Calculus Chapter 3
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Transcript of Calculus Chapter 3
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Calculus Chapter 3
Derivatives
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3.1 Informal definition of derivative
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3.1 Informal definition of derivative
A derivative is a formula for the rate at which a function changes.
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Formal Definitionof the Derivative of a function
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You’ll need to “snow” this
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Formal Definitionof the Derivative of a function
f’(x)= lim f(x+h) – f(x) h->0 h
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Notation for derivative
y’ dy/dx df/dx d/dx (f) f’(x) D (f)
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Rate of change and slope
Slope of a secant line
See diagram
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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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Rate of change and slope-slope of the tangent line to a curve
see diagram
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The slope of the tangent line gives the rate of change at that one point
i.e. the instantaneous change.
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compare Slope= y-y x-x Slope of secant line
m= f ’(x) Slope of tangent line
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Time for examples Finding the derivative
using the formal definition
This is music to my ears!
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A function has a derivative at a point
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A function has a derivative at a point
iff the function’s right-hand and left-hand derivatives exist and are equal.
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Theorem
If f (x) has a derivative at x=c,
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Theorem
If f (x) has a derivative at x=c,
then f(x) is continuous at x=c.
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Finding points where horizontal tangents to a curve occur
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3.3 Differentiation Rules
1. Derivative of a constant
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3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
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3.3 Differentiation Rules
1. Derivative of a constant
2. Power Rule for derivatives
3. Derivative of a constant multiple
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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
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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
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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
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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
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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
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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
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3.4 Definition
Average velocity of a “body”
moving along a line
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Defintion
Instantaneous Velocity
is the derivative of
the position function
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Def. speed
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Definition
Speed
The absolute value of velocity
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Definition
Acceleration
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acceleration Don’t drop the ball on this
one.
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Definition
Acceleration
The derivative of velocity,
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Definition
Acceleration
The derivative of velocity,
Also ,the second derivative of position
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3.5 Derivatives of trig functions
Y= sin x
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3.5 Derivatives of trig functions
Y= sin x Y= cos x
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3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x
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3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x Y= csc x
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3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x
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3.5 Derivatives of trig functions
Y= sin x Y= cos x Y= tan x Y= csc x Y= sec x Y= cot x
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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
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Whereas
The slope of the secant line gives the change between 2 distinct points on a
curve.
i.e. average rate of change