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Calculus III
Here are a set of practice problems for the Calculus III notes. Click on the "Solution" link for each problem to go to the page containing the solution.
Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section.
Here is a listing of sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.
3-Dimensional Space - In this chapter we will start looking at three dimensional space. This chapter is generally prep work for Calculus III and so we will cover the standard 3D coordinate system as well as a couple of alternative coordinate systems. We will also discuss how to find the equations of lines and planes in three dimensional space. We will look at some standard 3D surfaces and their equations. In addition we will introduce vector functions and some of their applications (tangent and normal vectors, arc length, curvature and velocity and acceleration).
The 3-D Coordinate System – In this section we will introduce the standard three dimensional coordinate system as well as some common notation and concepts needed to work in three dimensions.
Equations of Lines – In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.
Equations of Planes – In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane.
Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids.
Functions of Several Variables – In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.
Vector Functions – In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times.
Calculus with Vector Functions – In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions.
Tangent, Normal and Binormal Vectors – In this section we will define the tangent, normal and binormal vectors.
Arc Length with Vector Functions – In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we’ve already seen.
Curvature – In this section we give two formulas for computing the curvature (i.e. how fast the function is changing at a given point) of a vector function.
Velocity and Acceleration – In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function. For the acceleration we give formulas for both the normal acceleration and the tangential acceleration.
Cylindrical Coordinates – In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see cylindrical coordinates are really nothing more than a very natural extension of polar coordinates into a three dimensional setting.
Spherical Coordinates – In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the two).
Equations of Lines – In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.
Equations of Planes – In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane.
Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids.
Functions of Several Variables – In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.
Vector Functions – In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times.
Calculus with Vector Functions – In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions.
Tangent, Normal and Binormal Vectors – In this section we will define the tangent, normal and binormal vectors.
Arc Length with Vector Functions – In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we’ve already seen.
Curvature – In this section we give two formulas for computing the curvature (i.e. how fast the function is changing at a given point) of a vector function.
Velocity and Acceleration – In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function. For the acceleration we give formulas for both the normal acceleration and the tangential acceleration.
Cylindrical Coordinates – In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see cylindrical coordinates are really nothing more than a very natural extension of polar coordinates into a three dimensional setting.
Spherical Coordinates – In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the two).
Partial Derivatives - In this chapter we’ll take a brief look at limits of functions of more than one variable and then move into derivatives of functions of more than one variable. As we’ll see if we can do derivatives of functions with one variable it isn’t much more difficult to do derivatives of functions of more than one variable (with a very important subtlety). We will also discuss interpretations of partial derivatives, higher order partial derivatives and the chain rule as applied to functions of more than one variable. We will also define and discuss directional derivatives.
Limits – In the section we’ll take a quick look at evaluating limits of functions of several variables. We will also see a fairly quick method that can be used, on occasion, for showing that some limits do not exist.
Partial Derivatives – In this section we will look at the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. without the use of the definition). As you will see if you can do derivatives of functions of one variable you won’t have much of an issue with partial derivatives. There is only one (very important) subtlety that you need to always keep in mind while computing partial derivatives.
Interpretations of Partial Derivatives – In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I). We will also see that partial derivatives give the slope of tangent lines to the traces of the function.
Higher Order Partial Derivatives – In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives.
Differentials – In this section we extend the idea of differentials we first saw in Calculus I to functions of several variables.
Chain Rule – In the section we extend the idea of the chain rule to functions of several variables. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. We will also give a nice method for writing down the chain rule for pretty much any situation you might run into when dealing with functions of multiple variables. In addition, we will derive a very quick way of doing implicit differentiation so we no longer need to go through the process we first did back in Calculus I.
Directional Derivatives – In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives. In addition, we will define the gradient vector to help with some of the notation and work here. The gradient vector will be very useful in some later sections as well. We will also give a nice fact that will allow us to determine the direction in which a given function is changing the fastest.
Partial Derivatives – In this section we will look at the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. without the use of the definition). As you will see if you can do derivatives of functions of one variable you won’t have much of an issue with partial derivatives. There is only one (very important) subtlety that you need to always keep in mind while computing partial derivatives.
Interpretations of Partial Derivatives – In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I). We will also see that partial derivatives give the slope of tangent lines to the traces of the function.
Higher Order Partial Derivatives – In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives.
Differentials – In this section we extend the idea of differentials we first saw in Calculus I to functions of several variables.
