11.1 Modified 2015 test

(The paper below is a slightly modified version of the coursework which was set in place of the end-of-module test when flooding in Lancaster caused the cancellation of the last week of lectures in December 2015)

1. Evaluate the following integrals:

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2. Let $C$ be the parametrized curve given by $(x(t),y(t))=((t-1)^{2}e^{t},2(t-1)e^{t})$ for $t\in{\mathbb{R}}$.

i) Find the equations of the tangent and normal lines to $C$ at $Q=(x(2),y(2))$.

ii) Determine the length along the curve from $P=(x(0),y(0))$ to $Q$.

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3. State whether each of the following statements is true or false. Briefly (i.e. in about one sentence) justify your answer.

i) The general solution of the differential equation $\frac{d^{2}y}{dx^{2}}-3\frac{dy}{dx}+2y=0$ is $y=Ae^{x}+Be^{2x}$.

ii) The gradient of the curve $y^{2}-x^{3}=0$ at a point $(x,y)$ is $\frac{3x^{2}}{2y}$.

iii) $u(t,x)=e^{t}\cos x$ is a solution of the equation $\frac{\partial^{2}u}{\partial t^{2}}+\frac{\partial^{2}u}{\partial x^{2}}=0$.

iv) If $f(0)=0$, $f(1)=1$ and $f(2)=2$ then Simpson’s rule gives an estimate for the integral $\int_{0}^{2}f(x)\,dx$ of 6.

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4. Find and classify the stationary points of the function $f(x,y)={{x^{5}}\over 5}+2y-xy^{2}$.

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5. Solve the initial-value problem:

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Total: 50 marks.