Derivative Applications MAC 2233 Instantaneous Rates of Change of a - PDF document

3/16/2010 Derivative Applications MAC 2233 Instantaneous Rates of Change of a Function • The derivative is: ▫ The slope of the tangent line at a point ▫ The instantaneous rate of change of the function Marginal Analysis • The study of the amount of change that results in a function (cost, revenue, profit) from ______ ______________________. 1

3/16/2010 Marginal Functions • The marginal cost function, C ’( x ), is the approximate cost of ___________________ _________. • The marginal revenue function, R ’( x ), is the approximate gain or loss in revenue by ______________________________. • The marginal profit function, P ’( x ), is the approximate gain or loss in profit by _______ _______________________. Example Find the marginal cost, marginal revenue, and marginal profit if and Solution • The marginal cost is • The marginal revenue is • The marginal profit is 2

3/16/2010 Example The price-demand function for a collectable doll is found to be where x represents the number of collectable dolls produced in hundreds and p is the price of the dolls in dollars. a) Determine the revenue function. b) Determine the marginal revenue function. c) Evaluate and interpret R ’(15). From Brief Calculus,2 nd ed. By Armstrong & Davis, 2003, problem 78, p.272. a) Determine the revenue function • ______________! • For the next part, when we differentiate, we will need a ______________! b) Determine the marginal revenue function 3

3/16/2010 c) Evaluate and interpret R’ (15) • Plug 15 into the derivative! • Producing the ____________________ will _______ revenue by approximately $_____. Homework • p. 163 problems 1, 3, 5, 11, 13 • p. 337 problem 77 Increasing/Decreasing • A function is increasing on an interval if _____ _______________________________. • A function is decreasing on an interval if _____ _______________________________. 4

3/16/2010 Graphically Critical Values The critical values of a function f are the values that make the derivative ______________. Note: These are the only places where ________ ________________________! Example • Determine where the function is increasing and decreasing: 5

3/16/2010 Solution • Take ________ and find the ___________! • __________: Solution continued • Plot the _________ on a number line and test the signs around each. • The function is increasing on • The function is decreasing on Relative Extrema Relative Maximum — f has a relative (local) maximum at x = c if there exists an open interval ( a, b ) containing c such that f ( x ) ≤ f ( c ) for all x in ( a, b ). Relative Minimum — f has a relative (local) minimum at x = c if there exists an open interval ( a, b ) containing c such that f ( x ) ≥ f ( c ) for all x in ( a, b ). 6

3/16/2010 First Derivative Test To find the relative extrema: 1. Find the critical values of f 2. Determine the sign of f ’ on each side of each critical value a. If f ’ changes from _____________, then f ( c ) is a relative maximum b. If f ’ changes from _____________, then f ( c ) is a relative minimum c. If f ’ ________________, then f ( c ) is not a relative extremum Example • Find the relative extrema of • The derivative is • The critical values are Solution continued • Plot the critical values on a number line and test the signs around each. • Because the sign changed from + to — across __, there is _______________. • Because the sign changed from — to + across __, there is _______________. 7

3/16/2010 Solution continued • To find what the max and min are, plug into ___ ___________________! • The maximum is • The minimum is Example Linguini’s Pizza Palace is starting an all -you-can- eat pizza buffet from 5:00 to 9:00 p.m. on Friday evenings. A survey of local residents produced the price-demand function where x represents the quantity demanded and p represents the price in dollars. Determine where the revenue is increasing and decreasing. Find and interpret the relative extremum. From Brief Calculus,2 nd ed. By Armstrong & Davis, 2003, problem 58, p.335. Solution • Find the revenue function: • Take the derivative • Find the critical values: 8

3/16/2010 Solution continued • Plot the critical value on a number line and test the signs: • The revenue is _______ until we sell __ buffet tickets. After that, the revenue is _________. • The pizza place will have a _______ revenue of $______ when ____ buffets are sold. Homework • p. 202 problems 1-8, 11, 13, 15, 17, 23-29 odd, 45, 47, 53, 55, 59, 61, 63, 69, 71 • p. 238 problem 49 • P. 254 problem 19 • p. 337 problems 69, 71 • p. 352 problem 39 Absolute Extrema of a function Absolute Maximum — f has an absolute (global) maximum at x = c if f ( x ) ≤ f ( c ) for all x in the domain of f . Absolute Minimum — f has a absolute (global) minimum at x = c if f ( x ) ≥ f ( c ) for all x in the domain of f . 9

