This Exam Is Based on 200 Points. There Are 240 Points Available

Final Exam -- MTH 231-01

March 17, 2000

Show all work. Write answers on separate sheet(s) of paper. Keep this sheet for yourself.

This exam is based on 200 points. There are 240 points available.

(15) 1.(a)Find the center and radius of the sphere represented by the equation .

(b)Give parametric equations for this sphere.

(40) 2.Suppose v = (1, –1, 2) and w = (3, 1, 0).

(a)Find the angle (in radians) between v and w.

(b)Find the projection of v onto w.

(c)Find the area of the parallelogram that has v and w as two of its sides.

(d)Find the fourth vertex of the parallelogram that has v and w as two of its sides. (The origin and the endpoints of v and w are the other three vertices).

(e)Find an algebraic equation for the plane containing v and w (the origin and the two endpoints).

(25) 3.(a)Give parametric equations for the polar coordinate equation r = cos(2).

(b)Graph the polar equation r = cos(2). (You may use your graphing calculator or just draw it by hand.)

(c)Give but do not evaluate a definite integral for the arc length of the graph of r = cos(2).

(d)Give and evaluate a definite integral for the area inside the graph of r = cos(2).

(20) 4.The motion of a particle is given by the vector function .

(a)Find vector functions for the velocity vector v(t) and acceleration vector a(t).

(b)at time t = 1, find the tangential component of acceleration aT and the normal component of acceleration aN.

(c)Is the particle speeding up or slowing down at t = 1? Explain your answer.

(25) 5.In this problem we rotate the curve in the yz-plane about the z-axis to obtain a surface.

(a)Give parametric equations for the curve in the yz-plane, using parameter s.

(b)Give the formulas for a rotation of angle t about the z-axis.

(c)Carry out the rotation of the curve about the z-axis by substituting the parametric equations in (a) into the rotation formulas in (b). This should give parametric equations for the resulting surface.

(d)Give an algebraic equation for the surface, and identify it by name.

(25) 6.(a)Describe and name the surface defined by the equation .

(b)Give parametric equations for this surface.

(c)Convert the equation into both cylindrical and spherical coordinates.

(20) 7.(a)Give the first five nonzero terms of the Taylor series at x = 2 for .

(b)Write this Taylor series in summation form.

(c)Use the Ratio Test to find the interval of convergence for this Taylor series.

(20) 8.The Bessel function of order 0 is defined by the Taylor series

(a)Find the Taylor series for and write out the first three nonzero terms.

(b)Use your answer to (b) to give a numerical approximation of .

(20) 9.Suppose a function f(x) is defined on the interval [–π,π]. Its Fourier Series is given by

(a)Give formulas for finding the coefficients .

(b)Use these formulas to find if f(x) = | sin(x) |.

(15) 10.(a)Write both and in terms of cos() and sin().

(b)Use the equations in (a) to write cos() in terms of and .

(15) 11.(a)Show that for any complex number z with | z | = 1, .

(b)What is the reciprocal of ?