Syllabus

WeBWork (online homework)

(Step 1 due 27 October) **Presentation Project**

**Homework**

__For Exam 1 (all due by 18 September 2023)__

Homework 1: Review of functions

Homework 2: Limits (not limits as $x\rightarrow \pm\infty$)

Homework 3: Continuity and intermediate value theorem (only continuity part)

__For Exam 2 (all due by 23 October)__

Homework 4: Definition of differentiation and differentiation of polynomials

Homework 5: Nonpolynomial functions and the product and quotient rules

Homework 6: Chain rule and impiicit differentiation

Homework 7: Rates of change and related rates

Homework 8: Increasing and decreasing and concavity and 1st derivative test

__For Exam 3 (all due by 27 November)__

Homework 9: Absolute extrema and 2nd derivative test and applications

Homework 10: Mean Value Theorem

Homework 11: Limits revisited

Homework 12: Antidifferentiation

Homework 13: Riemann sums

**Quizzes**

__For Exam 1__

**Quiz 1** (29 Aug 2023) [soln]: #1: Solve $x^2+3x+1=0$. #2: Solve $2^{x+1}=3^{3x-2}$.

**Quiz 2** (31 Aug 2023) [soln]: Compute limits based on a graph (see pdf)

**Quiz 3** (7 Sep 2023) [soln]: Compute $\displaystyle\lim_{t \rightarrow 2} \dfrac{t^2-t-2}{t^2+3t-10}$.

__For Exam 2__

**Quiz 4** (12 Sep 2023) [soln]: Use the intermediate value theorem to estimate a root of the polynomial $f(x)=x^7-5x^6+x^5-x^2+5x-1$. Your solution should contain at least two steps in the root-finding process. (note: $x=1$ is a root -- don't use that as an endpoint of any interval!)

**Quiz 5** (19 Sep 2023) [soln]: Let $f(t)=2t^2+3t$. Use the limit definition of the derivative to compute $\dfrac{\mathrm{d}f}{\mathrm{d}t}$.

**Quiz 6** (22 Sep 2023) [soln]: Compute $\dfrac{\mathrm{d}}{\mathrm{d}x} \Big[ x^7 - 5x^6 + x^5 - x^2 + 5x -1 \Big]$.

**Quiz 7** () [soln]: Compute $\dfrac{\mathrm{d}}{\mathrm{d}x}[x^3 \sin(x)]$ and $\dfrac{\mathrm{d}}{\mathrm{d}x} e^x\cos(x)$.

**Quiz 8/8.5** (due 13 Oct) [soln]: Find an equation for the tangent line to the function $f(t)=e^{\sin(t)}$ at $t=\frac{\pi}{2}$. Include a drawing of the graph and its tangent line generated by software (e.g. Desmos).

**Quiz 8/8.5** (due 13 Oct) [soln]: Find an equation for the tangent line to the function $f(t)=e^t\sin(t)$ at $t=\dfrac{3\pi}{4}$. Include a drawing of the graph and its tangent line generated by software (e.g. Desmos).

**Quiz 9** (due 13 Oct) [soln]: Find an equation for the tangent line to the so-called "Witch of Agnesi" given by $y=\dfrac{1}{1+x^2}$ (see more here) at the point with $x$-coordinate $2$ (note: you have to find the $y$-value of the point by plugging this $x$ into the equation). Include in your solution a plot of the curve and the tangent line using software (e.g. Desmos).

**Quiz 10** (due 13 Oct) [soln]: Using the technique from the 4 October lecture video, derive the derivative of the $\arcsin$ function, i.e. figure out what $\dfrac{\mathrm{d}}{\mathrm{d}x} \arcsin(x)$ is using implicit differentiation.

**Quiz 11** (due 13 Oct) [soln]: The Stefan-Boltzmann law is an equation that relates the power radiated from a black body (e.g. black hole) in terms of its temperature. It is given by $J=\sigma T^4$ where $J$ represents the total energy radiated per unit surface area of the black body per unit time, $\sigma \approx 5.670$ is a constant called the Stefan-Boltzmann constant, and $T$ represents temperature (measured in Kelvin, $K$).

If the temperature is decreasing at a rate of $3 \dfrac{K}{hr}$, then how is the total energy radiated per unit surface area per unit time changing when the temperature is $T=100K$?

**Quiz 12** (due 13 Oct) [soln]: Find and classify all local extrema of the cubic function $f(x)=2x^3+3x^2-36x+5$.

__For Exam 3__

**Quiz 13** (due 31 Oct) []: The partition function $p$ has asymptotic formula $p(n) \approx \dfrac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2}{3}}\sqrt{n}}$. Use this asymptotic formula to determine the following two limits:
\[\displaystyle\lim_{n\rightarrow\infty} e^{-n}p(n),\]
and
\[\displaystyle\lim_{n\rightarrow\infty} \dfrac{1}{n^2}p(n).\]
**Quiz 14** (due 7 Nov) []: Compute $\displaystyle\sum_{k=3}^7 \big(k^2+k-1\big)$.

**Quiz 15** (due 8 Nov) []: Find left and right Riemann sums for $f(x)=\ln(x)$ over $[1,2]$ using $n=3$ rectangles. Give the final answers as decimals.

**Quiz 16** (due before break) []: Find local extrema of $f(x) = \displaystyle\int_0^x \dfrac{t^2+2t+1}{1+\sin^2(t)} \mathrm{d}t$.

**Exams**

Exam 1 (15 September) [soln]

Exam 2 (20 October)

Exam 3 (17 November)