3/16/2010 Extreme Value Theorem If a function f is continuous on [ a, b ], then f has an absolute maximum value and an absolute minimum value of f on [ a, b ]. To find Absolute Extrema on a Closed Interval 1. Verify _____________ on [ a, b ]. 2. Determine _______________ of f in ( a, b ). 3. Evaluate f ( x ) at ____________ and find _______________ . 4. The biggest number in step 3 is the absolute maximum & the smallest is the absolute minimum. Example • Find the absolute extrema of the function on the interval: 10

3/16/2010 Homework • p. 254 problems 1-15 odd • p. 336 problems 39-45 odd Optimization • We can use calculus to find the maximum or minimum of a quantity! • Procedure: ▫ Form an equation to describe the situation (a picture often helps!) ▫ Take the derivative ▫ Find the critical values ▫ Locate the maximum or minimum by  The first derivative test or  The Extreme Value Theorem Example For a rectangle with area 100 ft 2 to have the smallest perimeter, what dimensions should it have? From Applied Calculus,4 th ed. By Waner & Costenoble, 2007, problem 10, p.370. 11

3/16/2010 Example My orchid garden abuts my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. If I have a budget of $80 for the project, what is the largest area I can enclose? From Applied Calculus,4 th ed. By Waner & Costenoble, 2007, problem 18, p.371. Solution • Let x be the length of the southern fence. ▫ The fence here costs $4 per foot! • Let y be the length of the eastern & western y fences. ▫ The fence here costs $2 per foot! x Example Vanilla Box Company is going to make open- topped boxes out of 12” x 12” rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way? From Applied Calculus,4 th ed. By Waner & Costenoble, 2007, problem 32, p.372. 12

3/16/2010 Solution 12 12 Example A manufacturer of medical monitoring devices uses 36,000 cases of components per year. The ordering cost is $54 per shipment, and the annual cost of storage is $1.20 per case. The components are used at a constant rate throughout the year, and each shipment arrives just as the preceding shipment is being used up. How many cases should be ordered in each shipment in order to minimize total cost? From Calculus for Business, Economics, and the Social and Life Sciences,10 th ed. by Hoffman & Bradley, 2007, problem 32, p.273. Solution • Let x = ___________________ • Total Cost = ________+ ___________ • Storage Cost: ▫ If we reorder x every time we hit 0, __________ ______________________. ▫ Storage Cost = (_____________)*(________ ____________) 13

3/16/2010 Solution • Let x = ____________________ • Total Cost = ________+ _________ • Reorder Cost: ▫ Reorder Cost = (____________)*(________ _________) Solution • Let x = ___________________ • Total Cost = _________+ ____________ • So, the total cost is: Homework • p. 255 problems 31, 33 • p. 270 problems 5, 7, 9, 11, 17, 21, 23, 27, 31, 33, 39 14

3/16/2010 Concavity If f is differentiable on ( a, b ) then 1. f is concave up on ( a, b ) if _____________ on ( a, b ). 2. f is concave down on ( a, b ) if ___________ on ( a, b ). Theorem 1. If __________ for each x in ( a, b ) then f is concave up on ( a, b ). 2. If __________ for each x in ( a, b ) then f is concave down on ( a, b ). • Procedure to find intervals of concavity: ▫ Find all values for which _________________ ____________________ ▫ Determine the sign of _____ on the interval between each value in (1) by testing points. • The point where concavity changes is called an inflection point. Examples Determine where the function is concave up and where it is concave down: 15

3/16/2010 Point of Diminishing Returns • Where the rate of change of sales starts to decrease • This corresponds to _____________! Example The Sucre Cola Company estimates that total sales of its new cola, TS( x ), when spending x million dollars on advertising, can be modeled by Locate the point of diminishing returns for TS( x ) and interpret its meaning. From Brief Calculus,2 nd ed. by Armstrong & Davis, 2003, problem 60, p.350. Homework • p. 220 problems 1-4, 5, 7, 11, 39, 41, 45, 47, 53, 55, 61 • p. 237 problems 39, 41 16