# the mileage of a certain car can be approximated by: at what speed should you drive the car to...

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Of course, this problem isnt entirely realistic, since it is unlikely that you would have an equation like this for your car.

We could solve the problem graphically:

We could solve the problem graphically:On the TI-84, we use 2nd CALC 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

We could solve the problem graphically:On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

Notice that at the top of the curve, the horizontal tangent has a slope of zero.Traditionally, this fact has been used both as an aid to graphing by hand and as a method to find maximum (and minimum) values of functions.

Slope of the Tangent Linef (a) is the slope of the tangent line to the graph of f at x = a.This tangent slope interpretation of f has several useful geometric implications.The values of the derivative enables us to detect when a function is increasing or decreasing.

If f (x) > 0 on an interval (a, b) then f is increasing on that interval.If f (x) < 0 on an interval (a, b) then f is decreasing on that interval.

A number c in the interior of the domain of a function f is called a critical number if either:f (c) = 0f (c) does not existccPoint (c, f(c)) is called a critical point of the graph of f.

The only inputs, x = c, at which the derivative of a function can change sign (+ or +) are where f (c) is at the critical numbers of:0 orundefined

A local maximum (or relative maximum) for a function f occurs at a point x = c if f(c) is the largest value of f in some interval (a < c < b) centered at x = c.local max.A global maximum (or absolute maximum) for a function f occurs at a point x = c if f(c) is the largest value of f for every x in the domain.absolute max.

A local minimum (or relative minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f in some interval (a < c < b) centered at x = c.local min.A global minimum (or absolute minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f for every x in the domain.absolute min.

A function can have at most one global max. and one global min., though this value can be assumed at many points.Example: f(x) = sin x has a global max. value of 1 at many inputs x.Maximum and minimum values of a function are collectively referred to as extreme values (or extrema).A critical point with zero derivative but no maximum or min. is called a plateau point.

Example 1: Find critical numbers for the givenRemember, critical numbers occur if the derivative does not exist or is zero.Our critical numbers!!

Example 1:A number-line graph for f and f helps. It is a convenient way to sketch a graph.Remember, if f 0 f(x) is increasing.x110f '(x)++

Example 1:Recall, local min. at x=c if f(c) is the smallest value in the interval.x110f '(x)++Recall, local max. at x=c if f(c) is the largest value in the interval.min.min.max.f (x)

Extreme values can be in the interior or the end points of a function.Absolute MinimumNo AbsoluteMaximum

Absolute MinimumAbsoluteMaximum

No MinimumAbsoluteMaximum

No MinimumNoMaximum

Extreme Value Theorem:If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.Maximum & minimumat interior pointsMaximum & minimumat endpointsMaximum at interior point, minimum at endpoint

Local Extreme Values:A local maximum is the maximum value within some open interval.A local minimum is the minimum value within some open interval.

Absolute minimum(also local minimum)Local maximumLocal minimumAbsolute maximum(also local maximum)Local minimum

Local maximumLocal minimumAbsolute maximum(also local maximum)

There are no values of x that will makethe first derivative equal to zero.The first derivative is undefined at x=0,so (0,0) is a critical point.Because the function is defined over aclosed interval, we also must check theendpoints.

- To determine if this critical point isactually a maximum or minimum, wetry points on either side, withoutpassing other critical points.Since 0
- To determine if this critical point isactually a maximum or minimum, wetry points on either side, withoutpassing other critical points.Since 0
Absolute minimum (0,0)Absolute maximum (3,2.08)

Finding Maximums and Minimums Analytically:

Critical points are not always extremes!(not an extreme)

(not an extreme)p

Find the critical numbers of each function

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Locate the absolute extrema of each function on the closed interval.

Mean Value Theorem for DerivativesThe Mean Value Theorem only applies over a closed interval.

Slope of chord:Slope of tangent:Tangent parallel to chord.

Find all values of c in the interval (a,b) such that

Find all values of c in the interval (a,b) such that

Find all values of c in the interval (a,b) such that

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