Math 150 Homework

Below is a tentative list of homework problems for Math 150. In general, HW is due at the start of each class, and we will typically cover on the order of one section a day. All problems are worth 10 points; I will drop whatever assignment helps your HW average the most. Homework solutions are available here.

HOMEWORK: Homework problems listed below; suggested problems collected together at the end. Note dates MAY change (original dates are from 2018)

Due Monday, Feb 22 : Handout from first day of class is here Videos: 2/7/20: Sec 1: https://youtu.be/gfeXKy3fk1M
- Read: Section 11.1, 11.2. Use that time to read the material and make sure your Calc I/II is fresh. Feel free to check out the review videos: part 1 part 2. I am trying to get a rotated version posted on glow.
- Video of the week: Fibonacci numbers (click here for more on the Fibonacci numbers).
- Slides on the course mechanics.
- HW problems: (1) What is wrong with the following argument (from Mathematical Fallacies, Flaws, and Flimflam - by Edward Barbeau): There is no point on the parabola 16y = x² closest to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on the parabola is x² + (y-5)². As 16y = x²the distance-squared is f(y) = 16y + (y-5)². As df / dy = 2y+6, there is only one critical point, at y = -3; however, there is no x such that (x,-3) is on the parabola. Thus there is no shortest distance! (2) Compute the derivative of cos(sin(3x² + 2x ln x)). Note that if you can do this derivative correctly, your knowledge of derivatives should be fine for the course. (3) Let f(x) = x² + 8x + 16 and g(x) = x²+2x-8. Compute the limits as x goes to 0, 3 and ∞ of f(x)+g(x), f(x)g(x) and f(x)/g(x).
- Read Uslan's graduation speech, and email me something about it that resonates with you: http://www.graduationwisdom.com/speeches/0018-uslan.htm (do not include this with the HW you hand in to the TAs).
- Think about the following: The sum of two numbers is 8 and their product is 15. What is the sum of their reciprocals? How would you solve this?
- Extra Credit (due 2/12/20): Assume team A wins p percent of their games, and team B wins q percent of their games. Which formula do you think does a good job of predicting the probability that team A beats team B? Why? (a) (p+pq) / (p+q+2pq), (b) (p+pq) / (p+q-2pq), (c) (p-pq) / (p+q+2pq), or (d) (p-pq) / (p+q-2pq). Hand this in to me, NOT to the graders.
Due Friday, February 26:
- Read: 11.2, 11.3, 11.4. NOTE: we will skip sections 11.5, 11.6, 11.7
- HW #2 problems: Section 11.1: Page 823: #9, #18, #38, #42. Section 11.2: Page 833: #1, #39, and also find the cosine of the angle between a = <2, 5, -4> and b = <1, -2, -3>.
Due Monday, March 1:
- Read: 11.4, 11.8. NOTE: we will skip sections 11.5, 11.6, 11.7
- HW #3 problems: Section 11.2: Question 1: The corollary on page 830 states two vectors are perpendicular if and only if their dot product is zero. Find a non-zero vector, say u, that is perpendicular to <1,1,1>. (Extra credit: find another vector perpendicular to <1,1,1> and the vector u that you just found. This extra credit should be written right after this problem, or as part of this problem.) Question 2: Consider a triangle with sides of length 4, 5 and 6. Which two sides surround the largest angle, and what is the cosine of that angle? Section 11.3: Question 3: Find the determinant of the 2x2 matrix \(\left(\begin{array}{cc}1 & 2 \\
  3 & 4 \end{array}\right)\); in other words, we filled in the entries with the numbers 1, 2, 3 and 4 in that order, row by row. Similarly, find the determinant of the 3x3 matrix \(\left(\begin{array}{ccc}
  1 & 2 & 3 \\
  4 & 5 & 6 \\
  7 & 8 & 9
  \end{array}\right)\); in other words, we fill in the numbers by 1, 2, 3, 4, 5, 6, 7, 8, 9. (Extra credit: find a nice formula for the determinant of the n x n matrix where the entries are \(1, 2, \dots, n^2\) filled as above, and prove your claim. This extra credit should be turned in on a separate sheet of paper.) Question 4: Find the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10). FOR FUN -- DO NOT SUBMIT: Here is a great website with 10 excellent commencement speeches. It's worth the time reading these; I particularly liked the one by Uslan (on how it's not enough to just have a good idea, but how to get noticed).
BONUS LECTURE
Due Wednesday, March 3:
- Read: 11.8, 12.1, 12.2.
- Video of the week: light cycle scene from Tron (the original).
- Pictures of the week: lines and art.
- HW #4 problems: Section 11.3: Page 842: #1, #5, #11, #12. Section 11.4: Page 849: #1, #2, #3, #22.
Due Friday, March 5:
- Video: Watch: http://www.youtube.com/watch?v=71_8kHSAE4w (February 21, 2014: Limits, Defn Partial Derivatives)
- Read: 12.3, 12.4, 12.5.
- HW #5: Section 11.8: Page 893: #1, #26. Section 12.2: Page 908: #2, #4, #5, #27, #32. (Extra Credit: Section 11.8: #55.)
Due Monday, March 8: EXAM - IN CLASS
- There is a video to watch of an introductory lecture I gave (http://www.youtube.com/watch?v=g1oj7CIqGM8), and a template (http://web.williams.edu/Mathematics/sjmiller/public_html/math/LaTexMathematica/MathematicaIntroVer6.nb). There are more links to these on my handouts page http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm, For a detailed description of Mathematica see the solutions to HW 5, http://web.williams.edu/Mathematics/sjmiller/public_html/150/hwsolns/HWSolns_Math150_Sp2014.pdf .
- BONUS LECTURE: Programming: Sec 1: https://youtu.be/2xmH6w7xdyo Sec 2: https://youtu.be/2xmH6w7xdyo
Due Wednesday, March 10:
- Video: Watch: http://www.youtube.com/watch?v=S6wiYRCiQhs (March 3, 2014: Derivatives)
- Read 12.4, 12.5, 12.6, 12.7.
- HW #6: Section 12.3: Page 917: #1, #8, #10, #24, #38, #54. Section 12.4: Page 928: #1, #4, #5.

