MATH 2400 - Calculus 3, Spring 2019

Homework is due once a week, on Thursdays at the start of your recitation section. These problems should be written using complete sentences to explain your steps. Except where noted, problems refer to Stewart's "Calculus Concepts and Contexts", 4th edition. Here is some information about how to do a good job writing up your homework, (contributed by Cliff Bridges). Please always staple your homework and label it with your section number. Late homework will not be accepted.

  • HW 0
    • For Monday of the first week: complete the first two pages of Project 1
    • For Tuesday of the first week: complete pages 3 and 4 of Project 1
    (The first four pages of Project 1 will be collected in class on Thursday of week 1, along with HW 1.)

  • HW 1 (due Thursday, Jan. 17):
    • Read Guidelines for Graphing in 3D.
    • Print Project 1 and do the first four pages. Bring the Project, including the completed and blank pages, to recitation.
    • Section 9.1: 6, 41
      Section 9.2: 30 (see example 7 in 9.2), 35
    • A1) Write inequalities that describe the following 3D regions
      (a) The region between \( z=4 \) and the \(xy\)-plane
      (b) The solid infinite cylinder with radius 6 and whose axis runs along the \(y\)-axis
      (c) The solid portion of a sphere of radius \( 2 \) and center \( (0,0,0) \) contained in the first octant
    • Recommended: Memorize various forms for equations of points, lines, spheres and cylinders. For example, make sure you know the equation of a sphere centered at any point, with any radius. Make sure you know the equation of a cylinder whose main axis runs along any of the three coordinate axes. Perhaps exchange problems with a classmate and get a study group going!

  • HW 2 (due Thursday, Jan. 24):
    • Section 9.3: 42, 44
    • A1) The orthogonal projection of \( \mathbf{a} \) onto \( \mathbf{b}\) is the vector \( \textbf{orth}_\mathbf{b} \mathbf{a} = \mathbf{a} - \textbf{proj}_\mathbf{b} \mathbf{a}\). For each pair of vectors below, draw the four vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\textbf{proj}_\mathbf{b} \mathbf{a} \), and \(\textbf{orth}_\mathbf{b} \mathbf{a}\) on the same set of axes.
      (a) \( \mathbf{a} = \langle 1,2 \rangle, \; \mathbf{b} = \langle 6,1 \rangle \)
      (b) \( \mathbf{a} = \langle 6,1 \rangle, \; \mathbf{b} = \langle 1,2 \rangle \)
      (c) \( \mathbf{a} = \langle -2,3 \rangle, \; \mathbf{b} = \langle 6,1 \rangle \)
    • A2) If \(\mathbf{a}\) and \(\mathbf{b}\) are both nonzero vectors, under what circumstances is \( \textbf{proj}_\mathbf{b} \mathbf{a} = \textbf{proj}_\mathbf{a} \mathbf{b}\)?
    • Section 9.4: 26, 32, 35
    • Optional challenge problems: Section 9.3: 49, 50 and Section 9.4: 39

  • HW 3 (due Thursday, Jan. 31):
    • Section 9.5: 10, 16, 36
      Section 9.6: 15, 27
      Section 9.7: 26, 28, 32
    • A1) For each of the following equations, what type of quadric surface is its graph? Where appropriate, also name the coordinate axis along which the quadric surface lies. You do not have to graph the surfaces.
      (a) \(x^2-4y^2 + 16z^2 = -16\)
      (b) \(x^2-4y^2 + 16z^2 = 0\)
      (c) \(x^2+4y^2 + 16z^2 = 0\)
      (d) \(x^2-y-z^2=0\)
      (e) \(x^2-y+z^2=0\)
    • A2) A solid lies inside the sphere \( x^2 + y^2 + z^2 =25\) and inside the infinite cylinder \( y^2 + z^2 = 9 \). Write inequalities that describe the solid in an appropriate coordinate system.
    • Optional challenge problem: Section 9.6: 30

