MAT135Y5 Lecture 3: 1.14

2. Consider the following function of two variables: z f(x, y) = x/1 + y2 = (a) Find the gradient of the function (b) Find the directional derivative of the function at the point P(5, 3) in the direction of the point Q(8, 4). (c) Suppose that k=e24 andy-2t.|Use the ChainRule to comput4 , and evaluate the expression when t=0. dt'

5. Find the derivative of the function Answer: (1 + sec(t))'o dt 10 f(t) dt=-1 Answer (b) F(z) = Hunt: f(tdt . Answer (c) F(x)-10ond (t) dt (d) lf the Saints lose: F(x) =ht2+1 dt Hint. f(t)dt=.f(t) dt +Jof(t)dt Answer:

Let F = 〈z2, x, y) and let C be the curve that is given by r(t)-(3 + 8t2, 2-t2, t) for o t s 2. Calculate F(r(t)) and dr = r"(t) dt. F(r(t)) = dt Calculate the dot product F(r(t)) dr and evaluate the line integral. F(r(t)) . dr = dt F·dr =