MATH 16C

Final - Practice 1 | Spring '11 | Basu

1. Let $f(x, y) = \cos(1 + xy^2)$
(a) (4 pts) Compute $f_y$.

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(b) (6 pts) Compute $f_{yx}$.

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2. Consider the function $f(x, y) = 5 + x^2 - x^2y - y^2 - \frac{1}{3}y^3$.
(a) (8 pts) Find all critical points of $f(x, y)$.

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(b) (7 pts) Decide whether each critical point found in (a) is a relative minimum, relative maximum, saddle point or indeterminable.

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3. Write the $n$-th term in the following sequences:
(a) (2 pts) $-\frac{1}{2}, \frac{4}{3}, -\frac{9}{4}, \frac{16}{5}, \dots$

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(b) (2 pts) $2, 2, \frac{8}{6}, \frac{16}{24}, \frac{32}{120}, \dots$

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(c) (2 pts) $1, \frac{5}{8}, \frac{7}{15}, \frac{9}{24}, \frac{11}{35}, \frac{13}{48}, \dots$

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4. Determine if the following sequences converge or diverge. If the sequence converges, write the limit.
(a) (3 pts) $a_n = \frac{2^n + 1}{3 \cdot 2^n}$

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(b) (3 pts) $a_n = (-1)^n (\frac{1}{n^2 + 3})$

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(c) (3 pts) $a_n = (-1)^n (\frac{n + 1}{n})$

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5. Determine if the following series converge or diverge. Cleary explain why.
(a) (6 pts) $\sum_{n=0}^{\infty} \frac{n}{500n + 79}$

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(b) (6 pts) $\sum_{n=1}^{\infty} (2n)! (\frac{2}{3})^n$

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(c) (6 pts) $\sum_{n=1}^{\infty} \frac{1}{n^2\sqrt{n}}$

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6. (12 pts) Find the radius and interval of convergence for the power series $\sum_{n=1}^{\infty} \frac{3^{-n}}{n + 1}(x + 1)^n$

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7. (10 pts) Find the sum $\sum_{n=0}^{\infty} \frac{1}{3^n 2^{n-2}}$

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8. Evaluate the following double integrals. (Remember that sometimes it helps to change the order of integration).
(a) (12 pts) $\int_0^2 \int_0^\sqrt{x} y(x - y^2)^3 \; dy \; dx$

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(b) (12 pts) $\int_0^4 \int_\sqrt{x}^2 x \sin(1 + y^5) \; dy \; dx$

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9. Solve the following differential equations.
(a) (10 pts) $e^{x^2 - x}y' + y = 2xy$.

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(b) (10 pts) $xy' - 2y = x^2 \ln(x)$ with the initial conditions that $y = 0$ when $x = 1$.

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10. (15 pts) Approximate the definite integral $\int_0^1 xe^{-x^3} dx$ using a $7$th-degree Taylor polynomial for the function $xe^{-x^3}$.

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11. (11 pts) Minimize the function $f(x, y, z) = x^2 + 2y^2 + 3z^2$ subject to the constraint $3x - 2y + z = \frac{34}{6}$. Find $x, y, z$ at the minimum and the minimum value of the function.

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