Review Exercises 1 - 4 Previous Chapter Next Chapter

1. Given the following five vectors:A = (1, 2, 3); B = (2, -3, 5); C = (x, y, z); D = (cos t, sin t, t2); E = (-2, 1, 0)
Do each of the following:
a) Form the sum: A + B + C.
b) Compute AB.
c) Compute A(B+C).
d) Find values for x y and z for which CA = 0 and CB = 0.
e) Find the cosine of the angle between A and B. Between B and D (the answer will be a function of t).
f) Find the projection of E on B.
g) Find the determinant whose columns are A B and E; also find the determinant whose columns are A B and C.
h) Suppose the point p has coordinates x = 1, y = 2, z = 3. What are its spherical coordinates , and ?
i) What is the volume of the parallelepiped with edges A B and E?
j) Find the projection of D into the (x, y) plane. What is its length?

2. Consider the line containing the points A and B above.
a) Give a parametric representation of the points on that line.
b) Find a unit length "tangent vector" that points in the direction of the line.
c) Find two directions normal to that vector.
d) e) and f) consider the plane containing the points A B and E:
Find a (two parameter) parametric representation of the plane.
Find a normal to the plane.
Find an equation that points on the plane all obey.
g) Suppose we have a new and different product of Vectors V@W that has the property V@V = 0 for all Vand @ is linear in each argument so that you can apply the distributive law.
Deduce something about V@W + W@V by applying same to (V + W)@(V + W).

3. Differentiate the following functions with respect to the indicated variables:
a) sin (2exp(x)).
b) (exp (x+y))*sin (xy) with respect to x for fixed y.
c) x2 + y2-3xy with respect to y for fixed x.
d) (sin (y + s (sin t))*(exp -(x+s(cos t))) with respect to s everything else fixed.
e) Find the gradient of (sin y)exp(-x).
f) Find the directional derivative of this function in the direction whose unit vector is (cos t, sin t).
g) Find the linear approximation to sin (exp(x)) at x = 0.
h) Evaluate the derivative with respect to t of (rv) where v is ; suppose that is in the direction of r. What then is the answer?
i) Where is not differentiable? Where is tan x not differentiable? Where is not differentiable?
j) Find the derivative of an inverse function to sin (exp(x)) (to define an inverse function completely you have to specify a range; ignore that here).

4.
a)Find the gradient of the function