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					Calculus Chapter 3
Derivatives

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



You’ll need to “snow” this

Formal Definition of the Derivative of a function
 f’(x)=



lim h->0

f(x+h) – f(x) h

Notation for derivative
 y’

 dy/dx
 df/dx  d/dx  f’(x) D

(f)

(f)

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 slopeslope of the tangent line to a curve see diagram

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

compare
Slope= y-y  x-x  Slope of secant line


m= f ’(x)  Slope of tangent line


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
iff the function’s right-hand and lefthand derivatives exist and are equal.

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

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

3.3 Differentiation Rules
1.

Derivative of a constant

3.3 Differentiation Rules
1.

2.

Derivative of a constant Power Rule for derivatives

3.3 Differentiation Rules
1.

2.
3.

Derivative of a constant Power Rule for derivatives Derivative of a constant multiple

3.3 Differentiation Rules
1.

2.
3. 4.

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

3.3 Differentiation Rules
1.

2.
3. 4. 5.

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

3.3 Differentiation Rules
1.

2.
3. 4. 5. 6.

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
1.

2.
3. 4. 5. 6.

7.

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
1.

2.
3. 4.

5.
6. 7.

8.

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
1.

2.
3. 4. 5. 6. 7. 8. 9.

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

Average velocity of a “body” moving along a line

Defintion
Instantaneous Velocity is the derivative of the position function

Def. speed

Definition
Speed The absolute value of velocity

Definition
Acceleration

acceleration


Don’t drop the ball on this one.

Definition
Acceleration The derivative of velocity,

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

3.5 Derivatives of trig functions
 Y=

sin x

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

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

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

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

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

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


				
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