Due Friday, Mar 12:

Video: Watch: http://www.youtube.com/watch?v=L1WUWmjvnE0 (March 5, 2014: Optimization)
Read 12.4, 12.5, 12.6, 12.7. Read my notes on the Method of Least Squares and Sections 12.6, 12.7.
Video of the week: Flatland trailer. (The full movie is available here for class purposes only.) There's also projections of 4-dimensional cubes in our 3-dimensional space.
HW #7: Due Friday, Mar 12: Section 12.4: Page 928: #21, #25, #33, #36, #63 (is this surprising?).

Due Monday, Mar 15:

Video: Watch http://youtu.be/Da0cO905Aj8 (one of my favorite lectures in the course; we will NOT cover this material in class to free up time to do more optimization problems)

Videos of the week: The Battle of Trafalgar: Animated battle clip Facts on the battle Best animated re-enactment I could find (For more on this see the Additional Comments.)

HW #8: Due Wed March 17: Section 12.5: Page 940: #5, #11, #29, #61.

Doing more optimization problems in class

Due Wednesday, March 17:

Video: No video to watch
Read 12.4, 12.5, 12.6 and my notes on the Method of Least Squares.
HW #9: Due Friday March 19: Homework from handout http://www.williams.edu/Mathematics/sjmiller/public_html/105/handouts/MethodLeastSquares.pdf: Exercise 3.3: Consider the observed data \((0,0), (1,1), (2,2)\). Show that if we use (2.10) from the Least Squares handout to measure error then the line \(y=1\) yields zero error, and clearly this should not be the best fit line! Exercise 3.9: Show that the Method of Least Squares predicts the period of orbits of planets in our system is proportional to the length of the semi-major axis to the 3/2 power.

Due Friday March 19:

Read 12.7, 12.8

Video: Watch http://youtu.be/cgIoKYr11sY (March 12, 2014: Chain Rule)

HW #10: Due Wednesday March 24: (1) Page 949: #18: Use the exact value of \(f(P)\) and the differential \(df\) to approximate the value \(f(Q)\), where \(f(x,y)=\sqrt{x^2-y^2}\), with points \(P(13,5)\) and \(Q(13.2,4.9)\). (2) Page 949: #23: Use the exact value of \(f(P)\) and the differential \(df\) to approximate the value \(f(Q)\), where \(f(x,y,z)= e^{-xyz}\) with the points \(P=(1,0,-2)\) and \(Q=(1.02, 0.03, -2.02)\). (3) Briefly describe what Newton's Method is used for, and roughly how it works.
Extra Credit: to be handed in on a separate paper: Let \(f(x) = \exp(-1/x^2)\) if \(|x| > 0\) and 0 if \(x = 0\). Prove that \(f^{(n)}(0) = 0\) (i.e., that all the derivatives at the origin are zero). This implies the Taylor series approximation to \(f(x)\) is the function which is identically zero. As \(f(x) = 0\) only for \(x=0\), this means the Taylor series (which converges for all \(x\)) only agrees with the function at \(x=0\), a very unimpressive feat (as it is forced to agree there).