  • HW 4 (due Thursday, Feb. 7):
    • Section 10.1: 6, 13, 25
      Section 10.2: 2, 26, 32, 38
      Section 10.3: 14, 15
      Section 10.4: 4, 28
    • A1) Do problems 2(d) and 2(e) at the end of Project 3
    • A2) For each shape described below, sketch the shape, give a parametrization of the shape, and state how many parameters you used.
      (a) The curve of intersection between \(x^2+y^2=9\) and \(z=5\)
      (b) The surface on the plane \( z=5 \) inside the cylinder \( x^2 + y^2 = 9 \) and
      (c) The solid between the \(xy\)-plane and \(z=5\) inside the cylinder \(x^2+y^2 = 9\).
      (d) In each of the parametrizations above, how does the number of parameters used compare with the dimension of each object?
    • A3) Write the formula for finding the arclength of a parameterized curve. This formula can also be written as \(\int_{s=0}^{s=L} \,ds\), where \(s\) is the arclength parameterization. In this version of the formula, what does \(L\) represent, and what does \(ds\) represent geometrically?
    • Optional challenge problems: Section 10.2: 41-46

  • HW 5 (due Thursday, Feb. 14):
    • Section 10.5: 20, 22, 23
    • Section 11.1: 22, 28, 46
    • A1) Watch this this 14-minute screencast about the types of functions we study in Calc 3. Use the information in the screencast to answer these questions:
      (a) Is the graph of the function \(f(x,y) = \sqrt{x^2 + 9y^2}\) best described as a curve, a surface, or a solid? Explain your reasoning, and sketch the graph.
      (b) According to the screencast, what is one way to visualize the function \( \mathbf{F}(x,y) = \langle -y,x \rangle \)? Sketch the graph.
    • Section 11.2: 8, 24, 34
    • A2) Use the skills learned in Project 4 to complete the two problems below:
      (a) Find the area of a curved fence, whose base is the upper half of the circle of radius 4 centered at the origin, and whose height is given by the function \(f(x,y) = x^2y\)
      (b) Consider the line integral \(\int_C \rho(x,y)\,ds\), where \(C\) represents a curved wire in the \(xy\)-plane, and \(\rho(x,y)\) represents the density of the wire at any given point. Explain what \(ds\) represents geometrically, what physical meaning \(\rho(x,y)\,ds\) has, and what physical meaning the integral has.
    • Optional challenge problem: Section 10.5: 32

  • HW 6 (due Thursday, Feb. 21):
    • Section 11.3: 12, 43, 58, 70
      Section 11.4: 39, 42, 44
      Section 11.5: 20, 48, 50
    • A1) Does \( \lim \limits_{(x,y)\rightarrow (0,0)} \frac{2x^2y}{\sqrt{ x^2+y^2}}\) exist? Show a full solution.
    • A2) Consider the functions \(f(x,y)=x^2+y^2\), \(\mathbf{g}(t) = \langle t-1,e^t\rangle\) and \(\mathbf{k}(a,b) = \langle b,a \rangle\).
      (a) Is it possible to compose \(f\) and \(\mathbf{g}\) to form \(f \circ \mathbf{g} \)? Why or why not?
      (b) List all the other possible ways to compose the functions listed above.

  • HW 7 (due Thursday, Feb. 28):
    • Section 11.6: 27, 38, 54
    • Section 11.7: 4, 17, 37
    • A1) At what point on the paraboloid \(x=\dfrac{y^2}{9} + \dfrac{z^2}{16}\) is the tangent plane parallel to the plane \( 6x-4y+3z=2\)?
    • A2) We will look at an example of how the extrema of multivariable functions can be less well-behaved than those of single variable functions.
      (a) Draw a continuous scalar function of one variable with exactly two local maxima. Is it possible to draw one that has no local minima? Explain.
      (b) Show that the continuous scalar function of two variables given by \(f(x,y) = -(x^2 - 1)^2 - (x^2 y -x -1)^2\) has two local maxima and no local minima.
      (c) Create a graph of \(f(x,y)\) using Mathematica,'s 3D grapher, or some other tool. Confirm visually that there are two local maxima and no local minima.

