Calculus II, Final Exam, Fall 1998

The exam consists of nine questions: Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q9. (Note: they are all contained on this page.)

1. (30 pts total) Determine whether each of the following improper integrals converges or diverges. If it converges, find its value.

(a) [15 pts] $$\int^4_2 \frac{2}{x^2 - 9}\,$$dx

(b) [15 pts] $$\int^{\infty}_{2} 3x^5e^{-x^{3}}\,$$dx

2. [35 pts total]

(a) A curve C is given parametrically as x(t) = 3t⁴, y(t) = t³ - 4t; a sketch is shown below for a range of t.

(i) [ 4 pts] Add an arrow to show the direction of increasing t.

(ii) [6 pts] Find both values of t that give the point where C crosses itself.

ANS. $\; t = {\underline{\hspace*{1.0in}}},\, t = {\underline{\hspace*{1.0in}}}$

(iii) [6 pts] Find both values of t that correspond to the points on C where the tangent lines are horizontal.

ANS. $\; t = {\underline{\hspace*{1.0in}}},\, t = {\underline{\hspace*{1.0in}}}$

(iv) [7 pts] Find the equation for either one of the tangent lines to C at the point where C crosses itself.

(b) Consider the polar graph of $r = \frac{1}{\sqrt{2}} + \sin\,\theta, 0 \leq \theta \leq 2\pi$, which is shown below.

(i) [4 pts] Find the two values of $\theta$ where the curve passes through the origin.

ANS. $\; \theta = {\underline{\hspace*{1.0in}}},\, \theta = {\underline{\hspace*{1.0in}}}$

(ii) [8 pts] Find a definite integral for the area inside the small loop. [You need not evaluate the integral.]

3. [30 pts total]

(a) Consider the region R bounded by the curves y = f(x) = 2 -x² and y = g(x) = x.

(i) [3 pts] Sketch R.

(ii) [ 12 pts] Find the coordinates of the centroid of R.

ANS. $\; \bar{x} = {\underline{\hspace*{1.0in}}},\, \bar{y} = {\underline{\hspace*{1.0in}}}$

(b) Express the equations below in terms of rectangular coordinates.

(i) [6 pts] $\,z = r^2\cos\,2\theta$

(ii) [6 pts] $\rho^2\sin\phi\,\cos\phi\,\cos\theta = 1$

4. [40 pts total]

(a) [14 pts] For the series $$\sum^{\infty}_{n = 0}\frac{(-1)^{n+1}(3x + 2)^n}{4^n\sqrt{n+2}}$$, find the interval of absolute convergence.

ANS. $\;{\underline{\hspace*{.75in}}}\, < x < {\underline{\hspace*{.75in}}}$

(b) [8 pts] At the endpoints of the interval of convergence for the series in (a), determine whether the series converges absolutely, converges conditionally, or diverges.

(c) [12 pts] Suppose $f(x) = \sqrt{x+2}$. Find T₃(x), the Taylor polynomial of degree 3 for f(x) about x = 2.

(d) [6 pts] Suppose T(x) is the Taylor series for $f(x) = \sqrt{x+2}$ about x = 2. Based on f(x) and its graph, on what interval of x do you expect to be able to approximate f(x) by T(x)? Justify your answer briefly.

ANS. $\;{\underline{\hspace*{.75in}}} < x < {\underline{\hspace*{.75in}}}$

5. [25 pts total]

(a) A car on an amusement park ride travels along the path x = t³ - 12t² + 41t - 30, y(t) = t² - 10t + 21, which is shown below for a range of t values.

(i) [3 pts] Express the velocity of the car as a 2-D vector.

(ii) [4 pts] What is the speed of the car at t = 1?

ANS. $\;{\underline{\hspace*{2.0in}}}$

(iii) [6 pts] Let P be the point where the path crosses itself. Find a definite integral for the distance the car travels around the loop, stating and ending at P. [You need not evaluate the integral.]

(b) [15 pts] Consider the cardioid $r = 1 + \cos\theta$. Find the arc length of this curve between $\theta = 0$ and $\theta = 2\pi$.

Hint: Recall $$\cos^2(\theta/2) = \frac{1+\cos\,\theta}{2}$$. The cardioid is graphed below.

