Math 8
Calculus of functions of one and several variables

Last updated January 18, 2017

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

## Exam Practice

• #### Midterm 1

• Exam problems for Chapter 11

1. 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 $.00001$.

2. Does the following series converge? If so, find the sum: $$\sum_{n=1}^\infty 3^{n+3}2^{-2n-2}$$

3. Find the radius of convergence of the following power series: $$\sum_{n=2}^\infty \dfrac{(x-3)^n}{5^{2n}(n^3+1)}$$

4. Find the Taylor series for $\sin(x)$ centered at the point $a=\dfrac{\pi}{4}$.

5. Use the integral test to determine whether or not the series converges: $$\sum_{n=2}^\infty \dfrac{1}{n(\ln(n))^2}$$

6. 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: $$\sum_{n=1}^\infty \dfrac{1}{n(n+2)}$$

7. Evaluate the series: $$\sum_{n=1}^\infty n^2x^n$$

8. Use the alternating series remainder estimate to find the sum below to within $.0001$: $$\sum_{n=1}^\infty \dfrac{(-1)^n}{n\cdot10^n}$$

9. Determine whether the following two series converge or diverge: $$\sum_{n=1}^\infty \dfrac{\sin(n)}{n^2}$$
$$\sum_{n=3}^\infty\dfrac{\ln(n)}{n^2}$$

10. 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

1. Find the two unit vectors that are parallel to the tangent line to the parabola $y=x^2$ at the point $(3,9)$.

2. If $a=\langle3,0,0\rangle$ find the vector $b$ parallel to $\langle 1,2,-1\rangle$ such that $\operatorname{comp}_{a}b=-2$.

3. Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. If they intersect find the point of intersection. $$L_1:\dfrac{x-2}{1}=\dfrac{y-3}{-2}=\dfrac{z-1}{-3}$$ $$L_2:\dfrac{x-3}{1}=\dfrac{y+4}{3}=\dfrac{z-2}{-7}$$

4. 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 $\sqrt{3}x+y=1$.

5. Show that the limits do not exist: $$\lim_{(x,y)\rightarrow(0,0)}\dfrac{2xy}{x^2+2y^2} \qquad \lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^2y}{x^2+2y^4}$$

6. Find the distance from the point $(3,2,5)$ to the plane $x+y+2z=11$.

7. 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 $a\cdot b$.

8. Find the equation of the plane that is perpendicular to the plane $x-3y+2z=6$ and contains the line $r(t)=\langle t+5,2t-1,6\rangle$.

9. Calculate the length of the curve parametrized by $r(t)=\langle 1,t^2,t^3\rangle$ for $0\leq t\leq 2$.

10. 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