MATH 21C

Final | Fall '15 | Paramonov

1. Find the limits of the following sequences. (Explain your answers.)
a. (4) $\lim_{n \to \infty} \frac{\sin(n)}{n}$

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b. (4) $\lim_{n \to \infty} \frac{\ln(n)}{n}$

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c. (5) $\lim_{N \to \infty} \sum_{n = 0}^{N} \frac{2}{5^n}$

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2a. (7) Find the interval of convergence of the power series $\sum_{n=0}^{\infty} \frac{(x + 1)^n}{\sqrt{n + 1}}$

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b. (6) Determine for which values of $x$ the power serise above converges absolutely, for which values it converges conditionally, and for which values it diverges.

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3. (6) Use the formula for Taylor series $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$ to show that the Taylor series generated by the function $f(x) = \cos(x)$ at the point $a = 0$ is $\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$

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4a. (7) Let $P_N(x)$ be the Taylor polynomial of order $N$ generated by the function $f(x) = \cos(x)$ at $a = 0$. Show that for all values of $x$ $\lim_{N \to \infty} | \cos(x) - P_N(x) | = 0$ Hint: Here you should use the Taylor Remainder Theorem $f(x) - P_N(x) = \frac{f^{(N + 1)}(c)}{(N + 1)!}(x - a)^{N + 1}$

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b. (4) Prove that the Taylor series generated by $\cos(x)$ at $a = 0$ converges to $\cos(x)$ for all $x$.

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5a. (7) Find the powers series centered at $0$ for the function $\int_0^x [1 - \cos(t^2)] dt$

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b. (3) Estimate the value of $\int_0^1 [1 - \cos(t^2)]dt$ with an error of estimation less than $10^{-3}$.

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6. (6) Show that for any two vectors $\vec{a}$ and $\vec{b}$, $\frac{|\vec{a} + \vec{b}|^2 + |\vec{a} - \vec{b}|^2}{2} = |\vec{a}|^2 + |\vec{b}|^2$

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7. Given the line $L : x = 2t + 1, \quad y = -3t + 5, \quad z = t - 2$ and the point $S$ with coordinates $(1, 1, -1)$ in three dimensions,
a. (6) Find the distance between $L$ and $S$.

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b. (4) Find the equation of the plane that contains both $L$ and $S$.

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8. Given two planes $\pi_1$ and $\pi_2$ with equations $\pi_1 : 2x + y + 3z = 1, \quad \pi_2 : 5x + 2y + 6z = 2$ a. (5) Suppose $\theta$ is the angle between $\pi_1$ and $\pi_2$. Find $\cos(\theta)$.

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b. (7) Find the parametric equation for the line of intersection of $\pi_1$ and $\pi_2$.

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9. For the function of two variables $f(x, y) = \sqrt{x^2 + y^2 + 4}$ a. (4) Find the range and the domain of the function

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b. (4) Sketch the assortment of level curves $f(x, y) = c$ for values of $c = 0, 2, 4$.

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c. (6) Sketch the graph of the function.

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10. For the function $f(x, y) = \frac{\cos(xy) + y}{x + 1}$ a. (7) Find the direction in which $f(x, y)$ increases most rapdily and the direction in which $f(x, y)$ decreases most rapdily at the point $(0, 0)$.

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b. (4) Find the equation of the tangent line to the curve $\frac{\cos(xy) + y}{x + 1} = 1$ at the point $(0, 0)$.

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11. The resistance $R$ produced by wiring resistors of $R_1$ an $R_2$ ohms in parallel can be calculated from the formula $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ a. (5) Taking $R$ to be a function of $R_1$ and $R_2$, show that $dR = \left( \frac{R}{R_1} \right)^2 dR_1 + \left( \frac{R}{R_2} \right)^2 dR_2$

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b. (5) Suppose you plan to change $R_1$ from $20$ to $20.1$ ohms, and $R_2$ from $25$ to $24.8$ ohms. By about how much will this change $R$?

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12. Given the function $f(x, y) = e^{-y}(x^2 + y^2)$ a. (6) Find all critical points of the function inside the unit circle $x^2 + y^2 < 1$. Classify those points using the Second Derivative Test.

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b. (5) Find all local maxima and local minima of the function on the unit circle $x^2 + y^2 = 1$. (You might want to use the parametrization of the unit circle $x = \cos(t), y = \sin(t), t \in [0, 2\pi]$.)

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c. (3) Find absolute minimum and absolute maximum of the function on the domain $x^2 + y^2 \le 1$.

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