ANS. $\;L = {\underline{\hspace*{1.0in}}}$

6. [55 points total]

(a) Determine whether each of the following series converges absolutely, converges conditionally, or diverges. [8 pts each]

(i) $$\sum^{\infty}_{n = 1} \frac{(-1)^n n}{3n^2 - 2}$$

(ii) $$\sum^{\infty}_{n = 1} \frac{(-1)^n 2^n\sqrt{n}}{n!}$$

(iii) $$\sum^{\infty}_{n = 1} \frac{(-1)^n n^2}{e^-n}$$

(b) [9 pts each]

(i) The sum of the first five terms of the series $$\sum^{\infty}_{n=1}\frac{(-1)^n 4n}{e^n}$$ is about 0.8280. Give a bound for the error in using this value to approximate the infinite sum.

(ii) Suppose the function f(x) = e^8x is approximated on the interval $2 \leq x \leq 5$ by a Taylor polynomial of degree 5 centered at x = 3. What is the maximum possible error in using the polynomial to approximate values of f(x) on the given interval?

$$[\mbox{HINT}: R_n = \frac{1}{(n + 1)!}f^{(n+1)}(z)(x - a)^{n+1}.]$$

(c) The function $y = f(x) = \frac{1}{3}x^3 + x^2 - 3x -2$ is graphed below.

Suppose Newton's method is used to find the roots of f(x).

(i) [4 pts] Using x₀ = 3 as an initial value, find the first approximation x₁ to the root.

ANS. $\;x_1 = {\underline{\hspace*{1.0in}}}$

(ii) [5 pts] Again, using x₀ = 3, show on the graph where the next two approximations x₁ and x₂ lie. Place an ``R" on the graph at the root to which the sequence appears to converge.

(iii) [4 pts] Give two initial values x₀ for which the sequences generated by Newton's Method will not converge.

ANS. $\;x_0 = {\underline{\hspace*{1.0in}}}, \;x_0 = {\underline{\hspace*{1.0in}}}$

7. [30 pts]

(a) A particle moves along a curve C given by ${\bf r}(t) = <\sin(t/2), \pi^2 - t^2, \cos(t) >, -5 \leq t \leq 5$.

(i) [10 pts] Find the velocity vector of the particle at the point $P_1 = (-1/\sqrt{2}, 3\pi^2/4, 0)$.

(ii) [7 pts] Find an equation for the plane perpendicular to C at P₁.

(iii) [7 pts] Give an expression for the distance the particle travels from P₁ to $P_2 = (0, \pi^2, 1)$. (Do not evaluate your expression.)

(b) [6 pts] A magnetic quadrupole consists of two magnetic dipoles with moments of equal magnitude and opposite signs (call them m and -m), separated by a small distance d. It is known that the magnitude of the magnetic field B at the point P shown below, at a distance D from the nearest dipole, is

$$B = \frac{m}{D^3} - \frac{m}{(D + d)^3}.$$

Suppose that D is large compared to d, so that the ratio $x \equiv d/D \ll 1$. We can rewrite B as

$$B = \frac{m}{D^3} - \frac{m}{D^3}\frac{1}{(1 + x)^3} = \frac{m}{D^3}\left[1 - \frac{1}{(1 + x)^3}\right].$$

Use a two term Maclaurin series in x to approximate the function in B. Then show that B is approximately

$$B \approx \frac{k}{D^4}.$$

and find the constant k (known as the ``magnetic quadrupole") in terms of given quantities.

ANS. $\;k = {\underline{\hspace*{1.0in}}}$

8. [25 pts total]

(a) [5 pts] The function $z = f(x,y) = \mbox{ln}(9 - x^2 - y^2)^{3/2}$. What is the domain of f? Sketch this region.

(b) The function $u = u(x,y) = e^x\,\sin(x + y)$.

(i) [10 pts] Find the first partial derivatives of u at the point (x,y) = (2,3).

(ii) [10 pts] Is u_xy = u_yx for this function, for all values of x and y? Validate your answer by comparing u_xy and u_yx.

9. [30 pts total] The surface S has the equation 4x² + y² - 4y - z = -4.

(a) [8 pts] Reduce the equation to a standard form, and identify S.

(b) [6 pts] Find and identify the level curves of S.

(c) [8 pts] Sketch S.

(d) [8 pts] Find the equation of the tangent plane to S at the point $\left(\frac{1}{2}, 1, 2\right)$.

 

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