Problem 1. (10) Find the general solution of the differential equation
$\frac{dy}{dx} = \frac{x}{x^2 + 1}$

Problem 2. (10) Use the initial condition to find the particular solution of the following differential equation.
$\sqrt{x} + \sqrt{y} y' = 0 \; \text{and} \; y = 4 \; \text{when} \; x = 1$

Problem 3. (10) Solve the differential equation
$(x - 1)y' + y = x^2 - 1$

Problem 4. (10) Find the second partial derivatives $z_{xx}, z_{xy}, z_{yx}, z_{yy}$ for the function of two variables
$z = \frac{x}{x + y}$

Problem 5. (10) Examine the function for relative extrema and saddle points
$f(x, y) = e^{xy}$

Problem 6. (10) Find the minimum distance from the plane $x + y + z = 1$ to the point $(2, 1, 1)$. (Hint: Start by minimizing the square of the distance).

Problem 7. Sketch the region $R$ whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area.
$\int_0^4 \int_\sqrt{x}^2 dx \; dy$

Problem 9. (20) Test the series for convergence or divergence using any appropriate test. Identify the test and explain your reasoning. If the series is convergent, find the sum where possible.

(a) $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2 + 1}} = \frac{1}{\sqrt{2}} + \frac{2}{\sqrt{5}} + \frac{3}{\sqrt{10}} + \frac{4}{\sqrt{17}} + \cdots$

(a) $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2 + 1}} = \frac{1}{\sqrt{2}} + \frac{2}{\sqrt{5}} + \frac{3}{\sqrt{10}} + \frac{4}{\sqrt{17}} + \cdots$

(b) $\sum_{n = 0}^{\infty} \left( \frac{1}{2^n} - \frac{1}{3^n} \right)$

(c) $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots$

(d) $\sum_{n = 1}^{\infty} n(0.4)^n$

Problem 10. (10) Find the Taylor series of the function $f(x) = e^{-x}$ centered at $0$ by directly applying the formula for Taylor series.

Problem 11. (15) (a) Use the power series for $\ln(x + 1)$ to find the power series of the following function centered at $0$:
$f(x) = \ln(x^2 + 1)$

(b) Find the power series of the following function centered at $0$.
$g(x) = \int_0^x \ln(t^2 + 1)dt$

(c) Use the fourth degree Taylor polynomial to approximate
$\int_{-\frac{1}{4}}^{\frac{1}{4}} \ln(t^2 + 1) dt$

Problem 12. (10) Approximate, to two decimal places, the $x$-value of the point of intersection of the graphs
$f(x) = 4 - x \; \text{and} \; g(x) = \ln(x)$

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