Calculus Answers

What is the shortest distance from the surface xy+12x+z2=144 to the origin?

distance=?

Design a rectangular milk carton box of width w, length l, and height h which holds 452 cm3 of milk. The sides of the box cost 3 cent/cm2 and the top and bottom cost 4 cent/cm2. Find the dimensions of the box that minimize the total cost of materials used.

dimensions = ?

(Enter your answer as a comma separated list of lengths.)

Use Lagrange multipliers to find the point (a,b) on the graph of y=e8x, where the value ab is as small as possible.

P=?

Find the maximum and minimum values of the function f(x,y)=2x2+3y2−4x−5 on the domain x2+y2≤225.

1)What is the maximum value of f(x,y) ?

2) List the point(s) where the function attains its maximum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7).

3) What is the minimum value of f(x,y)?

4) List points where the function attains its minimum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7).

Use Lagrange multipliers to find the maximum and minimum values of f(x,y)=x−2y subject to the constraint x2+y2=5, if such values exist.

maximum = ?

minimum = ?

Find the maximum and minimum values of the function f(x,y)=2x2+3y2−4x−5 on the domain x2+y2≤324.

The maximum value of f(x,y). and List the point(s) where the function attains its maximum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7).

Suppose f(x,y)=(x−y)(16−xy). Find the saddle points of f.

Suppose f(x,y)=x2+y2−6x and D is the closed triangular region with vertices (6,0), (0,6), and (0,−6). Answer the following. Find the absolute maximum of f(x,y) on the region D. Find the absolute minimum of f(x,y) on the region D.

A conical tank that is 5 meters high has a radius of 2 meters, and is filled with a liquid that weights 800 kg per cubic meter. How much work is done in discharging all the liquid at point 3 meters above the top of the tank?

Find the volume generated by rotating the region bounded by y = x, x = 1 and y² = 4x, about the x axis.