Exam 2 Solutions
(30) 1. Given points P = (2, –1, 0), Q = (1, 3, 2), and R = (1, –2, 2),
(a) Give an algebraic equation for the plane containing P, Q, and R. (8)
An algebraic equation means a single equation in the variables x, y, and z.
First find two direction vectors: v = Q – P = (–1,4,2), w = R – P = (–1,–1,2).
, so we can use n = (10,0,5) as a normal vector.
The template equation for a plane is a(x–x0) + b(y–y0) + c(z–z0) = 0.
If we use n = (a,b,c) = (10,0,5) and P = (x0,y0,z0) = (2,–1,0), we get
10(x–2) + 0(y+1) + 5(z–0) = 0, or 10x + 5x = 20, or 2x + z = 4.
(b) Give parametric equations for the plane containing P, Q, and R. (8)
r(s,t) = p + sv + tw = (2,–1,0) + s (–1,4,2) + t (–1,–1,2). Written as parametric equations,
x = 2 – s – t, y = –1 + 4s – t, z = 2s + 2t.
(c) Give parametric equations for the line perpendicular to the plane containing P, Q, and R and containing the point P. (8)
Since the normal vector n is a direction vector for this line,
r(t) = p + tn = (2,–1,0) + t (10,0,5). Written as parametric equations,
x = 2 + 10t, y = –1, z = 5t.
(d) Find the area of the triangle with vertices P, Q and R. (6)
Use
(20) 2. Consider the surface with algebraic equation . Give parametric equations for this surface, and verify explicitly that your parametric equations satisfy the algebraic equation. Identify the type of surface, and give a brief description of it.
This is an ellipsoid with an x-radius of 1, a y-radius of 1/2, and a z-radius of 1/3. The parametric equations are just those for the unit sphere multiplied by these factors:
.
(16) 3. Find the first four nonzero terms of the Taylor series for at c = 1.
So the first four terms of the Taylor series are
(24) 4. For each function, give the Taylor series at c = 0 in summation notation, and give the first four nonzero terms of that series. No work is required; in particular you do not need to use the general Taylor series formula (unless you’ve forgotten some things you should have memorized).
(a) ;
(b) ;
(c) ;
(20) 5. We are interested in finding a Fourier series for the function.
(a) Is this function even or odd? Which of the Fourier coefficients a0, ak, and bk will be zero because of this? (6)
This is an even function, which is most easily seen by looking at the graph. As a result, the coefficients bk will be zero.
(b) Give the integral formulas for the coefficients a0, ak, and bk. Evaluate those that you don’t know will be zero. (10)
In working out these integrals it is helpful to use the fact that the integral of an even function over the interval [–a,a] is equal to 2 times the integral over [0,a].
(c) Give the resulting Fourier series for f(x). Attempt to give the series in summation form. If you can’t do that, write out the first four nonzero terms. (4)
For odd values of k, , and for even values of k, .
Therefore the first four nonzero terms are .
In summation form, the series is