Calc 3

M240 HOMEWORK AND PRACTICE PROBLEMS OCTOBER 25, 2018

1) Find the two partial derivatives ∂f/∂x and ∂f/∂y for the function �

2) For the function f(x,y) = √(x2+y2), find ∂f/∂x and ∂f/∂y at the point (3,4).

3) For the function � , find the second partials ∂²f/∂x², ∂²f/∂y² and ∂²f/∂x∂y.

4) Suppose � , and � , � , � . Find dw/dt using the chain rule.

5) The length of a rectangle, x, is increasing at a rate of 2 m/sec and the width, y, is increasing at a rate of 3 m/sec. Find the rate of change of the area, A =xy, when the length is 10 m and the width is 20 m.

6A) Find the equation of the tangent plane to the function z =xln(y) at the point (3, 1).

6B) Use the tangent plane to estimate the value of the z at the point (3.1,1.1).

7) Find the maximum rate of change of the function � at the point (2, 0).

8) Find the gradient of the function � .

9A) Find a unit vector in two dimensions that points in the direction 30 degrees counter clockwise from the x axis.

9B) Use the vector in part A to find the directional derivative of the function � in the direction 30 degrees clockwise from the x axis.

10) Refer to the page 779 of Stewart and do number 22, a problem on estimating wind speed. For those of you with the electronic copy, this problem which I assigned earlier starts:”The wind chill index W is the perceived temperature ….”

f (x, y) = x si n(2y)

f (x, y) = x2 + 3x y3

w(x, y, z) = x yez x = t + 1 y = t 2 z = 2t

z = f (x, y) = x si n(y)

f (x, y, z) = x y + yz + x z

f (x, y) = x2 + y2 − 4