Formula Quiz 3

Definition of a Critical Number and Critical Point

If f(c) is defined and f ‘(c) = 0 or is not differentiable, then…

• → c is a critical number of f(x)

• → (c , f(c)) is a critical point of f(x)

Extreme Value Theorem (EVT)

If f(x) is continuous on [a,b], then…

• f(x) has an absolute maximum value f(c) for c on [a.b]

• f(x) has an absolute minimum value f(d) for d on [a,b]

Candidates Test

Finds extrema on continuous function f on [a.b]

1. Find f ‘(x) and critical numbers of f on [a,b]

2. Evaluate f at each critical number in (a,b)

3. Evaluate f at each endpoint in [a,b]

4. The least of these values is the minimum. The greatest is the maximum.

Candidates Test Justification

f has an absolute max/min of f(c) at x = c by the Candidates Test

Mean Value Theorem (MVT)

If a function f is…

1. continuous on [a,b], and

2. differentiable on (a,b), then there is at least one number c in (a,b) such that

• → f ‘(c) = [f(b) - f(a)] / b - a

Rolle’s Theorem

If a function f is…

1. continuous on [a,b], and

2. differentiable on (a,b), and

3. f(a) = f(b), then there is at least one number c in (a,b) such that…

• → f ‘(c) = 0

Test for Increasing and Decreasing Functions

1. f ‘(x) > 0 for all x on (a,b) → f is increasing on (a,b)

2. f ‘(x) < 0 for all x on (a,b) → f is decreasing on (a,b)

3. f ‘(x) = 0 for all x on (a,b) → f is constant on (a,b)

First Derivative Test

f is continuous and differentiable (except possibly at c) on (a,b) containing critical number c…

1. If f ‘(c) changes from + → - at c, then (c, f(c)) is a relative maximum of f

2. If f' ‘(c) changes from - → + at c, then (c, f(c)) is a relative minimum of f

Linear Approximation

y - f(a) = f ‘(a)(x-a)

dy & dx

y = f(x) is differentiable.

• dx is a differential and an independent variable.

• dy is a differential and always dependent on x and dx

  • dy = f ‘(x)(dx)

L’Hopital’s Rule

Definition of Concavity

Let f be differentiable on (I)

1. f is concave up if f ‘(x) is increasing on I

2. f is concave down if f ‘(x) is decreasing on I

Test for Concavity

Let f be a function whose second derivative exists on (I)

1. f is concave up if f ‘‘(x) > 0 on I

2. f is concave down if f ‘‘(x) < 0 on I

Definition of an Inflection Point

f has an inflection point at (c , f(c)) if…

1. f ‘‘(c) = 0 or DNE

2. f ‘‘ changes from positive to negative or negative to positive at x = c

• or f' ‘(x) changes from increasing/decreasing or decreasing/increasing

Second Derivative Test

Let f be a function such that the second derivative exists on (a,b)

1. If f ‘(c) = 0 and f ‘‘(c) > 0, then f(c) is a relative minimum of f

2. If f ‘(c) = 0 and f ‘‘(c) < 0, then f(c) is a relative maximum of f

Evaluating Limits going towards infinity

1. Plug in infinity OR

2. Divide numerator and denominator by the highest power of x in the denominator

Even Function

f( -x ) = f(x)

• Y-axis symmetry

Odd Function

f( -x ) = -f(x)

• Origin symmetry