Slope fields and solution curves Problem set A

For \(\dfrac{dy}{dx}=x^2-y\), complete a table of the gradient at \(x=−1, 0, 1\) and \(y=−1, 0, 1, 2\).

1. (a) Complete the table of gradients.
2. (b) Which points in your table have a flat segment?

Sketch the slope field of \(\dfrac{dy}{dx}=x-2y\) at the integer points with \(-2\le x\le 2\) and \(-2\le y\le 2\), on the grid shown. Make a table of gradients first.

1. (a) Draw the slope field.
2. (b) Which points on the grid have a flat segment?

Use your calculator to plot the slope field of \(\dfrac{dy}{dx}=3-y\) for \(0\le x\le 4\) and \(0\le y\le 6\).

1. (a) Sketch the slope field you see. Describe the segments above, on and below the line \(y=3\).
2. (b) Add the solution curve that starts at \((0, 1)\). Describe what happens to it as \(x\) increases.
3. (c) Write down the constant solution.

The slope field of \(\dfrac{dy}{dx}=\dfrac{x-y}{2}\) is shown, with the points P\((−2, 1)\), Q\((0, −2)\) and R\((2, 2)\).

1. (a) Sketch the solution curve through each of P, Q and R.
2. (b) Explain why your curves do not cross.

The number of fish \(P\) (in thousands) in a farm pond after \(t\) years satisfies \(\dfrac{dP}{dt}=0.5P(8-P)\).

1. (a) Find the two constant solutions.
2. (b) Find the gradient when \(P=2\) and when \(P=10\).
3. (c) Describe what happens to a population of 10 thousand.

A mug of tea at \(50^\circ\text{C}\) cools in a room. Its temperature \(T\) after \(t\) minutes satisfies \(\dfrac{dT}{dt}=-0.5(T-20)\). The slope field is shown.

1. (a) Sketch the solution curve through \((0, 50)\).
2. (b) Estimate \(T\) when \(t=2\).
3. (c) State the temperature the tea approaches.

The number \(N\) (in thousands) of bacteria in a dish after \(t\) hours satisfies \(\dfrac{dN}{dt}=0.4N\left(1-\dfrac{N}{15}\right)\).

1. (a) Find the two constant solutions.
2. (b) Say which constant solution is stable. Give a reason using the sign of the rate of change on each side.
3. (c) Find the value of \(N\) at which the population grows fastest, and the greatest rate of growth.

A slope field is shown. It belongs to one of these equations: \(\text{P: }\dfrac{dy}{dx}=x^2+y\), \(\text{Q: }\dfrac{dy}{dx}=y-x^2\), \(\text{R: }\dfrac{dy}{dx}=x-y\).

1. (a) Work out the gradient of each equation at \((2, 0)\).
2. (b) Which equation gives the field shown? Use the segment at \((2, 0)\) to decide.

The number of rabbits \(P\) (in hundreds) on an island after \(t\) years satisfies \(\dfrac{dP}{dt}=0.1P(6-P)\). The slope field is shown.

1. (a) Show that \(P=0\) and \(P=6\) are constant solutions.
2. (b) Sketch the solution curve through \((0, 1)\) and estimate the time when \(P=3\).
3. (c) Sketch the solution curve through \((0, 7)\). Explain why it never goes below \(P=6\).