Ex 1: Identify the open intervals for which each function is increasing or decreasing.

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1 MATH 2040 Notes: Unit 4 Page 1 5.1/5.2 Increasing and Decreasing Functions Part a Relative Extrema Ex 1: Identify the open intervals for which each In algebra we defined increasing and decreasing behavior of functions: Definition: A function f is said to be increasing over an open interval (a, b) if for all x 1 < x 2 in the interval, f(x 1 ) < f(x 2 ). A function f is said to be decreasing over an open interval (a, b) if for all x 1 < x 2 in the interval, f(x 1 ) > f(x 2 ). Notice what happens at each x-value where the function changes from increasing to decreasing or vice versa. If the function has a point at one of these x-values, that point is called a turning point. The function value at each turning point is either the greatest or the least value in the neighborhood around that x-value. This leads to the concept of relative extrema. Definition: Let c be a number in the domain of f. f(c) is a relative maximum (or local maximum) for f if an open interval (a, b) containing c exists such that fx ( ) fc ( ) for all x in (a, b). f(c) is a relative minimum (or local minimum) for f if an open interval (a, b) containing c exists such that fx ( ) fc ( ) for all x in (a, b). The collective term for either being a relative maximum or relative minimum is relative extremum (and analogously, local extremum).

2 MATH 2040 Notes: Unit 2 Page 2 Now that we have defined derivatives in calculus, we can begin to connect the definitions of increasing and decreasing functions as well as relative extrema to the slope of a function. Observe that when a function is increasing, any tangent line within the open interval will have a positive slope. Similarly, when a function is decreasing, any tangent line within the open interval will have a negative slope. Behavior of f Nature of f Positive (+) Test for where f(x) is Increasing/Decreasing Test: Suppose a function is differentiable everywhere on an open interval; then 1. where f x) is positive for each x in the interval, f is increasing on the interval; 2. where f x) is negative for each x in the interval, f is decreasing on the interval 3. where f x) is zero for each x in the interval, f is constant on the interval Critical Numbers and Critical Points The open intervals where a function is increasing or decreasing can be established based on the limited number of possibilities where either a turning point exists or the function has a discontinuity. Focusing on where turning points occur, notice that either the derivative is zero or the derivative does not exist. Irrespective of whether a function actually has a turning point at a particular x-value, we explicitly define the x-values where one of those two possibilities occur. Definition: Critical numbers of a function f are those values c in the domain of f for which f c) = 0 or f c) does not exist. A critical point is a point (c, f(c)) if c is a critical number. Recognize that it is possible for a point to be a critical point without being a turning point.

3 MATH 2040 Notes: Unit 2 Page 3 The process for applying the test for where a function is increasing or decreasing will also lead us to the relative extrema, if they exist. 1. Locate the critical numbers for f on a number line as well as any values where f is undefined. Open intervals can then defined between these marks on the number line. 2. Choose a test value of x in each open interval. Evaluate f 3. For each test value whose value of f x) is positive, identify the open interval as increasing. Where f x) is negative, identify the open interval as decreasing. To establish where a function has a relative extrema, apply the First Derivative Test: The First Derivative Test Let c be a critical number for a function f. Assuming that f is continuous & differentiable over open intervals on either side of c, then: 1. f(c) is a relative maximum of f whenever is positive over the open interval less than c and f greater than c. 2. f(c) is a relative minimum of f whenever is negative over the open interval less than c and f interval greater than c. 3. No relative extremum of f exists whenever the signs of are the same on each side. Ex 2: Find the critical numbers of the function. 3 2 f x 2x 3x 12x 18

4 Ex 3: Find the critical numbers of the function f x x 4x 20x 10 MATH 2040 Notes: Unit 2 Page 4 Ex 4: Find the critical numbers of the function. 2 3 f x 6x 4x

5 Ex 5: Find the critical numbers of the function. f x x 3 x 4 MATH 2040 Notes: Unit 2 Page 5 Ex 6: Find the critical numbers of the function. f x 2xe 2 x

6 Ex 7: Find the critical numbers of the function. f x 2x lnx MATH 2040 Notes: Unit 2 Page 6 Ex 8: Find the critical numbers of the function f x x x

7 5.1/5.2 Part b Applications of Increasing/Decreasing Functions and Relative Extrema Ex 1: A manufacturer of CD players has determined that the profit P(x) (in thousands of dollars) is related to the quantity x of CD players produced (in hundreds) per month by x P x x 4 e 4, 0 x 3.9. Find the production levels where profit is increasing and where it is decreasing. MATH 2040 Notes: Unit 2 Page 7 Ex 2: A realty group estimates that the number of housing starts per year over the next three years can be projected by 500r H r, where r is the r mortgage rate (in percent). Find the interest rates where the number of housing starts is increasing and where it is decreasing.

