EML 5060Analysis in Mechanical Engineering Fall 2010
Test 1 Van Dommelen (http://www.eng.famu.fsu.edu/$\sim$dommelen) Due M 08/30/10

Hand in the solution to this test on the date stated above (5% of your final grade). Read carefully. And look it up. Answer questions in the order listed. You must work alone. You probably want to consult your 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.

$$1) \quad x-2 \qquad \qquad \qquad \qquad 2) \quad x^2 - 4 \qquad \qquad \qquad \qquad 3) \quad x^3 - x$$

$$4) \quad \sin(x)\qquad \qquad \qquad \qquad 5) \quad \arcsin(x)\qquad \qquad \qquad \qquad 6) \quad \sinh(x)$$

$$7) \quad \cos(x)\qquad \qquad \qquad \qquad 8) \quad \arccos(x)\qquad \qquad \qquad \qquad 9) \quad \cosh(x)$$

$$10) \quad \tan(x)\qquad \qquad \qquad \qquad 11) \quad \arctan(x)\qquad \qquad \qquad \qquad 12) \quad \tanh(x)$$

$$13) \quad \ln(x)\qquad \qquad \qquad \qquad 14) \quad e^x\qquad \qquad \qquad \qquad 15) \quad \sin(\pi x^2)$$

Find (include any integration constants and absolute signs):

$$16) \quad \int x^{-2} {\rm d} x= \qquad \qquad \qquad 17) \q... ...\qquad \qquad \qquad 18) \quad \int_1^x \xi^{-2} {\rm d} \xi =$$

$$19) \quad \int {{\rm d} x \over x} = \qquad \qquad \qquad 20... ...quad \qquad \qquad 21) \quad \int {1\over 1 + x^2} {\rm d} x =$$

$$22) \quad \int \ln(x) {\rm d} x = \qquad \qquad \qquad 23) \... ... x = \qquad \qquad \qquad 24) \quad \int x e^{x^2} {\rm d} x =$$

$$25) \quad \left\vert \matrix{ 1 & 4 & 7\cr 2 & 5 & 8\cr 3 & ... ...} \over {\rm d} x} \int_0^x \frac{\sin(x\xi)}{\xi} {\rm d}\xi =$$

$$28) \quad 1 + 2 + 3 + 4 \ldots + 1000 = \qquad \qquad\qquad\qquad\qquad 29) \quad x + x^2 + x^3 + x^4 + \ldots =$$

$$30) \quad {\rm Solve: }\quad{{\rm d} y\over {\rm d} x} = - y \qquad y(0)=1$$