Calculus Answers

Apply definition of antiderivative and find area under the curve of f(x) = x^1/2 between x=0 and

x=1

Find absolute maximum and minimum of the function f(x) = 2x^2- 5 in [-1, 2].

6. Determine whether each of the following statements about Fibonacci numbers is true or false. Note

The first 10 terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.

a. If n is even, then F is an odd number.

b. 2F-Fn-2 = Fn+1 for n 23

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}

Activity in Limit Theorems

Compute the following limits.

1. lim (4 • f(x))

x→c

2. lim (g(x) - h (x))

x→c ________

3. lim √12 • f(x)

x→c

4. lim (g(x) + h(x)) / f(x)

x→c

5. lim (f(x) + h(x))

x→c

Activity in Limit Theorems

Directions: Assume the following.

lim f(x) = 3/4;

x→c

lim g(x) = 12;

x→c

lim h(x) = -3;

x→c

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.

 QD099_September_2017

Page 7 of 10

   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)