Calculus Answers

3. An ideal shock absorption system would use a critically damped oscillator to absorb shock loads. The location of the absorbing piston (𝑥) is described by 𝑥 = 𝜏𝑒−𝛾𝑡 where:

- 𝜏 is the linear damping coefficient

- 𝛾 is the exponential damping constant

- 𝑡 is the time (𝑠)

- 𝑥 is the displacement of piston (𝑚)

The tasks are to:

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

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

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   c) Differentiate the function of 𝑥 and calculate the value of 𝑑𝑥 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠. 𝑑𝑡

d) Compare your answers for part b and part c. (M1)

e) Calculate the derivative for the velocity function(𝑑2𝑥).

Determine whether if

lim f(c) = f(c)

x→c

1. f(x) = x+2; c = -1

2. f(x) = x-2; c = 0

3. (at c = -1 )

f(x) = {x ² - 1 if x < -1}

f(x) = { (x - 1) ² - 4 if x ≥ -1}

4. (at c = 1 )

f(x) = {x³ - 1 if x < 1}

f(x) = { x² + 4 if x ≥ 1}

Using double integral fund the area of region enclosed by √x+√y=√a and x+y=a

The steady state temperature of certain medium is given by theta = e power 2x - 3 y. Find the linear approxmiation at (0,0)

find fourier integral for the following f(x)= e-x x>0

(a) Find the derivative of the function 𝑦 = 2𝑥^2+12/x^2, when 𝑥 = 2.

(b) Let 𝑓(𝑥) = −3/𝑥−7. Find the inverse of the function.

Question 6 [2]

State the mean Value Theorem.

Question 5 [2;2]

Investigate whether the following functions are odd or even.

(a) 𝑓(𝑥) = 𝑥^

3

(b) 𝑓(𝑥) = cos 𝑥

Question 4 [7]

Prove that 𝑓(𝑥) = 𝑥^2 + 2𝑥 is not injective.

Question 2 [7]

Use sign table to determine the values of 𝑥 for which

x^2 + 9𝑥 + 20 ≤ 0.