Rules of differentiation
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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). Antiderivative (indefinite integral).

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Monday sept 14th univariate calculus 2

Monday Sept 14th: Univariate Calculus 2

Integrals

ODEs

Exponential functions


Antiderivative indefinite integral
Antiderivative (indefinite integral)


Antiderivative indefinite integral1
Antiderivative (indefinite integral)





Another angle the upper limit as an argument
Another angle: the upper limit as an argument


Another angle the upper limit as an argument1
Another angle: the upper limit as an argument


Another angle the upper limit as an argument2
Another angle: the upper limit as an argument


Differential equations
Differential equations

Algebraicequation: involves functions; solutions are numbers.

Differential equation: involves derivatives; solutions are functions.

INITIAL CONDITION




Classification of odes
Classification of ODEs

Linearity:

Homogeneity:

Order:


Superposition linear homogeneous equations
Superposition(linear, homogeneous equations)

Can build a complex solution from the sum of two or more simpler solutions.


Superposition linear inhomogeneous equations
Superposition(linear, inhomogeneous equations)


Superposition linear inhomogeneous equations1
Superposition(linear, inhomogeneous equations)


Superposition nonlinear equations
Superposition(nonlinear equations)


ORDINARY differential equation (ODE): solutions are univariate functions

PARTIAL differential equation (PDE): solutions are multivariate functions


Exponential functions: start with ODE

Qualitative solution:

slope=1

1







Properties of the exponential function
Properties of the exponential function

Taylor series:

Sum rule:

Power rule:

Derivative

Indefinite integral




Homework
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


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