11.4 2012 test

1) (No longer relevant)

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2) Evaluate the improper integral $\int_{0}^{\infty}\frac{5x+4}{(x+3)(x^{2}+2)}\,dx$.

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3) Consider the parametrized curve $C$ given by

a) Find the gradient of $C$ at $(x(t),y(t))$.

b) Find the arc length along $C$ from $(x(0),y(0))$ to $(x(\pi),y(\pi))$.

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4) Let $f(x,y)$ be a continuous function defined for all real values of $x$ and $y$. Which of the following statements is true, and which is false? Justify your answers.

a) $\int_{c}^{d}\int_{a}^{b}f(x,y)\,dx\,dy=\int_{c}^{d}\int_{a}^{b}f(x,y)\,dy\,dx$ for any $a,b,c,d\in{\mathbb{R}}$ with $a\leq b$, $c\leq d$.

b) If $\frac{\partial^{2}f}{\partial x^{2}}=1$, $\frac{\partial^{2}f}{\partial x\partial y}=2$ and $\frac{\partial^{2}f}{\partial y^{2}}=1$ then $f(x,y)$ has no local maxima or minima.

c) The curve $C$ defined by $f(x,y)=0$ has gradient $\frac{dy}{dx}=\frac{(\partial f/\partial y)_{P}}{(\partial f/\partial x)_{P}}$ at a point $P$ on $C$.

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5) Calculate

and sketch the region of integration.

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Total: 30