane tangent to \( f(x,y)=\sin(xy)\) at \((\pi,1/2,1)\). (answer)

Ex 14.3.10 Find an equation for the plane tangent to \( f(x,y)=x^2+y^3\) at \((3,1,10)\). (answer)

Ex 14.3.11 Find an equation for the plane tangent to \( f(x,y)=x\ln(xy)\) at \((2,1/2,0)\). (answer)

Ex 14.3.12 Find an equation for the line normal to \( x^2+4y^2=2z\) at \((2,1,4)\). (answer)

Ex 14.3.13 Explain in your own words why, when taking a partial derivative of a function of multiple variables, we can treat the variables not being differentiated as constants.

Ex 14.3.14 Consider a differentiable function, \(f(x,y)\). Give physical interpretations of the meanings of \(f_x(a,b)\) and \(f_y(a,b)\) as they relate to the graph of \(f\).

Ex 14.3.15 In much the same way that we used the tangent line to approximate the value of a function from single variable calculus, we can use the tangent plane to approximate a function from multivariable calculus. Consider the tangent plane found in Exercise 11. Use this plane to approximate \(f(1.98, 0.4)\).

Ex 14.3.16 Suppose that one of your colleagues has calculated the partial derivatives of a given function, and reported to you that \(f_x(x,y)=2x+3y\) and that \(f_y(x,y)=4x+6y\). Do you believe them? Why or why not? If not, what answer might you have accepted for \(f_y\)?

Ex 14.3.17 Suppose \(f(t)\) and \(g(t)\) are single variable differentiable functions. Find \(\partial z/\partial x\) and \(\partial z/\partial y\) for each of the following two variable functions.

  • a. \(z=f(x)g(y)\) b. \(z=f(xy)\) c. \(z=f(x/y)\)

14.4: The Chain Rule

Ex 14.4.1 Use the chain rule to compute \(dz/dt\) for \(z=\sin(x^2+y^2)\), \(x=t^2+3\), \(y=t^3\). (answer)

Ex 14.4.2 Use the chain rule to compute \(dz/dt\) for \(z=x^2y\), \(x=\sin(t)\), \(y=t^2+1\). (answer)

Ex 14.4.3 Use the chain rule to compute \(\partial z/\partial s\) and \(\partial z/\partial t\) for \(z=x^2y\), \(x=\sin(st)\), \(y=t^2+s^2\). (answer)

Ex 14.4.4 Use the chain rule to compute \(\partial z/\partial s\) and \(\partial z/\partial t\) for \(z=x^2y^2\), \(x=st\), \(y=t^2-s^2\). (answer)

Ex 14.4.5 Use the chain rule to compute \(\partial z/\partial x\) and \(\partial z/\partial y\) for \(2x^2+3y^2-2z^2=9\). (answer)

Ex 14.4.6 Use the chain rule to compute \(\partial z/\partial x\) and \(\partial z/\partial y\) for \(2x^2+y^2+z^2=9\). (answer)

Ex 14.4.7 Chemistry students will recognize the ideal gas law, given by \(PV=nRT\) which relates the Pressure, Volume, and Temperature of \(n\) moles of gas. (R is the ideal gas constant). Thus, we can view pressure, volume, and temperature as variables, each one dependent on the other two.

  • a. If pressure of a gas is increasing at a rate of \(0.2 Pa/\hbox{min}\) and temperature is increasing at a rate of \(1 K/\hbox{min}\), how fast is the volume changing?
  • b. If the volume of a gas is decreasing at a rate of \(0.3 L/\hbox{min}\) and temperature is increasing at a rate of \(.5 K/\hbox{min}\), how fast is the pressure changing?
  • c. If the pressure of a gas is decreasing at a rate of \(0.4 Pa/\hbox{min}\) and the volume is increasing at a rate of \(3 L/\hbox{min}\), how fast is the temperature changing? (answer)

Ex 14.4.8 Verify the following identity in the case of the ideal gas law: \[{\partial P\over \partial V} {\partial V\over \partial T} {\partial T\over \partial P}=-1\]

Ex 14.4.9 The previous exercise was a special case of the following fact, which you are to verify here: If \(F(x,y,z)\) is a function of 3 variables, and the relation \(F(x,y,z)=0\) defines each of the variables in terms of the other two, namely \(x=f(y,z)\), \(y=g(x,z)\) and \(z=h(x,y)\), then \[{\partial x\over \partial y} {\partial y\over \partial z} {\partial z\over \partial x}=-1\]

