Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

IB Mathematics Analysis and Approaches HL Calculus Reference cheat sheet - grade 11-12

Click image to open full size

Calculus Grade 11-12

IB Mathematics Analysis and Approaches HL Calculus Reference Cheat Sheet

A printable reference covering limits, derivatives, integrals, differential equations, Maclaurin series, and IB HL calculus applications for grades 11-12.

TopicsCalculusDerivativesIntegralsFunctionsGraphsAlgebraStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

This cheat sheet covers the core calculus tools needed for IB Mathematics Analysis and Approaches HL. It is designed to help students quickly review essential derivative rules, integration techniques, series, and differential equations before solving exam-style problems. Students need these formulas because HL calculus questions often combine several skills in one multi-step solution.

A clear reference helps reduce mistakes and supports faster recognition of the right method.

The main ideas are limits, rates of change, accumulated change, and approximation. Differentiation focuses on rules such as the product rule, quotient rule, chain rule, implicit differentiation, and related rates. Integration includes substitution, integration by parts, partial fractions, definite integrals, volumes, and solving separable differential equations.

HL topics also include Maclaurin series, convergence ideas, and using calculus to model motion, growth, and optimization.

Key Facts

  • The derivative from first principles is f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
  • The product rule is ddx[uv]=u′v+uv′\frac{d}{dx}[u v]=u'v+uv' and the quotient rule is ddx(uv)=u′v−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}.
  • The chain rule is dydx=dydu⋅dudx\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} for a composite function y=f(u)y=f(u) and u=g(x)u=g(x).
  • For inverse trigonometric functions, ddx(sin⁡−1x)=11−x2\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}, ddx(cos⁡−1x)=−11−x2\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^2}}, and ddx(tan⁡−1x)=11+x2\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}.
  • The fundamental theorem of calculus states that if F′(x)=f(x)F'(x)=f(x), then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a).
  • Integration by parts is ∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du.
  • A separable differential equation can be solved by rearranging to g(y) dy=f(x) dxg(y)\,dy=f(x)\,dx and then integrating both sides.
  • The Maclaurin series for a function is f(x)=f(0)+f′(0)x+f′′(0)2!x2+f′′′(0)3!x3+⋯f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots when the series represents the function.

Vocabulary

Limit
A limit describes the value that a function approaches as the input approaches a particular value.
Derivative
A derivative measures the instantaneous rate of change of a function and the gradient of its tangent line.
Antiderivative
An antiderivative of f(x)f(x) is a function F(x)F(x) such that F′(x)=f(x)F'(x)=f(x).
Definite integral
A definite integral ∫abf(x) dx\int_a^b f(x)\,dx gives the signed area under f(x)f(x) from x=ax=a to x=bx=b.
Differential equation
A differential equation is an equation involving a function and one or more of its derivatives.
Maclaurin series
A Maclaurin series is a power series expansion of a function about x=0x=0.

Common Mistakes to Avoid

  • Forgetting the chain rule in composite functions is wrong because differentiating sin⁡(x2)\sin(x^2) as cos⁡(x2)\cos(x^2) misses the factor 2x2x.
  • Dropping the constant of integration in an indefinite integral is wrong because ∫f(x) dx=F(x)+C\int f(x)\,dx=F(x)+C represents a family of functions.
  • Using the product rule as ddx[uv]=u′v′\frac{d}{dx}[uv]=u'v' is wrong because each factor changes while the other factor is held in the two terms u′v+uv′u'v+uv'.
  • Treating signed area as total area is wrong because ∫abf(x) dx\int_a^b f(x)\,dx subtracts regions below the xx-axis unless absolute value or separate intervals are used.
  • Applying a Maclaurin series outside its interval of convergence is wrong because the infinite series may not equal the original function for those xx-values.

Practice Questions

  1. 1 Differentiate y=x2e3xsin⁡xy=x^2e^{3x}\sin x with respect to xx.
  2. 2 Evaluate ∫01x1+x2 dx\int_0^1 x\sqrt{1+x^2}\,dx exactly.
  3. 3 Solve the differential equation dydx=3xy\frac{dy}{dx}=3xy given that y(0)=2y(0)=2.
  4. 4 Explain how the first derivative and second derivative together can be used to classify a stationary point of a function.

Understanding IB Mathematics Analysis and Approaches HL Calculus Reference

Calculus becomes easier when each symbol is linked to a physical or graphical meaning. A derivative is not only a rule for changing powers. It describes how fast one quantity changes at one instant.

On a graph, it gives the slope of the tangent line. A positive derivative means the graph rises as the input increases. A negative derivative means it falls.

A derivative of zero marks a stationary point, but that point is not automatically a maximum or minimum. Students should check the sign of the derivative on either side, or use a second derivative, before making a conclusion.

An integral has a different job. It combines many tiny changes to find a total. The signed area under a graph can represent displacement when the graph shows velocity against time.

Areas below the horizontal axis count as negative, so displacement and total distance are often different. To find total distance, split the journey wherever velocity changes sign and add positive lengths. This distinction appears in motion problems, economics graphs, and any context where a rate can become negative.

Units provide a useful check. Integrating velocity in metres per second over seconds gives metres.

Many calculus errors come from choosing a method too quickly. Before differentiating, identify the structure of the expression. A product has two changing factors.

A quotient has a changing denominator. A function inside another function needs the chain rule. For integration, first look for a reverse chain rule pattern.

Substitution works when one part of the expression is closely related to the derivative of another part. Integration by parts is often useful when multiplying functions of different types, such as a polynomial and an exponential function. Partial fractions only begins after a rational expression has been made proper, meaning the degree of the numerator is smaller than the degree of the denominator.

Differential equations turn a rate relationship into a model for change over time. A population model may state that its growth rate depends on the current population. Separating the variables gives a family of possible solutions.

An initial condition selects the one model that matches a real starting value. Keep the constant of integration until this stage. Maclaurin series serve a related purpose by replacing a complicated function with a polynomial near zero.

More terms usually improve the approximation near zero, but they do not guarantee accuracy far away. In HL work, clear reasoning matters as much as the final answer. State restrictions, show substitutions, include constants, and interpret results in the context given.