Calculus Answers

After eight years of dating, a female broke up with her lover on a rainy day. They chose to split up in the same area where everything about them began. The boy is crying and rushing north at a speed of 5 feet per second, while the girl is strolling east at a speed of 1 foot per second, wondering if she made the right decision. 5 seconds after they started traveling to a new life without each other, how quickly are they detaching from each other?

A tube is spewing sand at a rate of one cubic meter per second. It generates a cone-shaped pile of material. The radius of the circle at the base of the cone is equal to its height. When the sand pile reaches 2 meters in height, how fast does it rise?

Solve the following Initial Value Problem using (i)R-K method of O(h²)

and (ii) R-K method of O(h⁴)

y' = x²y + x³ and y(0) = 1.

Find y(0.4) taking h = 0.2, where y' means dy/dx

For each of the following equations connecting x and y, if the rate of change of x is 2units, find the rate of change of y at the given instant

a) y= 3/(2x-3)³, x=2

b) y= x³ + 2 , y=10

Solve the B.V.P.

𝑢𝑡 = 𝑢𝑥𝑥 + 𝑢; { 𝑢(𝑥, 0) = cos 𝑥 0 < 𝑥 < 1 𝑢(0,𝑡) = 0, 𝑢(1,𝑡) = 0 

The ends 𝑥 = 0 and 𝑥 = 10 of a thin aluminum bar (𝛼 2 = 0.86) are kept at 0 𝑜𝐶, while the surface of the bar is insulated. Find an expression for the temperature 𝑢(𝑥,𝑡) in the bar if initially

𝑢(𝑥, 0) = { 10𝑥 0 < 𝑥 < 5 10(10 − 10𝑥) 5 ≤ 𝑥 < 10 

The heat equation in two space dimensions may be expressed in polar coordinates as

𝑢𝑡 = 𝛼 2 [𝑢𝑟𝑟 + 1/𝑟 𝑢𝑟 + 1/𝑟^2 𝑢𝜃𝜃]

Assuming that 𝑢(𝑟, 𝜃,𝑡) = 𝑅(𝑟)𝛩(𝜃)𝑇(𝑡),

find ordinary differential equations satisfied by 𝑅, 𝛩 and 𝑇. 

Find a solution u(x,t) to the problem 𝜕𝑢 𝜕𝑡 = 1.71 𝜕^2𝑢/𝜕𝑥^2 , 𝑢(𝑥, 0) = 𝑠𝑖𝑛 ( 𝜋𝑥/2 ) + 3 𝑠𝑖𝑛 ( 5𝜋𝑥/2 ) , 0 < 𝑥 < 2

A capacitor circuit has been charged up to 12V and the instantaneous voltage is 𝒗 = 𝟏𝟐 (𝟏 − 𝒆 − 𝒕 𝜏)

The tasks are to: 

a) Draw a graph of voltage against time for 𝑣 = 12𝑉 and 𝜏 = 2𝑠, between 𝑡 = 0𝑠 and 𝑡 = 10𝑠.

b) Calculate the gradient at 𝑡 = 2𝑠 and 𝑡 = 4𝑠

c) Differentiate 𝒗 = 𝟏𝟐 (𝟏 − 𝒆 − 𝒕 𝜏) and calculate the value of 𝑑𝑣 𝑑𝑡 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.

d) Compare your answers for part b and part c. 

e) Calculate the second derivative of the instantaneous voltage ( 𝑑^2𝑣/𝑑𝑡^2 ).