Exam 14: Partial Derivatives

Use the Chain Rule to find where .

Find the gradient of the function .

A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In the case of , , what can you say about f ?

The wind-chill index I is the perceived temperature when the actual temperature is T and the wind speed is v so we can write . The following table of values is an excerpt from a table compiled by the National Atmospheric and Oceanic Administration. Use the table to find a linear approximation to the wind chill index function when T is near and v is near 30 kmh.

Use implicit differentiation to find

The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.?

Find the linearization L(x, y) of the function at the given point. Round the answers to the nearest hundredth.

Find for the function

Find and sketch the domain of the function

Find the dimensions of the rectangular box with largest volume if the total surface area is given as .

Find three positive numbers whose sum is and whose product is a maximum.

Find the differential of the function

Let Find

Find the differential of the function

At what point is the following function a local minimum?

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints.

Find the absolute minimum value of the function on the set D. D is the region bounded by the parabola and the line

The length l, width w and height h of a box change with time. At a certain instant the dimensions are and , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing.

Let Find