14.5: Directional Derivatives

Ex 14.5.1 Find \(D_{\bf u} f\) for \( f=x^2+xy+y^2\) in the direction of \({\bf u}=\langle 2,1\rangle\) at the point \((1,1)\). (answer)

Ex 14.5.2 Find \(D_{\bf u} f\) for \( f=\sin(xy)\) in the direction of \({\bf u}=\langle -1,1\rangle\) at the point \((3,1)\). (answer)

Ex 14.5.3 Find \(D_{\bf u} f\) for \( f=e^x\cos(y)\) in the direction 30 degrees from the positive \(x\) axis at the point \((1,\pi/4)\). (answer)

Ex 14.5.4 The temperature of a thin plate in the \(x\)-\)y\) plane is \( T=x^2+y^2\). How fast does temperature change at the point \((1,5)\) moving in a direction 30 degrees from the positive \(x\) axis? (answer)

Ex 14.5.5 Suppose the density of a thin plate at \((x,y)\) is \( 1/\sqrt{x^2+y^2+1}\). Find the rate of change of the density at \((2,1)\) in a direction \(\pi/3\) radians from the positive \(x\) axis. (answer)

Ex 14.5.6 Suppose the electric potential at \((x,y)\) is \( \ln\sqrt{x^2+y^2}\). Find the rate of change of the potential at \((3,4)\) toward the origin and also in a direction at a right angle to the direction toward the origin. (answer)

Ex 14.5.7 A plane perpendicular to the \(x\)-\)y\) plane contains the point \((2,1,8)\) on the paraboloid \(z=x^2+4y^2\). The cross-section of the paraboloid created by this plane has slope 0 at this point. Find an equation of the plane. (answer)

Ex 14.5.8 A plane perpendicular to the \(x\)-\)y\) plane contains the point \((3,2,2)\) on the paraboloid \(36z=4x^2+9y^2\). The cross-section of the paraboloid created by this plane has slope 0 at this point. Find an equation of the plane. (answer)

Ex 14.5.9 Suppose the temperature at \((x,y,z)\) is given by \( T=xy+\sin(yz)\). In what direction should you go from the point \((1,1,1)\) to decrease the temperature as quickly as possible? What is the rate of change of temperature in this direction? (answer)

Ex 14.5.10 Suppose the temperature at \((x,y,z)\) is given by \( T=xyz\). In what direction can you go from the point \((1,1,1)\) to maintain the same temperature? (answer)

Ex 14.5.11 Find an equation for the plane tangent to \( x^2-3y^2+z^2=7\) at \((1,1,3)\). (answer)

Ex 14.5.12 Find an equation for the plane tangent to \( xyz=6\) at \((1,2,3)\). (answer)

Ex 14.5.13 Find an equation for the line normal to \( x^2+2y^2+4z^2=26 \) at \((2,-3,-1)\). (answer)

Ex 14.5.14 Find an equation for the line normal to \( x^2+y^2+9z^2=56\) at \((4,2,-2)\). (answer)

Ex 14.5.15 Find an equation for the line normal to \( x^2+5y^2-z^2=0\) at \((4,2,6)\). (answer)

Ex 14.5.16 Find the directions in which the directional derivative of \(f(x,y)=x^2+\sin(xy)\) at the point \((1,0)\) has the value 1. (answer)

Ex 14.5.17 Show that the curve \({\bf r}(t) = \langle\ln(t),t\ln(t),t\rangle\) is tangent to the surface \(xz^2-yz+\cos(xy) = 1\) at the point \((0,0,1)\).

Ex 14.5.18 A bug is crawling on the surface of a hot plate, the temperature of which at the point \(x\) units to the right of the lower left corner and \(y\) units up from the lower left corner is given by \(T(x,y)=100-x^2-3y^3\).

a. If the bug is at the point \((2,1)\), in what direction should it move to cool off the fastest? How fast will the temperature drop in this direction?

b. If the bug is at the point \((1,3)\), in what direction should it move in order to maintain its temperature? (answer)

Ex 14.5.19 The elevation on a portion of a hill is given by \(f(x,y) = 100 -4x^2 - 2y\). From the location above \((2,1)\), in which direction will water run? (answer)

Ex 14.5.20 Suppose that \(g(x,y)=y-x^2\). Find the gradient at the point \((-1, 3)\). Sketch the level curve to the graph of \(g\) when \(g(x,y)=2\), and plot both the tangent line and the gradient vector at the point \((-1,3)\). (Make your sketch large). What do you notice, geometrically? (answer)

Ex 14.5.21 The gradient \(\nabla f\) is a vector valued function of two variables. Prove the following gradient rules. Assume \(f(x,y)\) and \(g(x,y)\) are differentiable functions.

