Express the solution set of the given inequality in the

Question

Question 1:

In each of the following problems, express the solution set of the given inequality in the interval notation

a) 8x + 2 < 3x -18

b) 2x² + 5x -3 > 0

c) 3x+1/x + 4 + 1 ≥ 1

d) X² + 4x + 3/x-1 > 0

e) - 3 < 1- 6x ≤ 4

Question 2:

In each of the following problems, express the solution set of the given inequality in the interval notation

a) |2x + 3| ≤ 9

b) |3x -1| > 5x- 2

c) |(x -1)³|/2< 4

d) -2 ≤ |4x+3| < 5

e) |x + 1| ≥ x²/4

Question 3:

Differentiate the following expressions with respect to x.

a) y = x³e^x

b) y = e²x²-x

c) y = e^_x3lnx

d) y = e^_ex2

e) y = √e^_x2 + e√^_x2

Question 4

Differentiate the following expressions with respect to x.

a) y = ln(5x² + 9)

b) y = ln(5x² + 9)³

c) y = (ln(5x² + 9))³

d) y = ln(ln x + ln(ln 2))

e) y = √e^_x2 + e√^_x2

e) y = 1/lnx

Question 5

(a) Find and classify all the stationary points of f (x) = 3x(x + 4)2/3

(b) Find the Global Minimum and Maximum of f (x) = 3x(x + 4)^2/3 on [-5,-1].

Question 6

For the following function of two variables find -∂z/∂x and - ∂z/∂y.

a) z = 2x²y³ - x³y⁵

b) z = x² - y²/xy

c) z = ln(x² + xy + y²)

d) z = ex³ - Y²

Question 7

Find each of the following definite integrals

a) ∫_-1² (3x² - 2x+ 3)dx

b) ∫₀¹(³√x⁴ - 2⁴√x)dx

C) ∫₁⁴(x⁴ - 8)/x²dx

d) ∫₁⁴ x² +1/(√X³ + 3x)

e) ∫₁⁴ (√x -1)3/(√x)dx

Question 8

Find each of the following indefinite integrals

a) ∫³√(2x - 4dx)

b) ∫x² (x³ + 5)⁹ dx

c) ∫(x+ 3)e^{x2+ 6x} dx

d) ∫e^x/e^x -1 dx

e) ∫3e^2x/√(1 - e^2x) dx

Question 9

A closed box is to be constructed with a base length 6 times the base width and volume of 1500 m3. The cost of the material of the base is $3 per m² and the cost of the top and sides is $6 per m2. Determine the dimensions of the box that will minimise the cost.

Question 10:

The weekly cost to produce Q shoes is given by:

C(Q)= 75,000+100Q- 0.03Q² + 0.000004Q³ 0 ≤ Q ≤ 10,000

and the demand function for the shoes is given by:

D(Q) = 200 - 0.005Q Q ≤ 10,000

Determine the marginal cost dC / dQ and marginal profit dP/dQ when 2500 shoes are sold. Assume that the company sells exactly what they produce. [HINT: Revenue function: R(Q) = Qx D(Q) and profit function: P(Q) = R(Q)- C(Q)]