Chain Rule – In the section we extend the idea of the chain rule to functions of several variables. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. We will also give a nice method for writing down the chain rule for pretty much any situation you might run into when dealing with functions of multiple variables. In addition, we will derive a very quick way of doing implicit differentiation so we no longer need to go through the process we first did back in Calculus I.
Directional Derivatives – In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives. In addition, we will define the gradient vector to help with some of the notation and work here. The gradient vector will be very useful in some later sections as well. We will also give a nice fact that will allow us to determine the direction in which a given function is changing the fastest.
Applications of Partial Derivatives - In this chapter we will take a look at several applications of partial derivatives. We will find the equation of tangent planes to surfaces and we will revisit on of the more important applications of derivatives from earlier Calculus classes. We will spend a significant amount of time finding relative and absolute extrema of functions of multiple variables. We will also introduce Lagrange multipliers to find the absolute extrema of a function subject to one or more constraints.
Tangent Planes and Linear Approximations – In this section formally define just what a tangent plane to a surface is and how we use partial derivatives to find the equations of tangent planes to surfaces that can be written as \(z=f(x,y)\). We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point.
Gradient Vector, Tangent Planes and Normal Lines – In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line.
Relative Minimums and Maximums – In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points (i.e. neither a relative minimum or relative maximum).
Absolute Minimums and Maximums – In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded (i.e. no part of the region goes out to infinity) and closed (i.e. all of the points on the boundary are valid points that can be used in the process).
Lagrange Multipliers – In this section we’ll see discuss how to use the method of Lagrange Multipliers to find the absolute minimums and maximums of functions of two or three variables in which the independent variables are subject to one or more constraints. We also give a brief justification for how/why the method works.
Gradient Vector, Tangent Planes and Normal Lines – In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line.
Relative Minimums and Maximums – In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points (i.e. neither a relative minimum or relative maximum).
Absolute Minimums and Maximums – In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded (i.e. no part of the region goes out to infinity) and closed (i.e. all of the points on the boundary are valid points that can be used in the process).
Lagrange Multipliers – In this section we’ll see discuss how to use the method of Lagrange Multipliers to find the absolute minimums and maximums of functions of two or three variables in which the independent variables are subject to one or more constraints. We also give a brief justification for how/why the method works.
Multiple Integrals - In this chapter will be looking at double integrals, i.e. integrating functions of two variables in which the independent variables are from two dimensional regions, and triple integrals, i.e. integrating functions of three variables in which the independent variables are from three dimensional regions. Included will be double integrals in polar coordinates and triple integrals in cylindrical and spherical coordinates and more generally change in variables in double and triple integrals.
Double Integrals – In this section we will formally define the double integral as well as giving a quick interpretation of the double integral.
Iterated Integrals – In this section we will show how Fubini’s Theorem can be used to evaluate double integrals where the region of integration is a rectangle.
Double Integrals over General Regions – In this section we will start evaluating double integrals over general regions, i.e. regions that aren’t rectangles. We will illustrate how a double integral of a function can be interpreted as the net volume of the solid between the surface given by the function and the \(xy\)-plane.
Double Integrals in Polar Coordinates – In this section we will look at converting integrals (including \(dA\)) in Cartesian coordinates into Polar coordinates. The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates.
Triple Integrals – In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration. Getting the limits of integration is often the difficult part of these problems.
Triple Integrals in Cylindrical Coordinates – In this section we will look at converting integrals (including \(dV\)) in Cartesian coordinates into Cylindrical coordinates. We will also be converting the original Cartesian limits for these regions into Cylindrical coordinates.
Triple Integrals in Spherical Coordinates – In this section we will look at converting integrals (including \(dV\)) in Cartesian coordinates into Spherical coordinates. We will also be converting the original Cartesian limits for these regions into Spherical coordinates.
Change of Variables – In previous sections we’ve converted Cartesian coordinates in Polar, Cylindrical and Spherical coordinates. In this section we will generalize this idea and discuss how we convert integrals in Cartesian coordinates into alternate coordinate systems. Included will be a derivation of the \(dV\) conversion formula when converting to Spherical coordinates.
Surface Area – In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space.
Area and Volume Revisited – In this section we summarize the various area and volume formulas from this chapter.
Iterated Integrals – In this section we will show how Fubini’s Theorem can be used to evaluate double integrals where the region of integration is a rectangle.
Double Integrals over General Regions – In this section we will start evaluating double integrals over general regions, i.e. regions that aren’t rectangles. We will illustrate how a double integral of a function can be interpreted as the net volume of the solid between the surface given by the function and the \(xy\)-plane.
