IB Maths AI HL – Topic 5: Calculus

Solve $\dfrac{dy}{dx} = 2y$ given that $y = 3$ when $x = 0$.

Consider $\dfrac{dy}{dx} = x^2 – y$ with $y(0) = 1$. Use Euler’s method with step length $h = 0.2$ to find an approximation for $y(0.6)$. Give each intermediate value to 4 decimal places.

The slope field for $\dfrac{dy}{dx} = y – x$ is drawn on the region $-3 \le x \le 3$, $-3 \le y \le 3$.(a) Write down the equation of the line along which all the slope segments are horizontal.(b) Describe the gradient of the segments at points above and below this line.(c) Show that $y = x + 1$ is a solution, and describe the shape of the solution curve through $(0, 2)$.

Find the general solution of $\dfrac{dy}{dx} = \dfrac{\cos x}{y}$, and the particular solution with $y = 2$ when $x = 0$.

Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$, $y > 0$, given that $y = 3$ when $x = 0$, giving $y$ explicitly in terms of $x$.

A cup of coffee cools according to $\dfrac{dT}{dt} = -k(T – 20)$, where $T$ is the temperature in $^\circ$C after $t$ minutes. Initially $T = 90$ and after 10 minutes $T = 60$.(a) Show that $T = 20 + 70e^{-kt}$.(b) Find $k$.(c) Find the temperature after 30 minutes.

The number of fish $P$ in a lake satisfies $\dfrac{dP}{dt} = 0.4P\left(1 – \dfrac{P}{500}\right)$, with $P = 50$ when $t = 0$ (years).(a) State the carrying capacity.(b) The solution is $P = \dfrac{500}{1 + Ae^{-0.4t}}$. Find $A$.(c) Find $P$ after 5 years.(d) Find the time at which the population is growing fastest.

Consider the system $\dfrac{dx}{dt} = x + 4y$, $\dfrac{dy}{dt} = x + y$.(a) Find the eigenvalues and corresponding eigenvectors of the coefficient matrix.(b) Write down the general solution.(c) Find the solution satisfying $x(0) = 4$, $y(0) = 0$.(d) Describe the phase portrait and state the long-term behaviour of $\dfrac{y}{x}$.

Consider $\dfrac{dx}{dt} = y$, $\dfrac{dy}{dt} = -x$, with $x(0) = 1$, $y(0) = 0$.(a) Use Euler’s method with $h = 0.1$ to estimate $x(0.2)$ and $y(0.2)$.(b) The exact solution is $x = \cos t$, $y = -\sin t$. Comment on the accuracy of the estimates.(c) Find the eigenvalues of the system and hence describe the phase portrait.

A predator–prey model is $\dfrac{dx}{dt} = 0.5x – 0.02xy$, $\dfrac{dy}{dt} = -0.3y + 0.01xy$, where $x$ is the prey and $y$ the predator population (hundreds).(a) Find the equilibrium points.(b) At the non-zero equilibrium, describe the behaviour of the populations for a starting point close to it.(c) If initially $x = 60$ and $y = 25$, state whether $y$ is initially increasing or decreasing, and justify.

The displacement $x$ of a damped oscillator satisfies $\dfrac{d^2x}{dt^2} + 2\dfrac{dx}{dt} + 5x = 0$.(a) By letting $y = \dfrac{dx}{dt}$, write the equation as a system of two first-order equations.(b) Find the eigenvalues of the system and describe the phase portrait.(c) With $x(0) = 1$, $y(0) = 0$, use Euler’s method with $h = 0.1$ to estimate $x(0.2)$.

Salt water is pumped into a tank so that the mass $m$ kg of salt after $t$ minutes satisfies $\dfrac{dm}{dt} = 10 – 0.05m$, with $m = 0$ initially.(a) Solve the equation to find $m$ in terms of $t$.(b) Find the limiting mass of salt in the tank.(c) Find the time taken for the mass to reach 150 kg.