  • a. \(\nabla(fg)=f\nabla(g)+g\nabla(f)\) b. \(\nabla(f/g)=(g\nabla f - f \nabla g)/g^2\) c. \(\nabla((f(x,y))^n)=nf(x,y)^{n-1}\nabla f\)

14.6: Higher order Derivatives

Ex 14.6.1 Let \( f=xy/(x^2+y^2)\); compute \(f_{xx}\), \(f_{yx}\), and \(f_{yy}\). (answer)

Ex 14.6.2 Find all first and second partial derivatives of \(x^3y^2+y^5\). (answer)

Ex 14.6.3 Find all first and second partial derivatives of \(4x^3+xy^2+10\). (answer)

Ex 14.6.4 Find all first and second partial derivatives of \(x\sin y\). (answer)

Ex 14.6.5 Find all first and second partial derivatives of \(\sin(3x)\cos(2y)\). (answer)

Ex 14.6.6 Find all first and second partial derivatives of \(e^{x+y^2}\). (answer)

Ex 14.6.7 Find all first and second partial derivatives of \(\ln\sqrt{x^3+y^4}\). (answer)

Ex 14.6.8 Find all first and second partial derivatives of \(z\) with respect to \(x\) and \(y\) if \(x^2+4y^2+16z^2-64=0\). (answer)

Ex 14.6.9 Find all first and second partial derivatives of \(z\) with respect to \(x\) and \(y\) if \(xy+yz+xz=1\). (answer)

Ex 14.6.10 Let \(\alpha\) and \(k\) be constants. Prove that the function \(u(x,t)=e^{-\alpha^2k^2t}\sin(kx)\) is a solution to the heat equation \(u_t=\alpha^2u_{xx}\)

Ex 14.6.11 Let \(a\) be a constant. Prove that \(u=\sin(x-at)+\ln(x+at)\) is a solution to the wave equation \(u_{tt}=a^2u_{xx}\).

Ex 14.6.12 How many third-order derivatives does a function of 2 variables have? How many of these are distinct?

Ex 14.6.13 How many \(n\)th order derivatives does a function of 2 variables have? How many of these are distinct?

14.7: Maxima and minima

Ex 14.7.1 Find all local maximum and minimum points of \(f=x^2+4y^2-2x+8y-1\). (answer)

Ex 14.7.2 Find all local maximum and minimum points of \(f=x^2-y^2+6x-10y+2\). (answer)

Ex 14.7.3 Find all local maximum and minimum points of \(f=xy\). (answer)

Ex 14.7.4 Find all local maximum and minimum points of \(f=9+4x-y-2x^2-3y^2\). (answer)

Ex 14.7.5 Find all local maximum and minimum points of \(f=x^2+4xy+y^2-6y+1\). (answer)

Ex 14.7.6 Find all local maximum and minimum points of \(f=x^2-xy+2y^2-5x+6y-9\). (answer)

Ex 14.7.7 Find the absolute maximum and minimum points of \(f=x^2+3y-3xy\) over the region bounded by \(y=x\), \(y=0\), and \(x=2\). (answer)

Ex 14.7.8 A six-sided rectangular box is to hold \(1/2\) cubic meter; what shape should the box be to minimize surface area? (answer)

Ex 14.7.9 The post office will accept packages whose combined length and girth is at most 130 inches. (Girth is the maximum distance around the package perpendicular to the length; for a rectangular box, the length is the largest of the three dimensions.) What is the largest volume that can be sent in a rectangular box? (answer)

Ex 14.7.10 The bottom of a rectangular box costs twice as much per unit area as the sides and top. Find the shape for a given volume that will minimize cost. (answer)

Ex 14.7.11 Using the methods of this section, find the shortest distance from the origin to the plane \(x+y+z=10\). (answer)

Ex 14.7.12 Using the methods of this section, find the shortest distance from the point \((x_0,y_0,z_0)\) to the plane \(ax+by+cz=d\). You may assume that \(c\not=0\); use of Sage or similar software is recommended. (answer)

Ex 14.7.13 A trough is to be formed by bending up two sides of a long metal rectangle so that the cross-section of the trough is an isosceles trapezoid, as in figure 6.2.6. If the width of the metal sheet is 2 meters, how should it be bent to maximize the volume of the trough? (answer)

Ex 14.7.14 Given the three points \((1,4)\), \((5,2)\), and \((3,-2)\), \(\ds(x-1)^2+(y-4)^2+(x-5)^2+(y-2)^2+(x-3)^2+(y+2)^2\) is the sum of the squares of the distances from point \((x,y)\) to the three points. Find \(x\) and \(y\) so that this quantity is minimized. (answer)

Ex 14.7.15 Suppose that \(f(x,y)=x^2+y^2+kxy\). Find and classify the critical points, and discuss how they change when \(k\) takes on different values.