Due Monday, March 22:

Read 12.8, 12.9

Video: Watch: http://youtu.be/Da0cO905Aj8 (March 10, 2014: Linear Approximation)

NOTE: MOVING TO LATER: Pythagorean Formula: https://youtu.be/EH6PUS2OwUY (slides here)

HW #11: Due Wednesday March 24: Note: the notation for this homework is a bit annoying. For example, imagine we have a function \(f:\mathbb{R}^3\to\mathbb{R}\) and \(x, y, z:\mathbb{R}^2 \to \mathbb{R}\), so we have \(A(u,v) = f(x(u,v), y(u,v), z(u,v))\). If we want to figure out how this compound function changes with \(u\), I prefer to write \(\frac{\partial A}{\partial u}\); however, the book will often overload the notation and write \(\frac{\partial f}{\partial u}\). I think this greatly increases the chance of making an error, and strongly suggest introducing another function name. Page 960: #2: Find \(dw/dt\) both by using the chain rule and by expressing \(w\) explicitly as a function of \(t\) before differentiating, with \(w = \frac{1}{u^2 + v^2}\), \(u = \cos(2t)\), \(v = \sin(2t)\). Page 960: #5: Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = \ln(x^2 + y^2 + z^2)\), \(x = s - t\), \(y = s + t\), \(z = 2\sqrt{st}\). Page 960: #8: Find \(\partial w/\partial s\) and \(\partial w/\partial t\) with \(w = yz + zx + xy\), \(x = s^2 - t^2\), \(y = s^2 + t^2\), \(z = s^2t^2\). Page 960: #34: A rectangular box has a square base. Find the rate at which its volume and surface area are changing if its base is increasing at 2 cm/min and its height is decreasing at 3cm/min at the instant when each dimension is 1 meter. Page 960: #41: Suppose that \(w = f(u)\) and that \(u = x + y\). Show that \(\partial w/\partial x = \partial w/\partial y\).

Due Wednesday March 24:

Read 12.9, 3.1
Twelfth day lecture: Part I: http://youtu.be/FbwGZfkf9P8 Part II: http://youtu.be/RbhK3a308sg (March 14, 2014: Directional Derivatives, Exp Fn, Trig in a Day)
HW#12: Due Friday March 26: Page 971: Question 3: Find the gradient \(\nabla f\) at \(P\) where \(f(x,y) = \exp(-x^2-y^2)\) and \(P\) is (0,0). Page 971: Question 10: Find the gradient \(\nabla f\) at \(P\) where \(f(x,y,z) = (2x-3y+5z)^5\) and \(P\) is (-5,1,3). Page 971: Question 11: Find the directional derivative of \(f(x,y) = x^2+2xy+3y^2\) at \(P(2,1)\) in the direction \(\overrightarrow{v} = (1,1)\). In other words, compute \((D_{\overrightarrow{u}}f)(P)\) where \(\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|\). Page 971: Question 19: Find the directional derivative of \(f(x,y,z) = \exp(xyz)\) at \(P(4,0,-3)\) in the direction \(\overrightarrow{v} = (0,1,-1)\) (which is \(\textbf{j}-\textbf{k}\)). In other words, compute \((D_{\overrightarrow{u}}f)(P)\) where \(\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|\). Page 971: Question 21: Find the maximum directional derivative of \(f(x,y) = 2x^2+3xy+4y^2\) at \(P(1,1)\) and the direction in which it occurs.

