MATH 21C

Final - Practice 1 | Winter '11 | Saito

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series $\sum_{n = 0}^{\infty} \frac{n(x + 2)^n}{5^{n-1}}$

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Problem 2 (10 pts) Consider the function $f(x) = \ln x$.
(a) (5 pts) Approximate $f(x)$ by a Taylor polynomial of degree $2$ at $x = 2$.

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(b) (5 pts) How accurate is this approximation when $1 \le x \le 3$?

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Problem 3 (10 pts) State and prove the Cauchy-Schwarz Inequality. Note that you also need to state and prove the condition for the equality to hold.

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Problem 4 (10 pts) Let $P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)$.
(a) (5 pts) Find the area of the triangle $PQR$.

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(b) (5 pts) Find the distance from $P$ to the line $QR$.

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Problem 5 (10 pts) Find the limit if it exists, or show that the limit does not exist.
(a) (5 pts) $\lim_{(x, y) \to (0, 0)} \frac{y^4}{x^4 + 3y^4}$ Hint: Consider $(x, y) \to (0, 0)$ along the line $y = mx$.

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(b) (5 pts) $\lim_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2 + y^2}}$ Hint: Consider the limit in the polar coordinates $(r, \theta)$.

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Problem 6 (10 pts) Verify that the function of two variables $u(x, t) = e^{-\alpha^2 k^2 t} \sin kx$ where $k$ is an arbitrary positive integer is a solution of the heat condition equation over the interval $x \in [0, \pi]$. $u_t = \alpha^2 u_{xx}$ with the boundary condition $u(0, t) = u(\pi, t) = 0 \; \text{for all} \; t \ge 0$

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Problem 7 (10 pts) Assuming that the equation $\sin x + \cos y = \sin x \cos y$ defines $y$ as a differentiable function of $x$, use the Implicit Differentiation Theorem to find $\frac{dy}{dx}$.

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Problem 8 (10 pts) Find the direction in which $f(x, y, z) = \ln(xy^2z^3)$.
(a) (3 pts) Increases most rapidly at the point $(1, 2, 3)$.

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(b) (3 pts) Decreases most rapidly at the point $(1, 2, 3)$.

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(c) (4 pts) Does not change (i.e., is flat) at the point $(1, 2, 3)$.

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Problem 9 (10 pts)
(a) (5 pts) Show the equation of the tangent plnae at the point $P_0(x_0, y_0, z_0)$ on the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ is $\frac{x_0x}{a^2} + \frac{y_0y}{b^2} + \frac{z_0z}{c^2} = 1$

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(b) (5 pts) Compute the normal line to the same ellipsoid at the same point $P_0$. Furthermore, compute the point where this normal line intersects with the $xy$-plane. Assume that $z_0 \ne 0$.

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Problem 10 (10 pts) Find the linearization of $f(x, y, z) = \sqrt{x^2 + y^2 + z^2}$ at the point $(3, 2, 6)$. Then use it to approximate $f(3.1, 1.9, 6.2)$ by the three decimal point number. Note that the true value of $f(3.1, 1.9, 6.2)$ is $7.188$ in the three decimal point number.

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Problem 11 (10 pts) Find the absolute maximum and minimum values of $f(x, y) = x^2 - xy + y^2 + 1$ on the closed triangular domain bounded by the lines $x = 0, y = 2, y = 2x$, i.e., $\Omega = \{ (x, y) | x \ge 0, y \le 2, y \ge 2x \}$

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Problem 12 (10 pts) Use Lagrange multipliers to find the maximum and minimum values of the function $f(x, y) = e^{xy} \; \text{subject to} \; x^2 + y^2 = 1$ Note that we only consider the real values for $x$ and $y$, not the complex values.

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