Calculus and Differential Equations

Year 1 · 7 sections · 88 concepts · 293 practice questions

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What's covered

free1 · Functions and limits12 concepts

Overview 1 · The plan of this section
Definition 1.1 · Functions, domains, composition and inverses
Definition 1.2 · Exponentials and logarithms
Definition 1.3 · Inverse trigonometric functions
Definition 1.4 · Hyperbolic functions
Proposition 1.5 · Hyperbolic identities and inverse hyperbolic functions
Definition 1.6 · Limits
Technique 1.7 · Computing limits
Proposition 1.8 · Standard limits
Definition 1.9 · Continuity and the intermediate value theorem
Definition 1.10 · Limits at infinity and asymptotes
Check 1.11 · Check: evaluating a limit

2 · Differentiation techniques15 concepts

Overview 2 · The plan of this section
Definition 2.1 · The derivative
Proposition 2.2 · Derivatives of the basic functions
Theorem 2.3 · The product, quotient and chain rules
Proposition 2.4 · Derivatives of trig, inverse trig and hyperbolic functions
Technique 2.5 · Implicit differentiation
Technique 2.6 · Logarithmic differentiation
Proposition 2.7 · Parametric differentiation
Proposition 2.8 · Higher derivatives and Leibniz's rule
Proposition 2.9 · Stationary points and their nature
Technique 2.10 · Curve sketching
Technique 2.11 · Optimisation
Technique 2.12 · Related rates
Technique 2.13 · Newton's method
Check 2.14 · Check: differentiating correctly

3 · Integration techniques15 concepts

Overview 3 · The plan of this section
Definition 3.1 · Antiderivatives and definite integrals
Theorem 3.2 · The fundamental theorem of calculus
Proposition 3.3 · Standard integrals
Technique 3.4 · Integration by substitution
Technique 3.5 · Integration by parts
Technique 3.6 · Partial fractions
Technique 3.7 · Trigonometric integrals
Technique 3.8 · Trigonometric and hyperbolic substitutions
Technique 3.9 · Reduction formulae
Definition 3.10 · Improper integrals
Technique 3.11 · Areas between curves
Technique 3.12 · Volumes of revolution
Technique 3.13 · Arc length and surface area
Check 3.14 · Check: choosing an integration method

4 · Taylor series and approximation11 concepts

Overview 4 · The plan of this section
Definition 4.1 · Taylor polynomials
Theorem 4.2 · Taylor's theorem with remainder
Definition 4.3 · Taylor series
Proposition 4.4 · The standard Maclaurin series
Proposition 4.5 · The binomial series
Technique 4.6 · Building new series from old
Technique 4.7 · Limits using Taylor series
Technique 4.8 · Approximation with error bounds
Example 4.9 · Euler's formula from series
Check 4.10 · Check: series work

5 · First-order ODEs13 concepts

Overview 5 · The plan of this section
Definition 5.1 · Differential equations and initial value problems
Technique 5.2 · Separable equations
Theorem 5.3 · First-order linear equations
Technique 5.4 · Exact equations
Technique 5.5 · Homogeneous equations (y = vx)
Technique 5.6 · Bernoulli equations
Theorem 5.7 · Existence and uniqueness (Picard)
Proposition 5.8 · Autonomous equations and equilibria
Example 5.9 · Growth, decay and Newton's law of cooling
Example 5.10 · Mixing problems
Example 5.11 · The logistic equation
Check 5.12 · Check: solving a first-order ODE

6 · Second-order linear ODEs13 concepts

Overview 6 · The plan of this section
Proposition 6.1 · Superposition
Theorem 6.2 · The general solution of the homogeneous equation
Theorem 6.3 · Constant coefficients and the characteristic equation
Proposition 6.4 · General solution = complementary function + particular integral
Technique 6.5 · Undetermined coefficients
Theorem 6.6 · Variation of parameters
Example 6.7 · Simple harmonic motion
Example 6.8 · Damped oscillations
Example 6.9 · Forced oscillations and resonance
Technique 6.10 · Euler–Cauchy equations
Technique 6.11 · Reduction of order
Check 6.12 · Check: second-order linear ODEs

7 · Systems and difference equations9 concepts

Overview 7 · The plan of this section
Definition 7.1 · Linear systems of ODEs
Theorem 7.2 · Solving x' = Ax with eigenvectors
Proposition 7.3 · Complex eigenvalues give oscillations
Proposition 7.4 · Phase portraits and stability of 2D linear systems
Proposition 7.5 · First-order linear recurrences
Theorem 7.6 · Second-order linear recurrences
Technique 7.7 · Inhomogeneous recurrences
Check 7.8 · Check: systems and recurrences

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Proposition 1.5 · Hyperbolic identities and inverse hyperbolic functions

$\cosh^2 - \sinh^2 = 1$ (compare $\cos^2 + \sin^2 = 1$ ). Inverses are logs: $\operatorname{arsinh}x = \ln(x + \sqrt{x^2 + 1})$ . Method: set $y = \sinh x$ , get a quadratic in $e^x$ , keep the positive root.

Proof

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