Calculus Answers

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle.

How much wire (in m) should be used for the square in order to maximize the total area?

M = 0

How much wire (in m) should be used for the square in order to minimize the total area?

M= ?

I've tried 16.8 but its wrong.

A firm produces two commodities, ×, and y. The demand functions are

P, = 900 - 2x - 2 y

and

P, = 1400 - 2x -4y

respectively, where P, is the price of commodity x and P, is the price of commodity y. The costs

are given by

C, = 7000 + 100x + x2

and

C, = 10000 + 6 y*

a) Show that the firm's profit function is given by

7 (x, y) = -3x? - 10y? 4xy + 800x +1400y -17000

b) Suppose the firm is required to produce a total of exactly 60 units. Find the values of x

and y that maximize profits.

Use Green’s Theorem to evaluate

∮C(x − 2y2) dx + (y4 + 2xy) dy where C consists of the line segment

from (0, 2) to (0, 4), followed by the curve with parametric equations x = 4 cos t, y = 4 sin t from (0, 4) to (−2, 2√3), then the line segment from (−2, 2√3) to (−1, √3), and finally the curve with parametric equations x = 2 sin t, y = 2 cos t from (−1, √3) to (0, 2).

Find two numbers whose sum is 24 such that the sum of the square of one plus six times the other is a minimum.

A small jewelry box with square of base is to have a volume of 125 cu.cm. Find its dimensions to require the least amount of material.

Determine the second derivative of g(x)=sin(2x³-9x)

Use Green’s Theorem to evaluate

∮C(x − 2y2) dx + (y4 + 2xy) dy where C consists of the line segment

from (0, 2) to (0, 4), followed by the curve with parametric equations x = 4 cos t, y = 4 sin t from (0, 4) to (−2, 2√3), then the line segment from (−2, 2√3) to (−1, √3), and finally the curve with parametric equations x = 2 sin t, y = 2 cos t from (−1, √3) to (0, 2).

Consider the equation xe^x = cos x

(a) Apply the intermediate value theorem to show that the function has a root in the interval

[0, 1].

Find the lengths of the sides of an isosceles triangle with a given perimeter if its area is to be as great as possible.

Use logarithmic differentiation to prove D9: d/dx (1/u^n) =( -n/u^n+1)(du/dx)