Math 206A Quiz 05 page 1 03/04/2011 Name
1. Suppose that u(x) =
−−−−−−−−−−−−−−−−−−→
(f(x), g(x), h(x), k(x)) where the four component functions f,g,h,kare real-valued and
x=
−−−−−→
(x, y, z). Suppose furthermore that x=r2+s3+t,y=s2t5, and z=e8t+r4, and −→
m(r, s, t) is the vector-valued function
with these three component functions x,yand z.
1A. Find J(−→
m); simplify all entries.
1B. how many rows, and columns, does Juhave? rows = columns = .
1C. Let uNbe u◦
−→
m, and let uN,2be the second component function of uN. Find and simplify to the extent possible
a formula for ∂uN,2/∂t. (Hint: Fill out just enough of Juso that with your answer to 1A you can find the answer to this
problem)
2. Let P= (3,4) and Q= (15,9) be two points in the plane. Let f(x, y) = x3+xy. Let ube a unit vector pointing in
the direction from Pto Q.
2A. Find the directional derivative of fin the direction uat the point P. Show all your work. Express your final answer
as a decimal to five places after the decimal point.
2B. What is the approximation to this partial derivative, using h= 0.1 and the difference quotient (f(P+hu)−f(P))/h?
Show all your work to five places after the decimal point.