EML 5060Analysis in Mechanical Engineering Fall 2014
Test 1 Van Dommelen (http://www.eng.famu.fsu.edu/~dommelen) Due W 09/03/14

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 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.

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

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

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

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

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

Find (include any integration constants and absolute signs):

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

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

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

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

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

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