Calculus of functions of one and several variables
General Information  Syllabus  HW Assignments  Course Resources 

Announcements:
 New practice problems for the final exam are below.
 Some software for plotting 3d functions has been linked below.
 Notes and software for Taylor polynomials and Taylor error are below.
 Some software for looking at infinite series has been linked below.
Course Resources
Solutions
Software
 Here is a link to some notes about distances between points and planes.
 Here is a link to some code for doing gradient descent
 Here is a link to some code for computing level curves, linearizations, and gradient plots.
 Here is a link to some code for plotting 3d functions!
 Here is a link to some code for computing with Infinite Series and Infinite Series Error.
 Here is a link to some code for visualizing Taylor Error and some notes about computing Taylor Error including the solutions to HW #3.
 Here is a link to some code for visualizing Taylor Polynomials and some notes about computing Taylor Polynomials.
 Here is a link to some code for visualizing Interpolating Polynomials and some notes about Lagrange Interpolation
Documents
Exam Practice
Midterm 1

Exam problems for Chapter 11
 use Taylor's inequality to determine the number of terms of the Maclaurin series for $\sin(x)$ that should be used to estimate $\sin(.1)$ to within $.00001$. Find $\sin(.1)$ to within $.00001$.
 Does the following series converge? If so, find the sum: $$\sum_{n=1}^\infty 3^{n+3}2^{2n2}$$
 Find the radius of convergence of the following power series: $$\sum_{n=2}^\infty \dfrac{(x3)^n}{5^{2n}(n^3+1)}$$
 Find the Taylor series for $\sin(x)$ centered at the point $a=\dfrac{\pi}{4}$.
 Use the integral test to determine whether or not the series converges: $$\sum_{n=2}^\infty \dfrac{1}{n(\ln(n))^2}$$
 Find a formula for the partial sums $s_n$ of the telescoping series below and determine whether the series converges. If it does, find the sum: $$\sum_{n=1}^\infty \dfrac{1}{n(n+2)}$$
 Evaluate the series: $$\sum_{n=1}^\infty n^2x^n$$
 Use the alternating series remainder estimate to find the sum below to within $.0001$: $$\sum_{n=1}^\infty \dfrac{(1)^n}{n\cdot10^n}$$
 Determine whether the following two series converge or diverge:
$$\sum_{n=1}^\infty \dfrac{\sin(n)}{n^2}$$
$$\sum_{n=3}^\infty\dfrac{\ln(n)}{n^2}$$  Determine whether the integral is convergent or divergent and evaluate it if convergent: $$\int_0^1(1+x)^{\frac23}dx$$
 New Practice Problems
 Old Practice Problems #4 and #6
 2015 Exam #2, #3, and #5

Midterm 2

Exam problems for Exam 2
 Find the two unit vectors that are parallel to the tangent line to the parabola $y=x^2$ at the point $(3,9)$.
 If $a=\langle3,0,0\rangle$ find the vector $b$ parallel to $\langle 1,2,1\rangle$ such that $\operatorname{comp}_{a}b=2$.
 Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. If they intersect find the point of intersection. $$L_1:\dfrac{x2}{1}=\dfrac{y3}{2}=\dfrac{z1}{3}$$ $$L_2:\dfrac{x3}{1}=\dfrac{y+4}{3}=\dfrac{z2}{7}$$
 Find the point on the curve $r(t)=\langle 2\cos(t),2\sin(t),e^t\rangle$, $0\leq t\leq\pi$ where the tangent line is parallel to the plane $\sqrt{3}x+y=1$.
 Show that the limits do not exist: $$ \lim_{(x,y)\rightarrow(0,0)}\dfrac{2xy}{x^2+2y^2} \qquad \lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^2y}{x^2+2y^4}$$
 Find the distance from the point $(3,2,5)$ to the plane $x+y+2z=11$.
 If $a$ and $b$ are unit vectors and $a\times b=\langle\frac14,\frac14,\frac{\sqrt{6}}4\rangle$ what are the possible value(s) of $a\cdot b$.
 Find the equation of the plane that is perpendicular to the plane $x3y+2z=6$ and contains the line $r(t)=\langle t+5,2t1,6\rangle$.
 Calculate the length of the curve parametrized by $r(t)=\langle 1,t^2,t^3\rangle$ for $0\leq t\leq 2$.
 A river of width $500m$ flows at a constant velocity of $3km/h$ downstream. A rower can row at $5 km/h$. Determine the velocity vector that will cause her to travel directly across the river (that is, orthogonal to the flow of the river).
 New Practice Problems
 Old Practice Problems
 2015 Exam
Final Exam
Dartmouth Math Department
Last updated January 18, 2017