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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. - PowerPoint PPT Presentation
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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 Definitionof the Derivative of a function
You’ll need to “snow” this
Formal Definitionof the Derivative of a function
f’(x)= lim f(x+h) – f(x) h->0 h
Notation for derivative
y’ dy/dx df/dx d/dx (f) f’(x) D (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 slope-slope 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 left-hand 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. Derivative of a constant
2. Power Rule for derivatives
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
4. Sum and difference rules
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
4. Sum and difference rules
5. Higher order derivatives
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
4. Sum and difference rules
5. Higher order derivatives
6. Product rule
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
4. Sum and difference rules
5. Higher order derivatives
6. Product rule
7. Quotient rule
3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
4. Sum and difference rules
5. Higher order derivatives
6. Product rule
7. Quotient rule
8. Negative integer power rule
3.3 Differentiation Rules
1. Derivative of a constant2. Power Rule for derivatives3. Derivative of a constant multiple4. Sum and difference rules5. Higher order derivatives6. Product rule7. Quotient rule8. Negative integer power rule9. 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
Y= sin x Y= cos x
3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x
3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x Y= csc x
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
Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x Y= cot x
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