Rules of differentiation REVIEW:

The Chain Rule

Taylor series

Approximating the derivative

Monday Sept 14th: Univariate Calculus 2 Integrals ODEs Exponential functions

Antiderivative (indefinite integral)

Area under a curve = definite integral

Integrating data: the trapezoidal rule Very similar!

Example: integrating a linear function

Another angle: the upper limit as an argument

Differential equations Algebraic equation: involves functions; solutions are numbers. Differential equation: involves derivatives; solutions are functions. INITIAL CONDITION

e.g. dead reckoning

Example

Classification of ODEs Linearity: Homogeneity: Order:

Superposition (linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.

Superposition (linear, inhomogeneous equations)

Superposition (nonlinear equations)

ORDINARY differential equation (ODE): solutions are univariate functions PARTIAL differential equation (PDE): solutions are multivariate functions

1 slope=1 Exponential functions: start with ODE Qualitative solution:

Exponential functions: start with ODE Analytical solution

Exponential functions: start with ODE Analytical solution

Rules for addition, multiplication, exponentiation

Differentiation, integration (chain rule)

Properties of the exponential function Sum rule: Power rule: Taylor series: Derivative Indefinite integral

Cooling

Sinking

Homework: Do exercises for section 2.6, 2.8 and 2.9. Omit 2.9, #1. This will include: Exercise with antiderivatives and classifying ODEs. Carbon dating (for Thursday field trip) Derive further well-known functions from f’’=-f