Due Friday, March 26:

Reading: Read 13.1, 13.2
Thirteenth day lecture: http://youtu.be/pgwC2vOwRuE (March 17, 2014: Lagrange Multipliers)
Homework #13: Due Mon Mar 29: Question 1: Use Newton's Method to find a rational number that estimates the square-root of 5 correctly to at least 4 decimal places. Question 2: Let \(w(r,s,t) = f(u(r,s,t), v(r,s,t))\) with \(f(u,v) = u^2 + v^2, u(r,s,t) = t \cos(rs)\) and \(v(r,s,t) = t \sin(rs)\). Find the partial derivatives of \(w\) with respect to \(r\), \(s\) and \(t\) both by direct substitution (which is very nice here!) and by the chain rule. Question 3: Write \((1/2, \sqrt{3}/2)\) in polar coordinates. Question 4: Find the tangent plane to \(z = f(x,y)\) with \(f(x,y) = x^2 y + \sqrt{x+y}\) at \((1,3)\), and approximate the function at \((.9,1.2)\). General comments: These problems are all done the same way. Let's say we have functions of three variables, \(x,y,z\). Find the function to maximize \(f\), the constraint function \(g\), and then solve \(\nabla f(x,y,z) = \lambda \nabla g(x,y,z)\) and \(g(x,y,z) = c\). Explicitly, solve \(\frac{\partial f}{\partial x}(x,y,z) = \lambda \frac{\partial g}{\partial x}(x,y,z)\), \(\frac{\partial f}{\partial y}(x,y,z) = \lambda \frac{\partial g}{\partial y}(x,y,z)\), \(\frac{\partial f}{\partial z}(x,y,z) = \lambda \frac{\partial g}{\partial z}(x,y,z)\), and \(g(x,y,z) = c\). For example, if we want to maximize \(xy^2z^3\) subject to \(x+y+z = 4\), then \(f(x,y,z) = xy^2z^3\) and \(g(x,y,z) = x+y+z = 4\). The hardest part is the algebra to solve the system of equations. Remember to be on the lookout for dividing by zero. That is never allowed, and thus you need to deal with those cases separately. Specifically, if the quantity you want to divide by can be zero, you have to consider as a separate case what happens when it is zero, and as another case what happens when it is not zero. Page 981: Question 1: Find the maximum and minimum values, if any, of \(f(x,y)=2x+y\) subject to the constraint \(x^2+y^2=1\). Page 981: Question 14: Find the maximum and minimum values, if any, of \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint \(x^4+y^4+z^4=3\).

Due Monday, March 29:

Read 13.1, 13.2.
DO NOT WATCH: Fourteenth day lecture: http://www.youtube.com/watch?v=Q1TQtH6POyI (March 19, 2014: Fundamental Theorem of Calculus in a Day)
Optional: Supplemental lecture: Fractals: From Khan to Frozen (March 20, 2014: Mathematics of Fractals and applications to films)

For additional reading on some of the background and related material, see the following links. If you're interested in a math major, I strongly urge you to read these.

Intermediate and Mean Value Theorems and Taylor Series (we'll get to the Taylor series part later in the semester).

Proofs by Induction (as well as other items, including the notes above!).

Homework #14: Due Wed Mar 31: Page 981: Question 19: Find the point on the line \(3x+4y=100\) that is closest to the origin. Use Lagrange multipliers to minimize the SQUARE of the distance. Page 981: Question 35: Find the point or points of the surface \(z=xy+5\) closest to the origin. Page 981: Question 51: Find the point on the parabola \(y= (x-1)^2\) that is closest to the origin. Note: after some algebra you'll get that \(x\) satisfies \(2(x-1)^3+x=0\) (depending on how you do the algebra it may look slightly different). You may use a calculator, computer program, ... to numerically approximate the solution.

Due Wednesday, March 31:

Exam in class.... Bring in Problem #1 (to be done closed book at home)

Due Friday, April 2:

Read 13.2, 13.3, 13.4.
Fifteenth day lecture: http://youtu.be/N8nFFWG_6J4 (April 7, 2014: Integration in two variables)
Homework #15: Due Monday, April 5: Page 1004: Question 15: Evaluate \(\int^3_0 \int^3_0 (xy+7x+y) dx dy\). Page 1004: Question 24: Evaluate \(\int^1_0 \int^1_0 e^{x+y} dx dy\). Page 1004: Question 25: Evaluate \(\int^\pi _0 \int^\pi _0 (xy+\sin x) dx dy\). Page 1005: Question 37: Use Riemann sums to show, without calculating the value of the integral, that \(0\leq \int^\pi_0 \int^\pi_0 \sin \sqrt{xy}dxdy\leq \pi^2\). Extra credit: Let \(G(x) = \int_{t = 0}^{x^3} g(t) dt\). Find a nice formula for G'(x) in terms of the functions in this problem.

