Calculus — IB Math AI HL Study Guide

For: IB Mathematics Applications & Interpretation HL candidates.

Covers: IB AI HL Topic 5 — derivatives and rate of change in context, integration for area/volume, separable and first-order differential equations, Euler's method (HL extension), applied optimisation.

You should already know: Functions (Topic 2), basic limits.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB AI HL style for educational use. They are not reproductions of past IBO papers.

1. Why Calculus in AI HL

Topic 5 makes up about 22-28% of AI HL — the heaviest topic. The "AI" emphasis means most problems are applied — projectile motion, growth/decay, optimisation in business, accumulation of quantities. HL extension topics not in AI SL: differential equations, Euler's method, slope fields, chain/product/quotient rule applications.

2. Derivatives in context

For $y=f(x)$, the derivative $f^{\prime }(x)=\frac{dy}{dx}$ measures the rate of change at a point. In applied problems:

• $f^{\prime }(x)$ tangent slope; $f^{\prime }(x)>0$ increasing, $f^{\prime }(x)<0$ decreasing.
• $f^{\prime \prime }(x)>0$ concave up; $f^{\prime \prime }(x)<0$ concave down.
• Maxima where $f^{\prime }=0$ and $f^{\prime \prime }<0$; minima where $f^{\prime }=0$ and $f^{\prime \prime }>0$.

Common derivatives (memorise):

• $(x^n{)}^{\prime }=n{x}^{n-1}$
• $(e^x{)}^{\prime }=e^x$, $(\ln x{)}^{\prime }=1/x$
• $(\sin x{)}^{\prime }=\cos x$, $(\cos x{)}^{\prime }=-\sin x$
• Chain rule: $(f(g(x)){)}^{\prime }=f^{\prime }(g(x))\cdot g^{\prime }(x)$
• Product rule: $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$

3. Integration in context

Definite integral ${\int }_a^{b}f(x)dx$ measures the accumulated total as $x$ goes from $a$ to $b$. In applied problems:

• Area under a positive curve.
• Total distance from velocity: $\int vdt$.
• Net change of a quantity given its rate of change.

Common antiderivatives:

• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ ($n\ne -1$)
• $\int e^{x}dx=e^x+C$, $\int \frac{1}{x}dx=\ln |x|+C$
• $\int \sin xdx=-\cos x+C$

Volume of revolution: $V=\pi {\int }_a^b[f(x){]}^{2}dx$ (rotating $y=f(x)$ about the x-axis).

4. Differential equations (HL extension)

A separable ODE has the form $\frac{dy}{dx}=g(x)h(y)$. Solve by separating: $\int \frac{1}{h(y)}dy=\int g(x)dx$.

First-order linear ODEs of the form $y^{\prime }+p(x)y=q(x)$ are solved by integrating factor $e^{\int pdx}$. (HL syllabus stops short of this in some years — check IBO curriculum.)

Euler's method approximates an ODE solution numerically: $$y_{n+1}=y_n+h\cdot f(x_n,y_n)$$ where $h$ is the step size. Useful when an analytic solution doesn't exist.

Slope fields visualise an ODE: at each point, draw a small line segment with slope $dy/dx$. Solution curves follow these segments.

5. Optimisation

To maximise/minimise a function $f$ on $[a,b]$:

1. Compute $f^{\prime }(x)$ and find critical points (where $f^{\prime }=0$ or undefined).
2. Evaluate $f$ at each critical point and at the endpoints.
3. Largest is the max; smallest is the min.

In applied problems, often the constraint involves another variable — use it to express the objective in one variable before differentiating.

6. Worked Example

A bacterial population is modelled by $\frac{dP}{dt}=0.5P(1-P/100)$, with $P(0)=5$ (in thousands).

(a) Verify this is logistic with carrying capacity 100 and growth rate 0.5. (b) Use Euler's method with step size $h=0.5$ to estimate $P(1.0)$. (c) Find the equilibria and classify their stability.

Solution.

(a) Standard logistic form $dP/dt=rP(1-P/K)$ with $r=0.5$, $K=100$ ✓.

(b) At $t=0$, $P=5$: $dP/dt=0.5(5)(1-5/100)=2.5\cdot 0.95=2.375$. $P(0.5)=5+0.5\cdot 2.375=6.19$. At $t=0.5$, $P=6.19$: $dP/dt=0.5(6.19)(1-6.19/100)=3.095\cdot 0.9381=2.904$. $P(1.0)=6.19+0.5\cdot 2.904=7.64$ thousand.

(c) Equilibria where $dP/dt=0$: $P=0$ or $P=100$.

• At $P=0$: small positive perturbation gives $dP/dt>0$, so it grows away — unstable.
• At $P=100$: small positive perturbation $dP/dt<0$ (returns); small negative perturbation $dP/dt>0$ (returns) — stable.

7. Common Pitfalls

• Forgetting the chain rule: $\frac{d}{dx}\sin (2x)=2\cos (2x)$, not $\cos (2x)$.
• Wrong antiderivative for $1/x$: it's $\ln |x|$, with absolute value.
• Volumes of revolution sign: when the function takes negative values, $[f(x){]}^2$ keeps it positive — just compute as is.
• Euler's method local error: each step has $O(h^2)$ error, accumulating to $O(h)$ globally. Halving $h$ halves error.

8. Practice Questions

1. A car's velocity (m/s) is $v(t)=6t-t^2$ for $0\le t\le 6$. Find the maximum velocity, the total distance travelled, and the time when velocity is half the maximum.
2. Solve $\frac{dy}{dx}=3y$ with $y(0)=2$.
3. Use Euler's method with $h=0.2$ to estimate $y(0.6)$ for $y^{\prime }=x+y$, $y(0)=1$.

9. Quick Reference Cheatsheet

• $f^{\prime }$ slope; $f^{\prime \prime }>0$ concave up.
• Power rule: $(x^n{)}^{\prime }=n{x}^{n-1}$, $\int x^{n}dx=x^{n+1}/(n+1)+C$.
• Chain: $(f(g){)}^{\prime }=f^{\prime }(g)\cdot g^{\prime }$.
• Volume of revolution about x-axis: $\pi \int [f(x){]}^{2}dx$.
• Logistic ODE: $dP/dt=rP(1-P/K)$; equilibria at 0 (unstable) and $K$ (stable).
• Euler: $y_{n+1}=y_n+hf(x_n,y_n)$.

10. What's Next

Calculus in AI HL is closely linked to Topic 4 (Statistics) — for example, expectation as an integral of $xf(x)$. Use Ollie for any specific applied calculus problem: "Walk me through this projectile optimisation" or "How does Euler's method differ from Runge-Kutta?"