• See here for the syllabus.

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Homework

  • Week 13 (Drop off at my office Monday 5/5) Section 10.1 #9, 10, 12, 13, 21, 36, 37; Section 10.6 #1, 2, 9; Section 10.7 #1, 2, 3, 17, 26.

    Writing: Pick one of the final exam problems and write a full explanation for it. (Extra Credit) Do the same for two more problems.

  • Week 12 (Due Monday 4/28) Section 7.7 #2, 3, 5, 10, 12; Friday’s (4/25) worksheet Slopes #1–4, Area #1–3.

    Writing: Read section 7.10, about how to find the length of a curve given by a polar function. Write up an intuitive explanation for the formula given in that section. A picture goes a long way, but you should also use words! (Extra Credit) Give a rigorous, symbolic explanation for the formula.

  • Week 10 (Due Monday 4/14) Section 9.7 #1, 2, 5, 6, 27; Section 9.8 #1, 5, 12, 17, 18, 23; Section 9.10 #9, 10, 12; Section 9.11 #3, 7, 8, 11, 25.

    Writing: Derive the power series for $\sin x$ and $\cos x$. (Extra Credit) Use the formula $e^{iy} = \cos y + i \sin y$ to derive the angle sum formulas for sine and cosine.

  • Week 9 (Due Monday 4/7) Section 9.4 #2, 4, 5, 7, 23, 33, 40; Section 9.5 #1, 2, 4, 10, 11, 12, 15; Section 9.6 #1, 2, 4, 10, 11, 12, 15.

    Writing: Pick one of the convergence rules from this week and explain why it works. (Extra Credit) Pick two more convergence rules from this week and explain why they work.

  • Week 8 (Due Monday 3/31) Section 9.1 #9, 10, 12, 16, 27; Section 9.2 #1, 2, 5, 8, 14, 16; Section 9.3 #1, 3, 4, 6, 7.

    Writing: Show that the repeating decimal $0.142857142857\ldots$ is a rational number and calculate its value. [Hint: can you write it as a geometric series?] (Extra Credit) Prove the sum, difference, and constant rules for series.

  • Week 7 (Due Monday 3/24) Section 7.5 #2, 3, 5, 9, 22; Section 7.6 #1, 2, 9, 10

    Writing: Find the surface area generated by rotating the ellipse $x^2 + 4y^2 = 1$ around the $x$-axis. (Extra Credit) Do the same thing for the generic ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

  • Week 6 (Due Monday 3/10) Section 7.4: #1, 2, 3, 5, 11, 29, 34; Section 8.8: #1, 2, 12, 13, 21, 25

    Writing: Can you use integration by parts to evaluate $\int x^{1000}e^x \,\mathrm{d}x$? If so, explain what the process would be and why you know your process would eventually give an answer. If not, explain why the process won’t ever give an answer. (Please don’t actually try to compute this integral!) Do the same for $\int e^x \ln x \,\mathrm{d}x$. (Again, please don’t try to actually compute it!) Extra Credit: Consider the following seemingly paradoxical argument and explain what’s going on:

    “Use integration by parts on the integral $\int \frac{\mathrm{d}x}{x}$, with $u = \frac 1x$ and $\mathrm d v = \mathrm d x$. We get that $\mathrm d u = -\frac{1}{x^2}$ and $v = x$. Plugging those into the integration by parts formula and simplifying we get $\int \frac{\mathrm{d}x}{x} = 1 + \int \frac{\mathrm{d}x}{x}$. Cancel out the integrals on the two sides and conclude $0 = 1$.”

  • Week 4 (Due Wednesday 2/26) Section 6.6: (only find the mass and center of mass) #1, 2, 3, 39, 40, 41, 42; Section 6.7: #1, 2, 3, 6, 14, 16, 23, 33.

    Writing: The exponential distribution with parameter $\lambda$ is the probability distribution with density function $\rho_\lambda(x) = \lambda e^{-\lambda x}$, where $x \ge 0$. Confirm this really is a probability density function and determine its mean. [Hint: what is the derivative of $-e^{-x} - xe^{-x}$?] (Extra Credit) The Cauchy distribution with parameter $\gamma$ is the probability distribution with density function $\rho_\gamma(x) = \frac{1}{\pi\gamma[1 + (\frac{x}{\gamma})^2]}$, where $x$ can be any value. Confirm that this really is a probability density function and determine its mean. Explain.

  • Week 3 (Due Monday 2/17) Section 6.3: #1, 2, 3, 5; Section 6.4 #1, 2, 4, 13, 14; Section 6.5 #1, 2, 9, 10, 16.

    Writing: Using integration, give an explanation to justify the formula $A = 4\pi r^2$ for the surface area of a sphere of radius $r$. (Extra Credit) Write a computer program which uses numerical integration (Riemann sums, trapezoidal rule, or Simpson’s rule) to approximately calculate arc lengths of curves $y = f(x)$.

  • Week 2 (Due Monday 2/10) Section 6.1: #1, 2, 5, 6; Section 6.2 #1, 2, 3, 6, 10, 12, 27, 28, 47, 49, 50

    Writing: Using integration, give an explanation to justify the formula $V = \frac{\pi}{3}r^2 h$ for the volume of a circular cone of radius $r$ at the base and height $h$. (Extra Credit) A regular tetrahedron is the solid with four faces, each an equally sized equilateral triangles. Determine a formula for the volume of a tetrahedron with side length $s$, and give an explanation to justify it.

  • Week 1 (Due Monday 2/3) Extra Problems Page 41: #9, 12, 14, 17; Extra Problems Page 103: #31, 32, 34; Extra Problems Page 234: #7, 9, 10, 29, 30.

    Writing: Write the statement of the fundamental theorem of calculus (both parts!) and give an informal explanation for why it is true. (Extra Credit) The Transfer Principle says that any equality or inequality about real functions true in the reals is also true in the hyperreals. We could formulate the same principle but going from the rational numbers to the real numbers: “Any equality or inequality about rational functions true in the rationals is also true in the reals.” Is this rational to real transfer principle true or false? Justify your answer

Schedule

This course is organized into three units. Units 1 and 2 each end in a midterm, while unit 3 ends in the oral final. Homework is assigned weekly.

The schedule below is tentative; we might have small adjustments in the dates. For each week I’ve included which sections from the textbook we will be covering.

  • Unit 1 (1/27–2/28): Applications of integration

    • Week 1: Intro, review

    • Week 2: Volume [6.1, 6.2]

    • Week 3: Arc length, surface area, averages [6.3, 6.4, 6.5]

    • Week 4: Applications to physics, improper integrals [6.6, 6.7]

    • Week 5: Overspill, review, exam

  • Unit 2 (3/3–4/16): Advanced integration, series

    • Week 6: Integration by parts, rational functions [7.4, 8.8]

    • Week 7: Trig integrals, trig substitution, [7.5, 7.6]

    • Spring break!

    • Week 8: Intro to sequences and series [9.1, 9.2, 9.3]

    • Week 9: Tests for convergence and divergence [9.4, 9.5, 9.6]

    • Week 10: Power series [9.7, 9.8, 9.10, 9.11]

    • Week 11: Review, exam

  • Unit 3 (4/21–5/7): Calculus in other coordinates

    • Week 12: Polar calculus [7.7, 7.8, 7.9]

    • Week 13: Parametric calculus [10.1, 10.6, 10.7]

    • Week 14: Oral finals.

Important dates:

  • Friday, Feb 28: Midterm 1

  • Wednesday, Apr 16: Midterm 2

  • Oral Final: Week of 5/5