Due Monday, April 5:

Read 13.3, 13.4. We will not cover 13.5 or 13.6 (though we will of course discuss triple integrals).
Sixteenth day lecture: http://youtu.be/G9d9lcYevnM (April 9, 2014: Iterated integrals, changing order)
Video of the week: Coin Sorting. (We talked about this when doing the proof of the Fundamental Theorem of Calculus -- you can integrate by adding as you go, or by grouping by value).
Play with Mathematica (or go online to http://www.wolframalpha.com/).
Handout with correctly worked example from Monday's class on vertically/horizontally simple region
Here are some more problems (with solutions) in setting up double integrals.
Homework #16: Due Wednesday, April 7: Page 1011: Question 4: Evaluate \(\int_0^2 \int_{y/2}^1 (x+y) dxdy\). Page 1012: Question 11: Evaluate \(\int_0^1 \int_0^{x^3} \exp(y/x)dydx\). Additional Problem: Let \(f(x)=x^3-4x^2+ \cos(2x^3)+ \sin(x+1701)\). Find a finite \(B\) such that \(|f'(x)| \leq B\) for all \(x\) in \([2,3]\). Page 1011: #13: Evaluate the iterated integral \(\int_0^3 \int_0^y \sqrt{y^2 + 16}\ dx\ dy. \). Page 1011: #25: Sketch the region of integration for the integral \(\int_{-2}^2 \int_{x^2}^4 x^2y\ dy\ dx. \) Reverse the order of integration and evaluate the integral. Page 1011: #30: Sketch the region of integration for the integral \(\int_{0}^1 \int_{y}^1 \exp(-x^2)\ dx\ dy. \). Reverse the order of integration and evaluate the integral. Additional Problem: Give an example of a region in the plane that is neither horizontally simple nor vertically simple.

Due Wednesday, April 7:

Read 13.7: Just know the statements of cylindrical and spherical change of variables
Seventeenth day lecture: http://youtu.be/JP78q_ri-4o (April 11, 2014: Polar Change of Variables, Circles and Spheres)
Video of the week: Coin Sorting.
Play with Mathematica (or go online to http://www.wolframalpha.com/).
Here are some more problems (with solutions) in setting up double integrals.
Homework #17: Due Friday, April 9: Page 1018: #13: Find the volume of the solid that lies below the surface \(z=f(x,y)= y+e^x\) and above the region in the \(xy\)-plane bounded by the given curves: \(x=0\), \(x=1\), \(y=0\), \(y=2\). Page 1018: #42: Find the volume of the solid bounded by the two paraboloids \(z=x^2+2y^2\) and \(z=12-2x^2-y^2\). Page 1026; #13: Evaluate the given integral by first converting to polar coordinates: \(\int_0^1 \int_0^{\sqrt{1-y^2}} \frac{1}{1+x^2+y^2} dx dy.\). Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8 dx dy\). Additional Question 1: Find \(\int_{y = 0}^1 \int_{x=-y}^y x^9 y^8 dx dy\). Page 1026: Question 4: Evaluate \(\int_{-\pi/4}^{\pi/4} \int_0^{2\cos2\theta} r drd\theta\). Additional Question 2: Evaluate \(\int_0^1\int_{-y}^y \sin(xy) \cdot \exp(x^2y^2) dxdy\). Hint: in what way is this similar to an earlier problem on this homework assignment? Additional Question 3: Let \(f(x,y,z) = \cos(xy + z^2)\). Find D\(f(x,y,z)\). Additional Question 4: Find the maximum value of \(f(x,y) = xy\) given that \(g(x,y) = x^2 + 4 y^2 = 1\).

Due Friday, April 9:

Baseball lecture
Eighteenth day lecture: http://youtu.be/Nz7ahXOMTus (April 14, 2014: Monte Carlo Integration, Change of Variables for Ellipses)

Twenty-first day lecture: OPTIONAL: http://youtu.be/MN80UyR2Rh8 (April 21, 2014: Gamma Function and Calc I/II/III review)

Read: 13.9. Know the definition of Jacobian determinants. Read the statement of the Change of Variables formula. We will not deal with this theorem in its full generality, but I want you to at least be aware of its statement. We will concentrate on several special cases: polar coordinates, cylindrical coordinates, and spherical coordinates. If you want more information, click here for my handout on the Change of Variable formula.

