Calculus Answers

Show that the length of the portion of any tangent line to the asteroid a𝑥^2/3 + 𝑦^2/3 = 𝑎^2/3 ,cut off by the coordinate axes is constant.

A cone of radius 𝑟 centimeters and height ℎ centimeters is lowered point first at

a rate of 1 cm/s into a tall cylinder of radius 𝑅 centimeters that is partially filled with

water. How fast is the water level rising at the instant the cone is completely

submerged?

Suppose 𝑓 is odd and differentiable everywhere. Prove that for every positive

number 𝑏, there exists a number 𝑐 in (−𝑏, 𝑏) such that 𝑓 ′(𝑐) = 𝑓(𝑏)/𝑏.

At which point on the following curve does the tangent line has the largest slope?

𝑦 = 1 + 40𝑥^3 − 3𝑥^5

If you get a slice of a round pizza with perimeter 90 cm , what should be the diameter of the pizza for you to have gotten the largest slice?

Obtain the Fourier series for the following periodic function which has a period of 2π: f(x)=x for−π≤x≤π

A retailer receives a shipment of 10,000 kilograms of rice that will be used up over a 5-month period at the constant rate of 2,000 kilograms per month. If storage costs are 1 cent per kilogram per month, how much will the retailer pay in storage costs over the next 5 months?

It is estimated that x months from now the population of a certain town will be increasing at the rate of 2 + 6√𝑥 people per month. The current population is 5,000. What will be the population 9 months from now?