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
Rules for addition, multiplication, exponentiation
Differentiation, integration (chain rule)
Properties of the exponential function Sum rule: Power rule: Taylor series: Derivative Indefinite integral
Examples Add examples 6, 7 from notes.
Homework: Read examples 6 and 7 in text. (Should do in lecture) Do exercises for section 2.6, 2.7 and 2.8. This will include: Exercise with antiderivatives and classifying ODEs. Derivation of e x via compound interest. Carbon dating (for Tuesday field trip) Derive further well-known functions from f’’=-f