Calculus Answers

The base of the rectangle is changing at the rate of 3in/min. if its height remains constant, determine the rate of change of its perimeter with respect to time?

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.

The Altitude of a triangle is increasing at a rate of 8cm/s while its area is increasing at the rate of 12cm^2/s. At what rate is the base of the triangle changing when the altitude is 20 cm and the area is 100 cm^2 ?

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

If 𝑓(𝑥) is a differentiable and 𝑔(𝑥) = 𝑥 𝑓(𝑥) use the definition of the derivative to show

that 𝑔′(𝑥) = 𝑥𝑓'(𝑥) + 𝑓(𝑥).

Evaluate the following limits, if they exist, where ⌊𝑥⌋ is the greatest integer function.

(a)lim ⌊2𝑥⌋/𝑥

𝑥→0

(b) lim 𝑥 ⌊1/𝑥⌋

𝑥→0

Let 𝑓(𝑥) = ⌊𝑥⌋ + ⌊−𝑥⌋, where ⌊𝑥⌋ is the greatest integer less than or equal to 𝑥.

(𝑎) For what values of 𝑎, does lim𝑥→𝑎

𝑓(𝑥)exist?

(𝑏) At what numbers is 𝑓 discontinuous?

A firm's average revenue function 

𝐴𝑅=−18−7,5𝑄+𝑄

2

.

AR=−18−7,5Q+Q2.

Find the TR and 𝑀𝑅 functions.

a.

=−18𝑄−7,5𝑄2+𝑄3=−18Q−7,5Q2+Q3

b.=−18𝑄−7,5𝑄2+𝑄3=−18Q−7,5Q2+Q3

c.=−18−7,5𝑄+𝑄2=−18−7,5Q+Q2

d.=−18−7,5𝑄+𝑄2=−18−7,5Q+Q2