Calculus Answers

A piece of wire 36cm long is cut into two, one part being bent in the shape of an equilateral triangle and the other in the form of circle. Find the lengths of these two pieces of wire if the sum of the areas of these two figures is to be minimum.

Postal regulations require that a parcel post package be no greater than 3m in the sum of its length and perimeter of its cross-section (girth). What is the volume in cubic meter 9f the largest package allowed by the postal regulations if the package is to be rectangular in shape and has square ends?

The perimeter of a triangle is 60cm. Find the length of the sides of the triangle which gives a maximum area.

For each problem, find the equation of the line normal to the function at the given point. If the normal line is vertical line, indicate so. Otherwise, you answer should be in slope-intercept form.

1) y = -2x^2 - 4x at (-3, -6)

2) y = -x^3 + 2x^2 - 3 at (1, -2)

For each problem, find the equation of the line tangent to the function at the given point. Your answer should be a slope-form.

1) y = 2x^2 + 16x + 32 at (-3, 2)

2) y = (x-2)^(1/3) at (3,1)

3) y = -x^3 + 3x^2 -4 at (3, -4)

4) y = -3/ x^2 - 4 at (-1, 1)

For each problem, find the derivative of the function at the given value. Some will require a graphing calculator.

1) y= -(-x+1)^(1/2) at x= -1

2) y= 2sin (2x) at x= - π/2

3) y= x^2 + 4x + 1 at x= -1

4) y= -ln (x) at x = 2

Trace the curve x=a(theta + sin theta), y = a(1-cos theta). State the properties you use for tracing it also.

prove that xcosecx=1+(x)^2/6+(7/360)*x^4