# 香港數學代考|Financial Mathematics and Statistics

Problem 1.

a) Solve $e^x=9$. Write your answer correct to 2 decimal places,
b) Differentiate $y=x e^x$

c) Find the equation of the tangent to the curve $y=1+\ln (2 x)$ at $x=1$.
d)

The diagram above shows part of the graph of $y=\frac{2}{x-2}+2$.
The curve intersects the $x$-axis at $x=1$.
Find the exact value of the shaded area.

Problem 2.

a) Write $72^{\circ}$ in radians in terms of $\pi$
b) Solve $3 \tan ^2 x=1$ for $0 \leq x \leq 2 \pi$
c) Solve $\log _2(x)=1+\log _2(x-1)$
d) A company manufacturing electrical components finds that the cost (in \$) of$\quad 2$producing$x$components per month is given by$C(x)=72000+60 x$Find the marginal average cost at a production level of 300 components per month and interpret the result. Problem 3. A company produces$x$pillows per week. The cost equation is$C(x)=200-40 x$and revenue is given by$R(x)=600 x-2 x^2$. Which of the following gives the profit?$P(x)=200-2 x^2-640 xP(x)=560 x-2 x^2-200P(x)=640 x-2 x^2-200P(x)=2 x^2-640 x+200$Problem 4. Which of the following best shows the solution of$3^x<0.5$?$x>-1.585x<-1.585x>-0.631x<-0.631$Problem 5. The share price$(\$P)$ of a renewable energy company is increasing such that $\frac{d P}{d t}=\frac{6}{e^t}$, where $t$ is the time in months since the listing of the company. The initial price of the shares is $\$ 2$. Which of the following is closest to the share price after 3 months?$\$8.30$
$\$ 2.30\$7.70$
$\$ 18.00\$

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