Due Monday, April 12: https://www.youtube.com/watch?v=qVmHBnisZW4 https://www.youtube.com/watch?v=Z9lcdk6hLOU

Nineteenth day lecture: http://youtu.be/t3Pt4E1BeUTwt (April 16, 2014: Cylindrical and Spherical Coordinates, Newton's Shell Theorem)
Twenty-second day lecture: OPTIONAL: http://youtu.be/tZ9lcdk6hLOUt (April 23, 2014: Review Class: Perimeter Ellipse, Word Problems, Viewing Algebra)
Read: 13.9. Know the definition of Jacobian determinants. Read the statement of the Change of Variables formula. We will not deal with this theorem in its full generality, but I want you to at least be aware of its statement. We will concentrate on several special cases: polar coordinates, cylindrical coordinates, and spherical coordinates. If you want more information, click here for my handout on the Change of Variable formula.
Homework: #18: Due Wednesday, April 14: Page 1056: #37a. Use spherical coordinates to evaluate the integral \(I \ = \ \int \int \int_B \exp(-\rho^3)\ dV\) where \(B\) is the solid ball of radius \(a\) centered at the origin. Page 1056: #37b. Let \(a \rightarrow \infty\) in the result of part (a) to show that \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-(x^2 + y^2 + z^2)^{3/2})\ dx\ dy\ dz = \frac{4}{3}\pi.\)

Due Wednesday, April 14:

Twenty-third day lecture: http://youtu.be/aigdKmu-5ow (April 28, 2014: Geometric and Harmonic Series, Memoryless Processes)

Read multivariable calculus (Cain and Herod) and my lecture notes.
Homework #19: Due Friday April 16: THIS ASSIGNMENT IS ENTIRELY EXTRA CREDIT! IT INVOLVES YOU WATCHING THE VIDEO AND DOING THESE PROBLEMS. IT IS OPTIONAL. Page 1071: Solve for \(x\) and \(y\) in terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\) with \(u = x - 2y, v = 3x + y\). Page 1071: #3: Solve for \(x\) and \(y\) in terms of \(u\) and \(v\), and compute the Jacobian \(\partial(x,y)/\partial(u,v)\) with \(u = xy, v = y/x\).

Due Friday, April 16:

Twenty-fourth day lecture: http://youtu.be/6bf9fjwMs2o (May 2, 2014: Comparison Test, Implications of Limits of Terms)

Twenty-fifth lecture: https://youtu.be/fa4X1AcGFOA (May 2, 2014: Comparison Test, Implications of Limits of Terms: II)

Read multivariable calculus (Cain and Herod) and my lecture notes.
Homework: no written HW due.

Due Monday, April 19:

NO CLASS DUE TO MIDTERM 3 - Exam due at the start of Friday's class

Twenty-sixth lecture: Watch before start of your section on Monday April 19th: https://youtu.be/jFgCKfUTOQ8

Read multivariable calculus (Cain and Herod) and my lecture notes.

Homework #20: Due Friday April 23: Problem 1. Give an example of a sequence \(\{a_n\}_{n = 1}^{\infty}\) that diverges. Problem 2. Give an example of a sequence of distinct terms \(a_n\) such that the sequence \(\{a_n\}_{n = 1}^\infty\) converges. Problem 3. Give an example of a sequence of distinct terms \(a_n\) such that \(|a_n| < 2013\) and the sequence \(\{a_n\}_{n = 1}^\infty\) does not converge. Problem 10-4 (Cain-Herod): Find the limit of the sequence \(a_n = 3/n^2\), or explain why it does not converge. Problem 10-5 (Cain-Herod): Find the limit of the sequence \(a_n = \frac{3n^2+2n-7}{n^2}\).

Wednesday April 21: Mental Health Day (aka Spring Break)

Friday April 23: Review Day (class optional, consider it an extended mental break day)

Due Monday, April 26:

Twenty-fifth day lecture: http://youtu.be/u3BbXey5dVY (May 5, 2014: Ratio Test, Patton Movie Clip)
Read multivariable calculus (Cain and Herod) and my lecture notes.

Due Wednesday, April 28:

Twenty-sixth day lecture: http://youtu.be/ujJbUpCab6M (May 7, 2014: Root Test, Integral Test)
Read multivariable calculus (Cain and Herod) and my lecture notes. (You should have already done this).

For Taylor series, see my handout here (essentially just pages 2 and 3).

