# Calc04_1 - 4.1 Extreme Values of Functions Greg Kelly...

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4.1 Extreme Values of Functions Greg Kelly, Hanford High School Richland, Washington

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The textbook gives the following example at the start of chapter 4: The mileage of a certain car can be approximated by: ( 29 3 2 0.00015 0.032 1.8 1.7 m v v v v = - + + At what speed should you drive the car to obtain the best gas mileage? Of course, this problem isn’t entirely realistic, since it is unlikely that you would have an equation like this for your car.
( 29 3 2 0.00015 0.032 1.8 1.7 m v v v v = - + + We could solve the problem graphically:

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( 29 3 2 0.00015 0.032 1.8 1.7 m v v v v = - + + 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.
( 29 3 2 0.00015 0.032 1.8 1.7 m v v v v = - + + 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. The car will get approximately 32 miles per gallon when driven at 38.6 miles per hour.

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( 29 3 2 0.00015 0.032 1.8 1.7 m v v v v = - + + 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.
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