The Mean Value Theorem

Given an equation for f (x),

determine where f  has relative extrema.

Find where f  has a critical number

( f ' (x) = 0 or undefined)

and look for sign changes about those numbers to indicate whether it's a max or min.

Given a graph of  f '' , how do you locate inflection points of  f ?

f  has an inflection point where  f '' changes sign (why?)

By definition, f  changes concavity (and so has an inflection point) whenever its 1st derivative has a relative extremum.  Sign changes in the second derivative indicate exactly this, and so transitively indicate inflection points

Given a graph of  f ', how do you locate inflection points of  f ?

Look for where f ' has relative extrema; f  has an inflection point where f ' has has a peak or a valley.

Given a graph of  f ', what information

can be determined about f ?

---Net changes in f can be computed by finding net signed areas between f '.

-- f  has relative extrema where f ' changes sign.

- f  has inflection points where  f ' has extrema.

The derivative of this function

[image]

is positive/negative  and  increasing/decreasing

Positive (because it is increasing)

and

increasing ( because it curves up)

The derivative of this function

[image]

is positive/negative  and  increasing/decreasing

Negative (because it is decreasing)

and

positive (because its slope going from largely negative to less negative)

The derivative of this function

[image]

is positive/negative and increasing/decreasing

Negative (it is decreasing)

and

decreasing (slope getting more negative)

The derivative of this function

[image]

is positive/negative  and  increasing/decreasing

Positive (it is increasing)

and

decreasing (it is becoming less steep)

What are the conditions for a function to be continuous at x = a ?

Why is this function

[image]

discontinuous at x = a?

f (a) does not exist

Why is this function

[image]

discontinuous at x = a?

 [image]

left- and right-handed limits disagree

[image]

1

The graphs of  y = x  and  y = sin x  are indistinguistable near the origin, so their

ratio is close to 1.

[image]

The average value of  f (x) on [a, b]

[image]

= f (x)

The derivative of a function defined by an integral is the function being integrated.

How is 

[image]

 evaluated without a graph?

F(b) - F(a),

where F is any any antiderivative of  f .

This is one part of the Fundamental Theorem of Calculus

[image]

f (p(x))· p'(x)

Chain rule is needed.  The outside function is the integral.  The inside function is p(x).

[image]

0

It can be derived from the special trig

limit for sine.

Definitions of a

derivative as a limit

 f ' (x) =

Under what conditions will a function

not be differentiable at a point?

Vertical Tangents

Corners

Discontinuities

Volume of a solid using

a definite integral

[image], where A(x) is area of the cross sections of the solid as a function of x.  A(x) could be the area of a square, annulus, semicircle, etc.

What function has differential equation:

[image]

The rate of change of y is proportional to y itself.

[image]

Separate variables, integrate, and plug in the initial condition.

Where is f  increasing?

[image]

(-∞,c)  U  (e, ∞)

A function is increasing wherever its derivative is positive.

Where is f concave up?

[image]

(a, b)  U  (d, ∞)

A function is concave up wherever

its derivative is increasing.

Does  f   have relative extrema?

[image]

f  has a relative maximum at x = c

and

a relative minimum at x = e.

A max occurs where f ' switches from + to -

and

a min occurs where f ' switches from - to +.

Where is f concave up?

[image]

(-∞, c)  U  (e, ∞)

f  is concave up where its second derivative is positive (because it means its first derivative is increasoing)

Where is f increasing?

[image]

Not enough information.

The only thing that can be determined is whether

f '  is increasing or decreasing.  This says nothing about the values of   f , only how fast it is changing.

A function is linear if its derivative is _____.

constant.

A constant derivative means the

function has a constant slope.

Suppose v(t) is a velocity function of a particle.  What is

[image]   ?

The displacement (or change in position) of the particle from t = a to t = b.  It is the net signed area between v(t) to the t-axis.

Suppose v(t) is a velocity function of a particle.  What is

[image]   ?

The total distance traveled by the particle from

t = a  to  t = b.  It is the total area between v(t) and the t-axis.  It does not tell the final position of the particle- only the distance it went.

What are the hypotheses of the

Mean Value Theorem?

The interval must be closed and the

function must be differentiable everywhere between the endpoints.

True or False?

If f '(a) = 0, then   f  has a relative extremum

at x = a.

False.

If f '(a) = 0, then   f  has a horizontal tangent

at x = a; not necessarily an extremum.

If  f '(a) = 0  and  f ''(a) > 0, then   f

has a  _________ at x = a.

Relative minimum.

f '(a) = 0  means f  has a horizontal tangent at

x = a, and  f ''(a) > 0 means f is concave up at

x = a.  Hence the relative minimum.  This is the Second Derivative Test.

If f '(a) = 0  and  f ''(a) < 0, then f

has a ____________ at x = a.

Relative maximum.

f '(a) = 0  means f  has a horizontal tangent at

x = a, and  f ''(a) < 0 means f  is concave down at

x = a.  Hence the relative maximum.  This is the Second Derivative Test.

If  f '(a) = 0  and  f ''(a) = 0, then f

has a ____________ at x = a.

Cannot be determined.

f   has a horizontal tangent at x = a, but since

f ''(a)=0, its concavity at that point cannot be determined.  Hence the Second Derivative Test cannot be used to determine what f  looks like at

x = a.