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

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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
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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.