Calculus Answers

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

A firm's average revenue function 

𝐴𝑅=−18−7,5𝑄+𝑄

2

.

AR=−18−7,5Q+Q2.

Determine the number of units to be produced and sold to maximise revenue.

a.−1

b.6

c.3,75

d.0

The demand function of a firm is 

𝑄=90−1,5𝑃,

Q=90−1,5P,

where 𝑃

P and 𝑄

Q represent the price and quantity, respectively. At what price is revenue a maximum?

a.15

b.30

c.270

d.90

ACTIVITY IN BASIC CALCULUS

BASIC RULES IN DERIVATIVE

 Complete the blanks of the given function below with a number (except 0 and 1) to create your own problem and find the derivative of the function. Show your complete solution to each problem.

1. f(x) = -4x5 + ______x-4- 2468

2. f (x) =____x-3- _____x1/4 - 12x

3.f(x)= ____ "\\sqrt[4]{x}" ³ - "\\underset{x^6}{=}" + "\\frac{2}{3}" x⁶

4.f (x) = "\\underset{x^-6}{=}" - ____x² + "\\sqrt[4]{x}"

Differentiate the following functions:

1.y=1/2(3x^2+1)^2

2.y=3+4x-x^2

Find the derivative of the following

1.y=√x^2-2/2x^3+1

2.y=x^2√6x^2/3x

3.y=2x^2√x

Evaluate the following integrals

(1) integral of {(10x^9+40x^4+3)√(x^10+8x^5+3x+5)} DX

(2) integral of {1/[(x-5)(x^2+4)]} dx