5.15 Slope fields questions for IB Maths AI

Find the equation of the tangent to this solution curve at $A$. Give your answer in the form $ax+by+d=0$, where $a,b,d\in \mathbb{Z}$.

On the diagram, sketch the solution curve that passes through $A$.

Write down the equation of the line on which every solution curve has a gradient of zero.

Show that $y=2x-2$ is a solution of the differential equation.

Show that the equilibrium solutions are $P=2$ and $P=8$.

State, with a reason, which of these equilibrium solutions is stable.

On the diagram, sketch the solution curves for which the initial number of fish is

(i) 1000;
(ii) 4000;
(iii) 10 000.

The lake initially contains 4000 fish. Use Euler's method with a step length of 0.5 to estimate the number of fish after 2 years.

The manager wants to increase the number of fish caught each year to $H$ thousand, with the model ${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}=0.5P(1-{\displaystyle \frac{P}{10}})-H$. Find the largest value of $H$ for which the model still has an equilibrium solution with $P>0$.

Explain why catching exactly this number of fish each year would be risky for the fishery.

Write down the equation of the curve on which every local maximum and local minimum of the solution curves lies.

Sketch this curve on the diagram.

On the diagram, sketch the solution curve that passes through the point $(0,1)$.

The solution curve through $(0,1)$ has a turning point at which $x<0$. Determine whether this turning point is a local maximum or a local minimum. Justify your answer.

Identify which diagram, A, B or C, shows the slope field of this differential equation. Give a reason for your answer.

Solve the differential equation, given that $y=0$ when $x=0$. Give your answer in the form $y=f(x)$.

Show that this solution is defined for all real values of $x$.

Find the minimum value of $y$ on this solution curve.

Use Euler's method with a step length of 0.2 to estimate $y$ when $x=1$, and find the percentage error in this estimate.

Match each differential equation to its slope field. Give a reason for each choice.

For differential equation (I), write down the equilibrium solution and describe the long-term behaviour of every other solution.

Find her terminal velocity according to this model.

On the diagram, sketch the solution curve for the skydiver.

Use Euler's method with a step length of 0.5 to estimate her velocity after 2 seconds.

The exact solution of the differential equation is $v={\displaystyle \frac{14({\mathrm{e}}^{1.4t}-1)}{{\mathrm{e}}^{1.4t}+1}}$. Find the percentage error in your answer to part (c).

With reference to the slope field, explain why Euler's method overestimates her velocity.

Let $s$ metres be the distance she has fallen. Using ${\displaystyle \frac{\mathrm{d}v}{\mathrm{d}t}}=v{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}s}}$, show that $v^2=196(1-{\mathrm{e}}^{-0.1s})$.

Find her velocity after she has fallen 50 m.