Review practice · Section 1 of 2

In a loan scheme, the amount owed is $M$ dollars at time $t$ months; both variables are continuous. Interest increases the amount at a rate proportional to the amount owed, while repayment occurs at a constant rate of $500 per month. Interest and repayment balance when $M=10000$. An initial loan of $8000 is taken on 1 March 2013.

(i) Show $dM/dt=(M-10000)/20$.2 marks

(ii) Find $M$ in terms of $t$.4 marks

(iii) Find the date when the loan is cleared.2 marks

Show that the general solution of $y(dy/dx)=2x-1$, $y\ge0$, is $y^2=2x^2-2x+C$, where $C$ is arbitrary. Sketch the curve passing through $(0,1)$.

(i) Use $z=(2x-1)dy/dx$ to express $(2x-1)d^2y/dx^2=2(1-dy/dx)$ as an equation in $z,x$.2 marks

(ii) Solve with $y=2$ and $dy/dx=2$ at $x=0$.5 marks

(iii) Describe the gradient as (a) $x\to\pm\infty$ 1 mark and (b) $x\to1/2$.1 mark

(iv) Sketch the solution from (ii).2 marks

Using $y=vx$, solve $dy/dx=y(x+y)/[x(y-x)]$.

(a) Find $y=f(x)$ through $(1,0)$ and $(-1,2)$ with $dy/dx=3x^2-x^{-2}+C$.3 marks

(b) Perry jogs a distance $x$ kilometres in $t$ hours. The difference between his initial speed and his speed at time $t$ is proportional to distance covered. He starts at $5$ km/h and travels at $4$ km/h after $2$ km. Find the exact time for a $5$ km jog.6 marks

A town has population $3000$. The number infected after $t$ days satisfies $dI/dt=kI(3000-I)$, $k>0$.

(i) Given $I(0)=30$, show $I=3000/(1+99e^{-3000kt})$.6 marks

(ii) Given $I(7)=240$, find when $90\%$ of the population is infected.3 marks

(iii) State one contextual assumption needed for the model.1 mark