EML 5060Analysis in Mechanical Engineering Fall 2006
Test 1 Van Dommelen (http://www.eng.famu.fsu.edu/$\sim$dommelen) Due 9/01/06

Hand in the solution to this test on 9/01/06 (5% of your final grade). Read carefully. And look it up. Answer questions in order from left to right, top to bottom. You must work alone. You probably want to consult a math handbook.

Neatly draw the graph of the following functions, showing the locations of 0 and $\pm 1$ on each axis. Give the derivative. Indicate non-principal values as a broken line. Make sure that you give enough of the curves to clearly demonstrate all features. Make sure that you have answered all parts, including derivatives.

$$x-2 \qquad \qquad \qquad \qquad x^2 - 4 \qquad \qquad \qquad \qquad x^3 - x$$

$$\sin(x)\qquad \qquad \qquad \qquad \arcsin(x)\qquad \qquad \qquad \qquad \sinh(x)$$

$$\cos(x)\qquad \qquad \qquad \qquad \arccos(x)\qquad \qquad \qquad \qquad \cosh(x)$$

$$\tan(x)\qquad \qquad \qquad \qquad \arctan(x)\qquad \qquad \qquad \qquad \tanh(x)$$

$$\ln(x)\qquad \qquad \qquad \qquad e^x\qquad \qquad \qquad \qquad \sin(\pi x^2)$$

Find (include any integration constants and absolute signs):

$$\int x^{-2} {\rm d} x= \qquad \qquad \qquad \int_1^2 x^{-2} {\rm d} x = \qquad \qquad \qquad \int_1^x \xi^{-2} {\rm d} \xi =$$

$$\int {{\rm d} x \over x} = \qquad \qquad \qquad \int {1\over... ... d} x = \qquad \qquad \qquad \int {1\over 1 + x^2} {\rm d} x =$$

$$\int \ln(x) {\rm d} x = \qquad \qquad \qquad \int x e^x {\rm d} x = \qquad \qquad \qquad \int x e^{x^2} {\rm d} x =$$

$$\left\vert \matrix{ 1 & 4 & 7\cr 2 & 5 & 8\cr 3 & 6 & 1} \rig... ... \over {\rm d} x} \int_0^x \frac{\sin(x\xi)}{\xi} {\rm d}\xi =$$

$$1 + 2 + 3 + 4 \ldots + 1000 = \qquad \qquad\qquad\qquad\qquad x + x^2 + x^3 + x^4 + \ldots =$$

$${\rm Solve: }\quad{{\rm d} y\over {\rm d} x} = - y \qquad y(0)=1$$