Calculus Answers

1. The cable of a suspension bridge hangs in the shape of a parabola. The towers supporting the cable are 500ft. apart and 200ft. high. If the cable as its lowest is 50ft above the at its midpoint, how high is the cable 100 ft. away (horizontally) from either tower.

The product of two positive numbers is 4 √ 3. Find the numbers so that the sum S of the square of one and the cube of the other is as small as possible. 

Show that 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝)

What are dimensions of rectangular prism with the largest possible volume that can be construed using 2400 square inches of surface area ?

True or false:

There is a function  "z"  whose partial derivatives are "\\displaystyle{\\frac{\\partial z}{\\partial y}}=x" and  "\\displaystyle{\\frac{\\partial z}{\\partial x}}=-y"

Find the partial derivative "\\displaystyle{\\frac{\\partial z}{\\partial y}}" of the function "\\displaystyle{z^{2}=x^2+y}"

Find the second partial derivative "\\displaystyle{\\frac{\\partial^2 z}{\\partial x^2}}" of the function "z=\\sin(xy^2)" at point "(\\pi, 1)"

True or False:

The mixed partial derivatives "\\displaystyle{\\frac{\\partial^2 z}{\\partial y \\partial x}}"  and "\\displaystyle{\\frac{\\partial^2 z}{\\partial x \\partial y}}"   are equal