+89 1. (a)What is the present value of an annuity of monthly payments of e 500 over 25 years earning 2% p.a. interest compounded monthly?
(b) Find the annual percentage rate (APR) for the following interest rates
(i) 3.39% p.a. compounded annually
(ii) 3.29% p.a. compounded semi-annually,
(iii) 3.35% p.a. compounded quarterly, (iv) 3.4% p.a. compounded monthly .
Which of these gives the cheapest option when taking out a loan for two years? How much interest would it cost to borrow e1,000 for two years under the cheapest option?
(c) Find the final value of a savings account into which e 1000 are paid every three months for 10 years, if the bank is paying 2.5% p.a. interest compounded quar- terly.
(d) Differentiate each of the following functions and find the value of the derivative when x = 1:
(i) f(x)=2−3ln(x)+3x, 1 + 2x2ex
(ii)g(x)= 1+x2 ,
√
(e) Letf(x)=x3 −2x2 −4x.
(i) Find f′(x) and f′′(x).
(ii) Determine the critical points of f.
(iii) Show that x = −2 is a local maximum of f. 3
+9664 1(a) $500 x (1+2%)25 x 12
= $500 x 1.0230
= $905.6807920516769
\(\approx\) $ 905.68
(b)(i)\(\quad(\frac{1+0.0339}{1})^1-1\\ = 0.0339\)
APR = 3.39%
(b)(ii) \(\quad (\frac{2+0.0329}{2})^2-1\\ = 0.033170602.....\)
APR = 3.32.....%
(b)(iii) \(\quad (\frac{4+0.0335}{4})^4-1\\ = 0.033923......\)
APR = 3.392.....%
(b)(iv) \(\quad (\frac{12+0.034}{12})^{12}-1\\=0.0345348......\)
APR = 3.45......%
Which of these gives the cheapest option when taking out a loan for two years?
3.29% p.a. compounded semi-annually.
How much interest would it cost to borrow $1,000 for two years under the cheapest option?
1000 x 3.32% x 2
= $66.4
(c) 1000 x (1+2.5%)4 x 10
= 1000 x 2.685063838
= $2685.1 approx.
(d) f(x) = 2 - 3ln(x) + 3x
\(f'(x)= \frac{3}{x}+3\\ f'(1)= \frac{3}{1}+3 = 6\)
f(x) = 2x2ex
\(f'(x)= 2xe^x(x+2)\\ f'(1)= 2(1)(e)(1+2)= 6e\)
g(x) = 1 + x2
\(g'(x)= 2x\\ g'(1)= 2(1)=2\)
(e) \(f(x)=x^3-2x^2-4x\\ f'(x)= 3x^2-4x-4\\ f''(x)= 6x-4\)
I don't know what's critical point and local maximum.......
Max: Everything appears to be right in your calculations except this one:
(c) Find the final value of a savings account into which e 1000 are paid every three months for 10 years, if the bank is paying 2.5% p.a. interest compounded quar- terly.
FV=P{[1 + R]^N - 1/ R}=FV OF $1 PER PERIOD.
FV=1,000[1 + .025/4]^(10*4) -1/(.025/4)}
FV=1,000{[1.283027 - 1] / .00625}
FV=1,000 x 45.284291
FV=45,284.29e
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