Mathematics B

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1B Review of Differentiation

Rules for Differentiation:

For K is a Constant

> If f(x)=x^n , then f'(x)knx^n-1

> If g(x)=Kf(x) , then g'(x)=Kf'(x)

> If f(x)=g(x) + h(x) , then f'(x)=g'(x) + h'(x)

First principles:

f'(x)=lim f(x+h)-f(x) / h

Calculator Check

Math > 8

enter equation

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1D-Chain Rule

If r(x)=f(g(x))

then r'(x)=f'(g(x)) * g'(x) > short Chain Rule

dy/dx=du/dx * dy/du > Long Chain Rule

1E-Derivative of Rational Powers

f(x)=x^p/q

f'(x)=p/q x^p-q/q

1F-Product Rule

f(x)=U V where U and V are functions of x

f'(x)=U(x) V + V(x) U

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1G-Quotient Rule

If Y=U/V where U and V are functions of x

then Y'=(U'V-UV')/V^2

1L-Tangents and Normals

Note: Mt X Mn =-1

1. find the coordinate for gradient

2. Find the gradient of Tangent

-Derivatives

3. Find Gradient of Normal

4. Find equation of Tangent

5. Find equation of Normal

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Angles Between Curves

Note:External angles equal the sum of interior opposite angles

m=change y/change x

m=O/A

m=tan@

@=tan-1(m)

Draw it/Graph it

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1N Rates of Change-Applications

Remember:

Instantaneous Rate of Change of y wrt x

dy/dx

Average rate of Change

Change y/change x = y2-y1/x2-x1

1O-Applications of related rates of change

1. State what needs to be found

2. list given information

3. Write equation of needed derivative

4. Re-arrange given information into chain rule

5. simplify and answer

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2A-Stationary Points

When f'(x)=0

2B-Types of Turning Points

First derivative test

1. find TP x value f'(x)=0

2. find gradient of close values

3. determine max or min (max= + 0 -) min (-0+)

Second derivative test

1. if f''(x)>0 then min (+)

2. if f''(x)<0 the max (-)

3. if f''(x)=0 then use first derivative test

Things used for graphing

-y-int

-x-int

-Stationary points

-nature of points

-global max and min

2C-Stationary Points of Inflection

Positive: +0+

Negative: - 0 -