  • HW 8 (due Thursday, Mar. 7):
    • A1) We will approximate the extrema of \(f(x,y)=x^2+y^2+xy+x+1\) subject to the constraint \(x^2+4y^2=4\) graphically.
      (a) Open this Desmos demonstration. Notice that the blue oval shape is the graph of \(f(x,y)=c\). Drag the slider for \(c\) back and forth and observe the changes. What do each of the blue oval shapes represent?
    • (b) Turn on the graph of the constraint \(x^2 + 4y^2=4\) by clicking the gray circle to the left of the equation. Using the slider for \(c\), find four approximate values of \(c\) where the blue ovals seem to be tangent to the constraint graph.
      (c) Which of the four values of \(c\) represents the max of \(f(x,y)\)? Which represents the min?
      (d) Using Mathematica,'s 3D grapher, or some other tool, produce a 3D graph of \(f(x,y)\) and the constraint. Print or draw the graph, and then label the max and min on the graph. Submit the marked graph with your homework.
    • Section 11.8: 29
      Section 12.1: 12, 14
      Section 12.2: 24, 32
    • A2) Find the volume of the solid enclosed by the paraboloid \(z=x^2+(y-2)^2+2\) and the planes \(z=1\), \(x=-1\), \(x=1\), \(y=0\), and \(y=3\)
    • Optional challenge problem: Find the global extrema of \(x^2+xy+y^2+3y\) on the disk \( D: x^2+y^2 \leq 9 \). [Hint: Find the extrema on the boundary separately from the extrema in the interior. It's possible to do this problem without Lagrange multipliers!]

  • HW 9 (due Thursday, Mar. 14):
    • Section 12.3: 16, 38, 45, 46, 48, 52, 60
      Section 12.4: 1-5, 24(a), 35
    • A1) You may have learned in earlier calculus classes that the indefinite integral \( \int e^{-x^2} \; dx \) cannot be expressed using elementary functions, so we cannot compute it by hand. But there is a way to demonstrate that the definite integral \( I = \int_{-\infty}^\infty e^{-x^2} \; dx = \sqrt{\pi}\). Let \(I^2 = \left ( \int_{-\infty}^\infty e^{-x^2} \; dx \right ) \left ( \int_{-\infty}^\infty e^{-y^2} \; dy \right )\).
      (a) Write \(I^2\) as a double integral, and then convert to polar coordinates. [Hint: The bounds of integration in polar coordinates will have one "endpoint" that is infinite. Should it be an endpoint of \(r\) or \(\theta\)?]
      (b) Evaluate the integral to find \(I^2\) and conclude that \( \int_{-\infty}^\infty e^{-x^2} \; dx = \sqrt{\pi} \). This is a very important result in probability and statistics!
    • Optional challenge problem: Use a Riemann sum to estimate the volume of the solid below \(z=16-x^2-y^2\) in the first octant. Use polar coordinates and divide the interval for \(r\) into 3 sub-intervals and the interval for \(\theta\) into two intervals. Use midpoints. Draw a picture showing the six sub-solids you found the volume of.

  • HW 10 (due Thursday, Mar. 21):
    • Section 12.5: 16
      Section 12.6: 7, 11, 14
      Section 12.7: 14, 28, 34
    • A1) Let \(E\) be the solid bounded by \( x^2+y^2-z^2=16 \) and the planes \(z=3\) and \(z=-3\).
      (a) Sketch or use technology to draw \(E\).
      (b) Write an iterated integral in rectangular coordinates that gives the volume of \(E\). You do not need to evaluate the integral.
      (c) Which of the 6 possible orders of integration allow you to express the volume of \(E\) as a single iterated integral (as opposed to the sum of iterated integrals)? Explain.
    • A2) If \(R\) is a region in the \(xy\) plane, then what does \(\iint_R 1 \, dA\) represent geometrically? There are at least two possible answers. If \(V\) is a solid in space, then what does \(\iiint_V 1 \,dV\) represent geometrically?
    • A3) Write down the formula for finding the surface area of a parameterized surface. The formula can also be written as \(\iint_R \,dS\). In this version of the formula, what does \(dS\) represent geometrically?