Double Integrals in Polar Coordinates – In this section we will look at converting integrals (including \(dA\)) in Cartesian coordinates into Polar coordinates. The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates.
Triple Integrals – In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration. Getting the limits of integration is often the difficult part of these problems.
Triple Integrals in Cylindrical Coordinates – In this section we will look at converting integrals (including \(dV\)) in Cartesian coordinates into Cylindrical coordinates. We will also be converting the original Cartesian limits for these regions into Cylindrical coordinates.
Triple Integrals in Spherical Coordinates – In this section we will look at converting integrals (including \(dV\)) in Cartesian coordinates into Spherical coordinates. We will also be converting the original Cartesian limits for these regions into Spherical coordinates.
Change of Variables – In previous sections we’ve converted Cartesian coordinates in Polar, Cylindrical and Spherical coordinates. In this section we will generalize this idea and discuss how we convert integrals in Cartesian coordinates into alternate coordinate systems. Included will be a derivation of the \(dV\) conversion formula when converting to Spherical coordinates.
Surface Area – In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space.
Area and Volume Revisited – In this section we summarize the various area and volume formulas from this chapter.
Line Integrals - In this chapter we will introduce a new kind of integral : Line Integrals. With Line Integrals we will be integrating functions of two or more variables where the independent variables now are defined by curves rather than regions as with double and triple integrals. We will also investigate conservative vector fields and discuss Green’s Theorem in this chapter.
Vector Fields – In this section we introduce the concept of a vector field and give several examples of graphing them. We also revisit the gradient that we first saw a few chapters ago.
Line Integrals – Part I – In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length.
Line Integrals – Part II – In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to \(x\), \(y\), and/or \(z\). We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion.
Line Integrals of Vector Fields – In this section we will define the third type of line integrals we’ll be looking at : line integrals of vector fields. We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z.
Fundamental Theorem for Line Integrals – In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful.
Conservative Vector Fields – In this section we will take a more detailed look at conservative vector fields than we’ve done in previous sections. We will also discuss how to find potential functions for conservative vector fields.
Green’s Theorem – In this section we will discuss Green’s Theorem as well as an interesting application of Green’s Theorem that we can use to find the area of a two dimensional region.
Line Integrals – Part I – In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length.
Line Integrals – Part II – In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to \(x\), \(y\), and/or \(z\). We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion.
Line Integrals of Vector Fields – In this section we will define the third type of line integrals we’ll be looking at : line integrals of vector fields. We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z.
Fundamental Theorem for Line Integrals – In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful.
Conservative Vector Fields – In this section we will take a more detailed look at conservative vector fields than we’ve done in previous sections. We will also discuss how to find potential functions for conservative vector fields.
Green’s Theorem – In this section we will discuss Green’s Theorem as well as an interesting application of Green’s Theorem that we can use to find the area of a two dimensional region.
Surface Integrals - In this chapter we look at yet another kind on integral : Surface Integrals. With Surface Integrals we will be integrating functions of two or more variables where the independent variables are now on the surface of three dimensional solids. We will also look at Stokes’ Theorem and the Divergence Theorem.
Curl and Divergence – In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem and show how the curl can be used to identify if a three dimensional vector field is conservative field or not.
Parametric Surfaces – In this section we will take a look at the basics of representing a surface with parametric equations. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface.
Surface Integrals – In this section we introduce the idea of a surface integral. With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Also, in this section we will be working with the first kind of surface integrals we’ll be looking at in this chapter : surface integrals of functions.
Surface Integrals of Vector Fields – In this section we will introduce the concept of an oriented surface and look at the second kind of surface integral we’ll be looking at : surface integrals of vector fields.
Stokes’ Theorem – In this section we will discuss Stokes’ Theorem.
Divergence Theorem – In this section we will discuss the Divergence Theorem.
Parametric Surfaces – In this section we will take a look at the basics of representing a surface with parametric equations. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface.
Surface Integrals – In this section we introduce the idea of a surface integral. With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Also, in this section we will be working with the first kind of surface integrals we’ll be looking at in this chapter : surface integrals of functions.
Surface Integrals of Vector Fields – In this section we will introduce the concept of an oriented surface and look at the second kind of surface integral we’ll be looking at : surface integrals of vector fields.
Stokes’ Theorem – In this section we will discuss Stokes’ Theorem.
Divergence Theorem – In this section we will discuss the Divergence Theorem.