8 Ex 3: A high-end sports apparel manufacturer produces hockey jerseys for sale in college bookstores. Its cost function, in 2 dollars, is Cx ( ) x 0.2x where x is the number of jersey manufactured. Find the production level that will produce the minimum average cost per unit Cx ( ). MATH 2040 Notes: Unit 2 Page 8 Ex 4: Find the profit function given demand function and cost function. Then determine the number, q, of units that produces the maximum profit, the price, p, per unit that produces the maximum profit, and the maximum profit, P. Cq ( ) 30q 5000, p q

9 MATH 2040 Notes: Unit 2 Page 9 Ex 5: In planning a restaurant it is estimated that revenues of $8 per seat can be made when the number of seats is limited to 50. If more than 50 seats are created, the revenue on each seat will decrease by 10 for each seat above 50. b. Find the number of seats that will maximize revenues and what the maximum revenue will be. a. Find an expression for the revenues earned if more than 50 seats are permitted. Ex 6: A used bicycle store estimates that on average it costs them $70 to acquire and recondition ten-speed bicycles. The store s weekly fixed expenses are $500. Based on experience they believe they can sell 7 bicycles per week if they price the reconditioned ten-speeds at $200. Raising the price by $10 will result in one fewer bicycle being sold each week. b. Find the price and number of bicycles sold that will maximize profits and what the maximum profit will be. a. Find an expression for the profits earned where x represents the number of times they raise the price from $200.

10 MATH 2040 Notes: Unit 2 Page Concavity & The Second Derivative Test Whereas the first derivative is used to show where the function is increasing or decreasing, the second derivative which is a rate of change of the first derivative indicates how fast the derivative is increasing or decreasing. Another perspective is to contrast the first derivative as describing how the function is behaving now and the second derivative as describing how the function will behave in the future. It indicates the trend beyond the current increasing or decreasing nature. For each graph below, we focus our attention on x > In this graph f' > 0, f 2. In this graph f' > 0, f 3. In this graph f' < 0, f < 0 4. In this graph f' < 0, f Visually the second derivative corresponds to concavity, what we informally describe as the curl of the function. No concavity Concave Up Concave Down Ex: Identify the intervals of the function which are concave up and concave down and find any inflection points. A point of a function where concavity changes is called an inflection point.

11 MATH 2040 Notes: Unit 2 Page 11 Test for Concavity Test: Suppose a function is twice differentiable everywhere on an open interval; then 1. where f x) is positive for each x in the interval, f is concave up on the interval; 2. where f x) is negative for each x in the interval, f is concave down on the interval Ex 1: Find the open intervals for which f is concave up or concave down. Then find any inflection points of f. f x x 9x 3 2 Inflection points occur where f not exist. Ex 2: Find the open intervals for which f is concave up or concave down. Then find any inflection points of f. f x xe 2x

12 Ex 3: Find the open intervals for which f is concave up or concave down. Then find any inflection points of f f x x 4x 48x 24 MATH 2040 Notes: Unit 2 Page 12 The Law of Diminishing Returns in economics is related to the idea of concavity. At first the rate at which revenues increase grows faster over the time than the rate of expenses. This coincides with having upward concavity. Eventually the increases in revenue slow down relative to those of expenses and the function behavior switches to downward concavity. The change must occur (by definition at an inflection point) and that inflection point is considered the point of diminishing returns. Ex: Revenues as a function of x thousands of dollars spent on advertising can be expressed by the function 3 2 Rx ( ) x 42x 800 x, 0 x 20, in thousands of dollars. Find the point of diminishing returns.

13 MATH 2040 Notes: Unit 2 Page 13 While values of the second derivative can be used to establish the type of concavity a function has over an interval, it can also be used with the critical numbers found by the first derivative to establish relative extrema. Where a function is concave up, the function can possible have a relative minimum. Where a function is concave down, the function can possibly have a relative maximum. The Second Derivative Test Let c be a critical number for a function f. Assuming that f is continuous and twice differentiable over open intervals on either side of c, then: 1. If f c) > 0, then f(c) is a relative minimum. 2. If f c) < 0, then f(c) is a relative maximum. 3. If f c) = 0 or f c) does not exist, then the test gives no information as to whether there is a relative extremum at c. The First Derivative Test must be applied. Ex: Find the critical numbers of f. Then apply the Second Derivative Test to determine whether a relative maximum or relative minimum exists at each critical number. 3 2 f x 2x 4x 2