math calculus

MATH 310, FALL 2017: ASSIGNMENT 3

Due: Monday, October 30, in class

Please answer the following questions, explaining all of the steps that you took to find solutions to the following problems.

1. Use implicit differentiation to find dy/dx for the following curves at the given points.

a.

cos(xy) − 1 − sin(y) = 0, (x,y) = (1, 0) b.

ey cos(y) −x−xy = 0, (x,y) = (0,π/2) 2. Let F(x,y) be a function of two variables whose partial derivatives

are continuous at (a,b). Assume that f(t0) = a and g(t0) = b. a. Compute the equation for the line tangent to

~r(t) = 〈f(t),g(t),F(f(t),g(t))〉 at 〈a,b,F(a,b)〉 (i.e. when t = t0).

b. Check that this line is contained in the plane tangent to the graph of F(x,y) at 〈a,b,F(a,b)〉.

3. Show that the line which is normal to the tangent plane each point of the sphere

x2 + y2 + z2 = r2

passes through (0, 0, 0) in R3. 4. Find the gradient vector of the following functions. Then evaluate

the gradient at the given point P and find the rate of change in the given direction ~v and the direction of maximal rate of change of f(x,y) at P .

a.

f(x,y) = x2 + cos(xy), P = (0, 1), ~v = 〈1/2, √

3/2〉 b.

f(x,y) = x2y + y2x + x + y, P = (1, 1), 〈1/ √

2, 1/ √

2〉. 5. Find the critical points of the following points and identify which are

local maxima which are local minima and which are saddle points. a.

f(x,y) = xy(1 −x−y) b.

f(x,y) = xy + 1

x +

1

y

MATH 310, FALL 2017: ASSIGNMENT 3

6. Use Lagrange multipliers to find the points on the sphere x2 + y2 + z2 = 1 closest and furthest from the point (2,−1, 2).