BC Unit 5

Rolle’s Theorem

Consider y=f(x) from a<=x<=b

• Must be continuous

• Must be differential

If f(a)=f(b)=0 → Exists at least 1 value C such that a<c<b, where f’(c)=0

Mean Value Theorem

Consider y=f(x) from a<=x<=b

• Must be continuous

• Must Be Differentiable

Then, there must exist at least c value where

f’(c)=( f(b)-f(a) )/( b-a )

MVT Corolaries

• f’(x) = g’(x) → f(x)=g(x)+c

• f’(x)=0 → f(x)=0

• If y=f(x) is Continuous & Diffable, (from a<=x<=b) & f(a) & f(b) have different signs, f’ is never between a & b)

  • will always rise or always fall

First Derivative Test

• Solve for x when f’(x)=0

• points given are maxs/mins

• Use number line test

Second Derivative Test

• Find maxs/mins using f’(x)=0

• Plug values into f’’(x)

• if f’’(x)>0, it is a minimum

• if f’’(x)<0, it is a maximum

Characteristics of Logistic Curve

• Grow slowly at first, then experience a period of rapid growth, and eventually slow down

Appropriate Procedure for Word Problem

1. Define variables/draw picture

2. Write appropriate equations

3. Take the derivative (if needed) & solve.

4. Check work & give units

Area of equilateral triangle

^√3/₄ s²

Regular Hexagon Area

Regular Octagon Formula

Linearization Equation

• Based on 2nd Definition of Derivatives

• L(x)=f’(a)(x-a)+f(a)

  • more or less slope intercept form with derivatives

Newtons Method

• Numerical Technique for approximating a zero of a function

• steps:

1. Write equation in form of f(x)=0

2. Define first approximation as x₀, where x=0, or x₀ is given

3. Use Recursive formula: x_n-f(x_n)/f’(x_n)

how to simplify equations with “d”

• replace d with dx, then take derivative of everything inside/multiplied to d