Calculus Chapter 3

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Calculus Chapter 3:

Calculus Chapter 3 Derivatives

3.1 Informal definition of derivative:

3.1 Informal definition of derivative

3.1 Informal definition of derivative:

3.1 Informal definition of derivative A derivative is a formula for the rate at which a function changes.

Formal Definition of the Derivative of a function:

Formal Definition of the Derivative of a function

Slide 5:

You’ll need to “snow” this

Formal Definition of the Derivative of a function:

Formal Definition of the Derivative of a function f’(x)= lim f(x+h) – f(x) h->0 h

Notation for derivative:

Notation for derivative y’ dy/dx df/dx d/dx (f) f’(x) D (f)

Rate of change and slope:

Rate of change and slope Slope of a secant line See diagram

:

The slope of the secant line gives the change between 2 distinct points on a curve. i.e. average rate of change

Rate of change and slope- slope of the tangent line to a curve see diagram:

Rate of change and slope- slope of the tangent line to a curve see diagram

Slide 11:

The slope of the tangent line gives the rate of change at that one point i.e. the instantaneous change.

compare:

compare Slope= y-y x-x Slope of secant line m= f ’(x) Slope of tangent line

Time for examples:

Time for examples Finding the derivative using the formal definition This is music to my ears!

A function has a derivative at a point:

A function has a derivative at a point

A function has a derivative at a point:

A function has a derivative at a point iff the function’s right-hand and left-hand derivatives exist and are equal.

Theorem:

Theorem If f (x) has a derivative at x=c,

Theorem:

Theorem If f (x) has a derivative at x=c, then f(x) is continuous at x=c.

Finding points where horizontal tangents to a curve occur:

Finding points where horizontal tangents to a curve occur

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules Higher order derivatives

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules Higher order derivatives Product rule

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules Higher order derivatives Product rule Quotient rule

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules Higher order derivatives Product rule Quotient rule Negative integer power rule

3.3 Differentiation Rules:

3.3 Differentiation Rules Derivative of a constant Power Rule for derivatives Derivative of a constant multiple Sum and difference rules Higher order derivatives Product rule Quotient rule Negative integer power rule Rational power rule

3.4 Definition:

3.4 Definition Average velocity of a “body” moving along a line

Defintion:

Defintion Instantaneous Velocity is the derivative of the position function

Def. speed:

Def. speed

Definition:

Definition Speed The absolute value of velocity

Definition:

Definition Acceleration

acceleration:

acceleration Don’t drop the ball on this one.

Definition:

Definition Acceleration The derivative of velocity,

Definition:

Definition Acceleration The derivative of velocity, Also ,the second derivative of position

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x Y= cos x

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x Y= cos x Y= tan x

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x Y= cos x Y= tan x Y= csc x

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x

3.5 Derivatives of trig functions:

3.5 Derivatives of trig functions Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x Y= cot x

TEST 3.1-3.5:

TEST 3.1-3.5 Formal def derivative Rules for derivatives Notation for derivatives Increasing/decreasing Eq of tangent line Position, vel, acc Graph of fct and der Anything else mentioned, assigned or results of these

Whereas :

Whereas The slope of the secant line gives the change between 2 distinct points on a curve. i.e. average rate of change