EML 5060Analysis in Mechanical Engineering Fall 1998
Test 1 Van Dommelen (dommelen@eng.famu.fsu.edu) Due 8/28/98

Hand in the solution to this test on 8/28 (5% of your final grade). If your performance is insufficient, you will need to hand in a corrected version; however, only your initial grade counts. Please note: This test must have been accepted before exam 1, on 9/16, or you also receive a 0 grade for exam 1. Read carefully. And look it up.

Neatly draw the graph of the following functions, showing the locations of 0 and $\pm 1$ on each axis. Also give the derivative. Indicate nonprincipal values as a broken line. Make sure that you draw enough of the curves to clearly demonstrate all features.

$$1-x \qquad \qquad \qquad \qquad x^2 - 2 \qquad \qquad \qquad \qquad x^3 - x$$

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

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

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

$$\ln(x)\qquad \qquad \qquad \qquad e^x\qquad \qquad \qquad \qquad \cos(x^2)$$

Find (include any integration constants and absolute signs):

$$\int x^4 {\rm d} x= \qquad \qquad \qquad \int_0^1 x^4 {\rm d} x = \qquad \qquad \qquad \int_0^x \xi^4 {\rm d} \xi =$$

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

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

$$\left\vert \matrix{ 1 & 2 & 3\cr 1 & 2 & 6\cr 1 & 4 & 6} \ri... ...uad {{\rm d} \over {\rm d} x} \int_a^x f(\xi) g(x) {\rm d}\xi =$$

$$2 + 4 + 6 + 8 + \ldots + 998 + 1000 = \qquad \qquad\qquad\qq... ...ver \sqrt2} + {1\over \sqrt2^2} + {1\over \sqrt2^3} + \ldots =$$

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