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 = x3ex
b) y = e2x2-x
c) y = ex3lnx
d) y = eex2
e) y = √ex2 + 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 = √ex2 + 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)ex2+ 6x dx
d) ∫ex/ex -1 dx
e) ∫3e2x/√(1 - e2x) 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)]