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

Download The mileage of a certain car can be approximated by: At what speed should you drive the car to obtain the best gas mileage? Of course, this problem isn’t

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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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