Knowledge organisers

Calculus: IB Maths AI SL knowledge organiser

Everything to know about calculus on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Derivative: The gradient function: how fast y changes as x changes, at each point.
Tangent: The straight line that touches the curve at a point and has the same gradient there.
Optimisation: Finding a maximum or minimum value, usually by solving f′(x) = 0.
Trapezoidal rule: Estimates the area under a curve by splitting it into trapezia of equal width.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Derivative of \(x^n\)In the formula booklet	\(f(x)=x^n\Rightarrow f'(x)=nx^{n-1}\)
Constant multiples, sums	\(f(x)=ax^n+bx^m\Rightarrow f'(x)=anx^{n-1}+bmx^{m-1}\)
Gradient of tangent at \(x=a\)	\(f'(a);\) \(\text{tangent }y-f(a)=f'(a)(x-a)\)
Gradient of normal	\(-\frac{1}{f'(a)}\)
Increasing / decreasing	\(f'(x)>0;\) \(f'(x)<0\)
Stationary points; optimisation	\(f'(x)=0;\) \(\text{check the sign of }f'\text{ either side}\)
Integral of \(x^n\), \(n\ne-1\)In the formula booklet	\(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Area, \(y\ge0\)In the formula booklet	\(A=\int_a^by\,dx\)
Trapezoidal rule, \(h=\tfrac{b-a}n\)In the formula booklet	\(\int_a^by\,dx\) \({}\approx\tfrac12h\big[(y_0+y_n)+2(y_1+\cdots+y_{n-1})\big]\)
Boundary condition: find \(C\) by substituting a known point into \(\int f'(x)\,dx\)

Worked example

f(x) = x³ − 3x. Find the gradient of the curve at x = 2 and the equation of the tangent there.

f′(x) = 3x² − 3, so f′(2) = 9
f(2) = 8 − 6 = 2, so the tangent passes through (2, 2)
y − 2 = 9(x − 2)

Answer: Gradient 9; tangent y = 9x − 16

Common mistakes

Calculus: not differentiating, f(a) for f′(a), and the derivative as a rate
Trapezoidal rule and integrals: heights, strip width and dx
Forgetting to multiply by the original exponent when differentiating xⁿ
Treating the derivative of a constant as anything other than 0

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the average rate of change of a function over an interval as the gradient of a chord, with units, in contexts such as speed.
Use the sign of f′(x) to find where a function increases or decreases, and sketch the graph of f′ from the graph of f.
Find indefinite integrals of axⁿ by reversing differentiation and use a boundary condition, such as a point on the curve, to find the constant c.
Find the points where a curve has a given gradient and write the equations of the tangent and normal there.
Interpret definite integrals as accumulated change, such as total water flow from a rate, and find areas with technology.
Use definite integrals with technology to find a total from a rate, for example a volume from a flow rate.

The printable sheet

Calculus knowledge organiser for IB Maths AI SL: one A4 page of key definitions, formulas, a worked example and common mistakes — Calculus knowledge organiser (IB Maths AI SL), A4. Download the PDF.

Revise it next

Other IB Maths AI SL topics: Number and algebra · Functions · Geometry and trigonometry · Statistics and probability · All IB Maths AI SL organisers