Homework #21: Dues Friday, April 30: (1) Cain-Herod: Find the limit of the series \(\sum_{n=1}^\infty \frac{1}{3^n}\). (2) Cain-Herod: Find a value of \(n\) that will insure that \(1+1/2+1/3+\cdots+1/n > 10^6\). Prove your value works. (3) Cain-Herod: Question 14: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2e^k+k}\) converges or diverges. (4) Cain-Herod: Question 15: Determine if the series \(\sum_{k=0}^\infty \frac{1}{2k+1}\) converges or diverges. (5) Let \(f(x)=\cos x\), and compute the first eight derivatives of \(f(x)\) at \(x=0\), and determine the \(n\)-th derivative.

Due Friday, April 30:

Twenty-seventh day lecture: http://youtu.be/yr01SLw9t4c (May 12, 2014: Taylor Series)
For Taylor series, see my handout here (essentially just pages 2 and 3).
Homework #22: Dues Monday May 3: (1) Cain-Herod 10-18: Is the series \(\left(\sum_{k=0}^n\frac{10^k}{k!}\right)\) convergent or divergent? (2) Cain-Herod 10-21: Is the following series convergent or divergent? \(\sum_{k=1}^n \frac{3^k}{5^k(k^4+k+1)}\). (3) Let \(a_n = \frac{1}{(n \ln n)}\) (one divided by \(n\) times the natural log of \(n\)). Prove that this series diverges. \emph{Hint: what is the derivative of the natural log of \(x\)? Use \(u\)-substitution.} (4) Let \(a_n = \frac{1}{ (n\ln^2 n)}\) (one divided by n times the square of the natural log of \(n\)). Prove that this series converges. \emph{Hint: use the same method as the previous problem. (5) Give an example of a sequence or series that you have seen in another class, in something you've read, in something you've observed in the world, ....

Due Monday, May 3: Review Class: These videos will also be the review videos for Friday May 12

Video on logarithms: https://youtu.be/-SsbkPaB6j8

Here are some videos you can watch to help review.

Review: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw

Review: Spherical Integral, Changing Orders, Series: https://youtu.be/Jgb3z3E58Zw

Review: Change of Variables, Odd/Even Functions, Bounding Functions, Comparison Test: http://youtu.be/8EZQKzoWxEo

Review: Mostly series test (one series and we apply all four tests!): http://youtu.be/HANb4mVLOOc

Due Wednesday, May 5: not assigned 2018

Twenty-ninth day lecture: http://youtu.be/4OcxtpxuSJw (May 14, 2014: Special Series, Alternating Series, Pi formulas, Birthday Problem: Not doing 2018)
For Taylor series, see my handout here (essentially just pages 2 and 3).

Due Friday, May 7: NO CLASS - MENTAL HEALTH DAY

Due Monday, May 10: Class is optional.

For Taylor series, see my handout here (essentially just pages 2 and 3).

Optional advanced material: Watch whatever you want and come to class to discuss. Class is optional.

Green's Theorem in a Day: https://youtu.be/Iq-Og1GAtOQ

Streaming Video: https://www.youtube.com/watch?v=WjnHNHPhJtU

M&M game: Lecture from Probability: M&M Game: https://youtu.be/DFobthridWI (slides) (paper)

Twenty-eighth day lecture: http://youtu.be/UfL8o_IetYQ (May 12, 2014: Review by Partitioning: Not doing 2018)

Twenty-ninth day lecture: http://youtu.be/4OcxtpxuSJw (May 14, 2014: Special Series, Alternating Series, Pi formulas, Birthday Problem)

Due Wednesday, May 12:

Watch: Green's Theorem in a Day: https://youtu.be/Iq-Og1GAtOQ
Homework #23: Due Friday, May 12: Cain-Herod: Question 20: Does the series \(\sum_{n=1}^\infty \frac{3^{2k+1}}{10^k}\) converge or diverge? Additional Question 1: Compute the first five terms of the Taylor series expansion of \(\ln (1-x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 2: Compute the first five terms of the Taylor series expansion of \(\ln (1+x)\) (the natural logarithm of x) about \(x = 0\), and conjecture the answer for the full Taylor series. Additional Question 3: Give an example of a sequence or series you like. Additional Question 4: Find the second order Taylor series expansion of \(\cos(xy)\) about \((0,0)\). Additional Question 5: Find the second order Taylor Series expansion of \(\cos(\sqrt{x+y})\) about \((0,0)\). Additional Question 6: Find the second order Taylor series expansion of \(\cos(x^3 y^4)\) about \((0,0)\). Extra Credit 1: Give a product of infinitely many distinct, positive terms such that the product converges to a number \(c\) with \(0 < c < \infty\). Extra Credit 2: Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive numbers such that \(\sum_{n = 1}^\infty 1/a_n\) converges. Let \(B_n = 1/n \sum_{k = 1}^n a_k\). Prove that \(\sum_{n = 1}^\infty 1/B_n\) converges.

