DP Mathematics: Applications and Interpretation Questionbank

(i)   the population of rabbits 1 year after the foxes were introduced.

(ii)  the population of foxes 1 year after the foxes were introduced.

point $\text{B}$.

point $\text{A}$.

Find the non-zero equilibrium point for the populations of rabbits and foxes.

Write down expressions for $M_{n+1}$ and $S_{n+1}$ in terms of $M_n$ and $S_n$.

Use Euler’s method to find an estimate for the mackerel population density after one year.

Show that $r\approx \frac{N\left(n+1\right)-N\left(n\right)}{N\left(n\right)}$

Hence find three approximations for the value of $r$.

Find the equation of the regression line of $h$ on $t$.

Suggest why Eva’s use of the linear regression equation in this way could be unreliable.

Find the equation of the least squares quadratic regression curve.

Find the value of $k$.

Hence, write down a suitable domain for Eva’s function $h\left(t\right)=p{t}^2+qt+r$.

Interpret the meaning of parameter $a$ in the context of the model.

By solving the differential equation $\frac{dh}{dt}=-R^2\sqrt{70 560h}$, show that the general solution is given by $h=17 640{\left(c-R^{2}t\right)}^2$, where $c\in \mathrm{ℝ}$.

Show that $\frac{dH}{dt}\approx 0.2514-0.009873t-0.1405\sqrt{H}$, where $0\le t\le T$.

Use the general solution from part (d) and the initial condition $h\left(0\right)=3.2$ to predict the value of $T$.

Find this new height.

Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$.

Show that $\frac{dh}{dt}=-R^2\sqrt{70 560h}$.

Find the equilibrium population of brown squirrels suggested by this model.

Show that $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=2{\text{e}}^x\cos x$.

Verify that $x=800$, $y=600$ is an equilibrium point.

Given that the initial populations are $x=100$, $y=100$, find the populations of each species of squirrel when $t=1$.

Use the same method to find the long-term populations of squirrels when the initial populations are $x=400$, $y=100$.

Show that the function $f$ has a local maximum value when $x=\frac{3\pi }{4}$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Use Euler’s method with step length 0.2 to sketch, on the same axes, the approximate trajectories for the populations with the following initial populations.

(i)      $x=1000$, $y=1500$

(ii)    $x=1500$, $y=1000$

Write down the general solution of $\frac{\text{d}y}{\text{d}t}=3y$.

Explain why the population of squirrels is increasing for values of $x$ less than this value.

Find the other three equilibrium points.

Given that the equilibrium point at (800, 600) is a saddle point, sketch the phase portrait for $x$ ≥ 0 , $y$ ≥ 0 on the same axes used in part (e).

Find the area of the region enclosed by the graph of $f$ and the $x$-axis.

Find the $x$-coordinate of the point of inflexion of the graph of $f$.

Find the value $\kappa$ for $x=\frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.

Find the value of the curvature of the graph of $f$ at the local maximum point.

By using separation of variables, show that the general solution of $\frac{\text{d}x}{\text{d}t}=2x$ is $x=A{\text{e}}^{2t}$.

Write down the expressions for $x_{n+1}$ and $y_{n+1}$ that the conservationists will use.

Sketch the graph of $f$, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.

If both populations contain 10 squirrels at $t=0$ use the solutions to parts (c) (i) and (ii) to estimate the number of black and brown squirrels when $t=0.2$. Give your answers to the nearest whole numbers.

Use further iterations of Euler’s method to find the long-term population for each species of squirrel from these initial values.