  • HW 11 (due Thursday, Apr. 4):
    • Section 12.8: 2, 4, 26
      Section 12.9: 7, 10, 12, 13, 16, 24
    • A1) Complete problems 5 and 6 from Project 10.
    • A2) Let \(S^1\) be the unit circle \( x^2 + y^2 = 1\) in \(\mathbb{R}^2\). Let \(S^2\) be the unit sphere \(x^2+y^2+z^2=1\) in \(\mathbb{R}^3\). Let \(S^n\) be the unit hypersphere \(x_1^2 + x_2^2 + \cdots + x_{n+1}^2 = 1\) in \(\mathbb{R}^{n+1}\).
      (a) Write an iterated double integral in rectangular coordinates that expresses the area inside \(S^1\). Write an iterated triple integral in rectangular coordinates that expresses the volume inside \(S^2\). Write an iterated quadruple integral in rectangular coordinates that expresses the hypervolume inside \(S^3\).
      (b) Use technology to evaluate the (hyper)volume inside \(S^n\) for \(n=4,5,6\). Which of these 3 hyperspheres has the largest hypervolume?

  • HW 12 (due Thursday, Apr. 11):
    • Section 13.1: 29-32, 34
    • Section 13.2: 18, 40, 46
    • Section 13.3: 11, 25, 29, 30
    • A1) Let \( \mathbf{F} \) be a vector field and let \( f \) be a scalar function with continuous partial derivatives. Determine if each statement below is true or false. Explain your reasoning or give a counterexample where appropriate.
      (a) If \( \oint _C \mathbf{F} \cdot d \mathbf{r}=0 \) for a particular closed curve \( C \), then \( \mathbf{F} \) is a conservative vector field.
      (b) If \(\mathbf{F}(x,y) = \langle P, Q \rangle \) and every line integral of F along a closed curve is zero, then \( Q_x - P_y=0 \) everywhere in \( \mathbb{R}^2 \).
      (c) Let \(C\) be a smooth curve given by the vector \(\mathbf{r}(t)\) for \(a \leq t \leq b\). Then \( \int_C \nabla f \cdot d \mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))\).
      (d) Let \(C\) be a smooth curve given by the vector \(\mathbf{r}(t)\) for \(a \leq t \leq b\). Then \( \int_C f \; ds = f(\mathbf{r}(b)) - f(\mathbf{r}(a))\).
      (e) A line integral of a conservative vector field along a smooth curve depends only on the endpoints of the curve.

  • HW 13 (due Thursday, Apr. 18):
    • Section 13.4: 4, 12, 18, 19
      Section 13.5: 10, 17, 22, 30
    • A1) Calculate the length of the curve \(C\), defined by \(\mathbf{r}(t)=\langle 2\cos(t),2\sin(t)\rangle \) with domain of \(-\pi/2 \leq t \leq \pi/2\). (This will be a good review for Thursday's Project 13)
    • A2) In Project 12, you considered geometric representations for line integrals over scalar fields versus over vector fields.
      (a) Go to the Wikipedia page for line integrals and view the animation for line integrals over scalar fields. Write a brief explanation of the geometric meaning of \( \int_C f \; ds\)
      (b) View the animation for line integrals over scalar fields. Write a brief explanation for the geometric meaning of \( \int_C \mathbf{F} \cdot d \mathbf{r} \).
    • Optional challenge problem: Section 13.4: 31

  • HW 14 (due Thursday, Apr. 25):
    • Section 13.6: 1, 21, 23, 42, 46

  • HW 15 (due Thursday, May 2):
    • Section 13.7: 1, 5, 11, 13
    • Section 13.8: 4, 5, 20, 22, 27
    • A1) On page 973 (Section 13.9) in the textbook, there is a summary of all the variations of the Fundamental Theorem of Calculus you've encountered so far. Write a brief summary - in your own words - of what each of the theorems mean and what they allow us to do in calculus.