Due Friday, May 14: REVIEW CLASS

Homework: Due at the start of class on Monday May 15: Take-home part of the final

Here are some videos you can watch to help review.

Video on logarithms: https://youtu.be/-SsbkPaB6j8 (slides here)
Review: Spherical Integral, Lagrange Multipliers, Level Sets: https://youtu.be/Jgb3z3E58Zw

Review: Spherical Integral, Changing Orders, Series: https://youtu.be/Jgb3z3E58Zw

Review: Change of Variables, Odd/Even Functions, Bounding Functions, Comparison Test: http://youtu.be/8EZQKzoWxEo

Review: Mostly series test (one series and we apply all four tests!): http://youtu.be/HANb4mVLOOc

Due Monday, May 17: Part 1 of the Final (in class)

Due Wednesday, May 19: Part 2 of the Final (in class)

DATES FROM HERE ON HAVE NOT BEEN UPDATED.

The following are almost surely the assignments, but the dates will change as these are from 2011.

Due Friday, May 13, 2011

If you plan on coming to the lecture, a good summary is my lecture notes on Green's theorem.
The video of the week is a TED lecture (these are great talks, and worth hearing). We'll do Malcolm Gladwell on Spaghetti Sauce (but only a snippet starting around 5 minutes).

Suggested Problems and Extra Credit Problems for Math 105: The suggested problems are not to be turned in, but are for your own personal edification or for additional practice, though of course I and the TAs are happy to chat about these (or any) problems. If you submit an extra credit problem, please clearly mark that it is an extra credit problem.

Introduction: THREE Extra Credit Problems: (1) Let N be a large integer. How should we divide N into positive integers a_i such that the product of the a_i is as large as possible. Redo the problem when N and the a_i need not be integers. (2) Without using any computer, calculator or computing by brute force, determine which is larger: e^π or π^e. (In other words, find out which is larger without actually determining the values of e^π or π^e). If you're interested in formulas for π, see also my paper A probabilistic proof of Wallis' formula for π, which appeared in the American Mathematical Monthly (there are a lot of good articles in this magazine, many of which are accessible to freshmen). (3) Prove that the product of the slopes of two perpendicular lines in the plane that are not parallel to the coordinate axes is -1. What is the generalization of this to lines in three-dimensional space? What is the analogue of the product of the slopes of the line equaling -1?
Section 11.1: Page 823: Is #38 true for all points (i.e., if you take any three points in the plane)?
Section 11.2: Page 833: #59, #61.
Section 11.3: Page 842: #7, #17a.
Section 11.4: Page 849: #25, #54, #58, #60.
Section 11.8: Page 893: #33, #53. Extra Credit: #55.
Section 12.2: Page 908: #41, #43, #45.
Section 12.3: Page 917: #41, #51, #55.
Section 12.4: Page 981: #55, #57, #58, #68.
Section 12.5: Page 940: #10, #17, #46.
Section 12.6: Extra Credit: Let f(x) = exp(-1/x²) if |x| > 0 and 0 if x = 0. Prove that f⁽ⁿ⁾(0) = 0 (i.e., that all the derivatives at the origin are zero). Show this implies the Taylor series approximation to f(x) is the function which is identically zero. As f(x) = 0 only for x=0, this means the Taylor series (which converges for all x) only agrees with the function at x=0, a very unimpressive feat (as it is forced to agree there).
Section 12.7: Page 960: #38, #53.
Section 12.8: Page 971: #29, #40, #41, #60.
Section 12.9: Page 981: #36, #37, #47, #49, #62 (important).
Section 13.1: Page 1004: #33.
Section 13.2: Page 1011: #41, #44, #49.
Section 13.3: Page 1018: #29.
Section 13.4: Page 1026: #7, #34.
Section 13.7: Page 1056: #47, #48. (Extra credit for solving both of these.)
Section 13.9: Page 1070: #10, #28, #29.
From multivariable calculus (Cain and Herod): Exercise 1 (page 10.3). Extra credit: Find a series where the ratio test provides no information on whether or not it converges but the root test says whether or not it converges or diverges.
Problems leading up to Green's Theorem TBD.