Ex 14.7.16 Find the shortest distance from the point \((0,b)\) to the parabola \(y=x^2\). (answer)

Ex 14.7.17 Find the shortest distance from the point \((0,0,b)\) to the paraboloid \(z=x^2+y^2\). (answer)

Ex 14.7.18 Consider the function \(f(x,y)=x^3-3x^2y+y^3\).

  1. Show that \((0,0)\) is the only critical point of \(f\).
  2. Show that the discriminant test is inconclusive for \(f\).
  3. Determine the cross-sections of \(f\) obtained by setting \(y=kx\) for various values of \(k\).
  4. What kind of critical point is \((0,0)\)?

Ex 14.7.19 Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid \(2x^2+72y^2+18z^2=288\). (answer)

14.8: Lagrange Multipliers

Ex 14.8.1 A six-sided rectangular box is to hold \(1/2\) cubic meter; what shape should the box be to minimize surface area? (answer)

Ex 14.8.2 The post office will accept packages whose combined length and girth are at most 130 inches (girth is the maximum distance around the package perpendicular to the length). What is the largest volume that can be sent in a rectangular box? (answer)

Ex 14.8.3 The bottom of a rectangular box costs twice as much per unit area as the sides and top. Find the shape for a given volume that will minimize cost. (answer)

Ex 14.8.4 Using Lagrange multipliers, find the shortest distance from the point \((x_0,y_0,z_0)\) to the plane \(ax+by+cz=d\). (answer)

Ex 14.8.5 Find all points on the surface \(xy-z^2+1=0\) that are closest to the origin. (answer)

Ex 14.8.6 The material for the bottom of an aquarium costs half as much as the high strength glass for the four sides. Find the shape of the cheapest aquarium that holds a given volume \(V\). (answer)

Ex 14.8.7 The plane \(x-y+z=2\) intersects the cylinder \(x^2+y^2=4\) in an ellipse. Find the points on the ellipse closest to and farthest from the origin. (answer)

Ex 14.8.8 Find three positive numbers whose sum is 48 and whose product is as large as possible. (answer)

Ex 14.8.9 Find all points on the plane \(x+y+z = 5\) in the first octant at which \( f(x,y,z) = xy^2z^2\) has a maximum value. (answer)

Ex 14.8.10 Find the points on the surface \(x^2 -yz = 5\) that are closest to the origin. (answer)

Ex 14.8.11 A manufacturer makes two models of an item, standard and deluxe. It costs \\)40 to manufacture the standard model and \\)60 for the deluxe. A market research firm estimates that if the standard model is priced at \(x\) dollars and the deluxe at \(y\) dollars, then the manufacturer will sell \(500(y-x)\) of the standard items and \(45,000+500(x-2y)\) of the deluxe each year. How should the items be priced to maximize profit? (answer)

Ex 14.8.12 A length of sheet metal is to be made into a water trough by bending up two sides as shown in figure 14.8.3. Find \(x\) and \(\phi\) so that the trapezoid--shaped cross section has maximum area, when the width of the metal sheet is 27 inches (that is, \(2x+y=27\)). (answer)

Figure 14.8.3. Cross-section of a trough.

 

Ex 14.8.13 Find the maximum and minimum values of \(f(x,y,z)=6x+3y+2z\) subject to the constraint \( g(x,y,z) = 4x^2+2y^2 + z^2 - 70 = 0\). (answer)

Ex 14.8.14 Find the maximum and minimum values of \(f(x,y)=e^{xy}\) subject to the constraint \(g(x,y) = x^3+y^3 - 16 = 0\). (answer)

Ex 14.8.15 Find the maximum and minimum values of \( f(x,y) = xy + \sqrt{9-x^2-y^2}\) when \( x^2+y^2 \leq 9\). (answer)

Ex 14.8.16 Find three real numbers whose sum is 9 and the sum of whose squares is a small as possible. (answer)

Ex 14.8.17 Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere. (answer)