Reference no: EM132310967
Algebra
1. Evaluate (i) log √e (ii) log1/e2 (iii) log bc/d, if log b = Π, log c = √-2, and log d = 11.
2. A knick-knack purchased in 1998 was worth $14.15 in 2007 and $7.85 in 2012. Assume that its value, V, is given by V = Ac-t, where is the number of years since purchase, with and being some positive constants. How much was the knick-knack worth when it was purchased?
How much was it worth in 2018?
3. a) Find all such that 2 +3 ≤ 4 +5 < 6-7.
(b) Solve the inequality (2x - 3)/(4x - 5) < 1
(c) Solve |4x - 9|≥|3x + 2|, including a sketch to illustrate your solution.
4. (a) Find the sixth term of a geometric progression of positive numbers, where the third term is 9 and the 9th term is 128.
(b) Find the sum of a 8-term geometric progression if the first term is 177147 and the common ratio is 2/3.
(c) Find four numbers in geometric progression, if the first two add up to 75, and the other two add up to 1200.
5. (a) What are the monthly payments on a 20-year, $200,000 loan having 6% annual interest, compounded monthly? Interest is computed based on the balance at the beginning of each month, and payments are made at the end of each month.
(b) How much must be deposited each month to achieve a balance of $200,000 at the end of 25 years, at 6% annual interest compounded monthly? Deposits are made at the beginning of each month, and interest is paid on the balance at the end of each month.
Calculus
6. Find the derivative of each of the following functions.
(a) R(Q) = 5√?Q - Q/20 -1/20
(b) K(v) = 1/v -3/√v
(c) G(w) = (w3 + 7w)1/3
7. Let f(x) = (x -2)/(x + 1)2
(a) Find the equation of the tangent to at = 4.
(b) Find and classify the stationary points of f.
(c) Find the global maximum and minimum of the function on the interval [1,4].
(d) Determine if there are any inflection points of f.
8. Exercise 7.7 from Elements of Mathematics for Economics and Finance, Mavron.
In addition, show that for the number of floors that minimises the average cost per floor, the marginal cost is the same as the average cost.
Explain why these two values must be the same in order to minimise the average cost per floor.
9. The diagram shows a contour diagram for the monthly payment as a function of the interest rate, r%, and the amount, L, of a 5-year loan.
(a) Estimate ∂P/∂r and ∂P/∂L at the values r = 8 and L = 4000.
(b) What is the financial interpretation of your result for in the previous part?
10. Exercise 9.6 from Elements of Mathematics for Economics and Finance, Mavron.
For part (b)(ii), compare your estimate (using the interpretation of λ-see the lecture notes or the textbook) with the answer obtained directly from solving the problem with the new constraint.