Calculus 2: exercises

Grading

This is a set of exercises for Calculus 2: course.

Techniques of integration

For the following integrals, suggest the first step: substitution, by parts, trigonometric identities, etc. (indicate specifically!):

1. $∫\sin 2x\cos 5xdx$
2. $∫2x(1+x²)^{3/2}dx $
3. $∫\sin ⁴x\cos ⁶xdx $
4. $∫(1+x²)^{3/2}dx $
5. $∫\sin ³xdx$

1. Fill in the blanks (marked with $``..."$) in the following table for partial fraction decomposition: $$ \begin{array}{ccc} \text{factors} & \text{multiplicities} & \text{partial fractions} \\ x-5 & 3 & ... \\ & & \\ ... & ... & \dfrac{B_1}{x+1}+\dfrac{B_2}{(x+1)^2} \\ & & \\ x^2+2 & 2 & ... \end{array} $$

2. Evaluate the integral $$\int \ln \frac {x}{x+1}dx, $$ if you know that $\int \ln xdx=x\ln x-x+C.$

1. Sketch the region bounded by the graphs of the equations $x=y²+1,x=y+3$ and determine the area of the region.
2. The region bounded by the graphs of $y=2√x,y=0$, and $x=3$ is revolved about the x-axis. Find the surface area of the solid generated.
3. (a) Set up the Riemann sum for the volume of the solid obtained by revolving the region under the graph of a continuous function $y=f(x)≥0,a≤x≤b,$ about the x-axis, provide an illustration and the integral formula. (b) Use the formula to set up the integral for the volume of the circular cone of radius 1 and height 1 (do not evaluate).
4. Evaluate the indefinite integral $∫\cos t/(\sin ²t+2\sin t+1)dt$.
5. Evaluate the indefinite integral $∫√(1-y²)dy.$
6. Evaluate the indefinite integral $∫\frac{x}{(x+1)(x²+1)}dx.$
7. Evaluate the indefinite integral $∫xe^{2x}dx.$

Sequences

Indicate if the following statements are true or false.

1. The sequence $\{1,1/2,1,1/3,1,1/4,1,...\}$ converges to 0.
2. The sequence $\{(-1)ⁿ\}$ diverges.
3. If the sequence $\{a_{n}\}$ converges then the sequence $\{1/a_{n}\}$ converges.
4. The sequence $a_{n}=\sin n$ diverges to ∞.
5. Every monotone sequence converges.

Series

Indicate if the following statements are true or false.

1. The sequence of partial sums of any series converges.
2. If the series $∑_{n=1}^{∞}a_{n}$ converges then the series $∑_{n=1}^{∞}(-a_{n})$ also converges.
3. $∑_{n=1}^{∞}(-1)ⁿ$ is a geometric series.
4. If the series $∑_{n=1}^{∞}a_{n}$ converges then $|a_{n}|→0$ as $n→∞$.
5. The series $1/7+1/8+1/9+1/10+…$ converges.

1. If the series $∑(-1)ⁿa_{n}$ with $a_{n}>0$ converges absolutely then the series $∑a_{n}$ converges absolutely.
2. The series $∑(-1)ⁿ/n$ converges conditionally.
3. If $\lim_{n→∞}|a_{n+1}/a_{n}|=1$ then the series $∑a_{n}$ converges absolutely.
4. $∑_{n=0}^{∞}x²ⁿ$ is a power series.
5. If a power series $∑c_{n}(x-a)ⁿ$ converges for x=0 and for x=2 then it converges for x=1.
6. $∑_{n=1}^{∞}1/2ⁿ$ is a p-series.
7. If a series converges then $\lim_{n→∞}R_{n}=0$, where $R_{n}$ is the remainder of the series.
8. If $a_{n}>b_{n}>0$ and $∑_{n=1}^{∞}a_{n}$ converges, then $∑_{n=1}^{∞}b_{n}$ also converges.
9. $∑_{n=1}^{∞}(-1)³ⁿ⁺¹1/2ⁿ$ is an alternating series.
10. The series $∑_{n=1}^{∞}(1+1/n)$ converges.

1. Evaluate the limit $\lim _{x}\frac{\ln x}{x²-1}$.
2. Write an expression for the nth term of the sequence $-1/2,3/4,-7/8,15/16,-31/32,....$
3. Find the sum of the series $∑_{n=0}^{∞}(1+2ⁿ)/3ⁿ$.
4. Apply the Integral Test to show that the p-series $∑1/n^{1/3}$ diverges.
5. Test the following series for convergence (including absolute/conditional): $∑\frac{2n^{1/2}}{n²-1}.$
6. Test the following series for convergence (including absolute/conditional): $∑(-1)ⁿ1/√n.$
7. Test for convergence (including absolute/conditional): $∑(-3)ⁿ/n!.$

1. Suppose the series $\sum_{k=1}^{\infty}$diverges$.$ Explain why $a_{k}>0$ for all $k\geq1$ implies that $\lim_{n\rightarrow\infty}S_{n}=\infty,$ where $S_{n}$ are the partial sums of the series.

Function series

1. Find the Taylor polynomial of order 2 centered at $c=π/2$ of the function $f(x)=\sin ²x$. How accurate is this approximation on the interval [0,π]? Answer: $P₂(x)=1-(x-π/2)².$
2. Find the radius and the interval of convergence of the series $∑\frac{(x-1)ⁿ}{√n2ⁿ}$. Answer: $[-1,3)$.
3. Find the Maclauren series of the function $f(x)=x²/(1-x²)$ and its interval of convergence. Answer: $∑_{n=0}^{∞}x²ⁿ⁺²$.
4. Use the fact that $\frac{d}{dx}xe^{x²}= e^{x²}+2x²e^{x²}$ to find the power series representation of the latter function. Answer: $∑_{n=0}^{∞}\frac{2n+1}{n!}x²ⁿ$.

Parametric curves

1. Suppose the parametric curve is given by $x=e^{-t},y=e^{3t}$. Eliminate the parameter, simplify and sketch the graph of the function. Answer: $y=1/x³$.

Indicate if the following statements are true or false.

1. In polar coordinates, $A=(1,\pi /2)$ and $B=(-1,-\pi /2)$ represent the same point.
2. The slope of the polar curve $r=0$ is equal to $0.$
3. The graph of the curve $r=\cos 2\theta $ is a spiral.
4. The parametric curve $x=t^2,y=\sin t$ is bounded.
5. The graph of $r=\theta ^2$ can be represented as a parametric curve in Cartesian coordinates.

3. Let $0<a<1.$ Use polar coordinates to find the area of the region inside the circle $r=1$ and to the right of $x=a.$

2. Suppose that a differential equation has the form (a) $y^{\prime}=f(x),$(b) $y^{\prime}=g(y).$ Explain how you could recognize this fact from the direction field\ of the differential equation.