2.1. Definition: If a < b, then f(a) < f(b) for every a and b in that interval. If a < b, then f(a) > f(b) for every a and b in that interval.

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1 1.1 Concepts: 1. f() is INCREASING on an interval: Definition: If a < b, then f(a) < f(b) for every a and b in that interval. A positive slope for the secant line. A positive slope for the tangent line. f () > 0. f() is DECREASING on an interval: If a < b, then f(a) > f(b) for every a and b in that interval. A negative slope for the secant line. A negative slope for the tangent line. f () < 0 E: Draw a graph the match the description. G() is decreasing over (,4) and (9, ), and increasing over (4, 9).

2 3. f() has a RELATIVE MAXIMUM on an interval: Assume that the relative maimum occurs at: (c, f(c)). Definition: If f(c) f() for every in that interval. f() is increasing to the left of c, and f() is decreasing to the right of c. The f () > 0 to the left of c, and f () < 0 to the right of c. 4. f() has a RELATIVE MINIMUM on an interval: Assume that the relative minimum occurs at: (c, f(c)). Definition: If f(c) f() for every in that interval. f() is decreasing to the left of c, and f() is increasing to the right of c. The f () < 0 to the left of c, and f () > 0 to the right of c.

3 3 Definition: Critical Value c is a critical value of f() if: 1. f (c) = 0. f (c) Does Not Eist. Think of critical values as candidate values for where relative etrema could occur. First Derivative Test for Relative Etrema f() must be a continuous function with eactly one critical value on an open interval (a, b). 1. f() has a relative maimum at c if f () > 0 on (a, c) and f () < 0 on (c, b).. f() has a relative minimum at c if f () < 0 on (a, c) and f () > 0 on (c, b). 3. f() has neither a maimum nor a minimum of f () has the same sign on (a, c) and (c, b).

4 4 E: Find the relative etrema of each function, if they eist. List each etremum along with the -value at which it occurs. Then sketch a graph of the function. a) 3 f ( ) g ( ) b) 4 1 c) g( ) 5

5 5 To graph a function f() using the first derivative f (): 1. Find f (). Solve for -values that make f () = 0 or make f () undefined. These are the critical values, the potential minimums/maimums of f(). 3. Determine which critical values are in fact zeros (intervals and test points). 4. f() is increasing where f () > f() is decreasing where f () < 0. E) The graph of a derivative f () is shown. Use the information in each graph to determine where f() is increasing or decreasing and the -values of any etrema. Sketch a possible graph of f(). E) Draw a graph to match the description. f() has a negative derivative over (,) and (5,9), and a positive derivative over (, 5) and (9, ).

6 6 I. Concavity: The state of being curved like the inner surface of a sphere..

7 7 How can we relate derivatives to the terms increasing/decreasing and concave up/concave down? f() is increasing when f () > 0. f() is decreasing when f () < 0. f() is concave up when f () > 0. f() is concave down when f () < 0. How can we use this information to locate etrema? First Derivative Test Second Derivative Test

8 8 Eample: Sketch a graph that possesses the characteristics listed. f() is increasing and concave up on (,4). f() is increasing and concave down on (4, ). Eample: Find all relative etrema and classify each as a maimum or minimum. Use the second derivative test where possible. 3 f ( ) 80 9

9 9 II. Inflection Point: a point where the concavity of f() changes from either concave down to concave up, or from concave up to concave down. Check that the inflection point is in the domain of the original function! Eample: Sketch a graph that possesses the characteristics listed. f() is concave down at (1, 5), concave up at (7, -), and has an inflection point at (4, 1).

10 10 To find the potential inflection points, solve for -values that make f () = 0 or make f () undefined. Let s call the solutions to these equations p. Then: 1. f (p) = 0, or. f (p) does not eist (check the domain). Eample: List the coordinates of where etrema or points of inflection occur. 3 f ( ) 3 4 Eample: The graphs of a function, its first derivative, and its second derivative are shown below. Identify each graph.

11 11 III. Steps to graph a function using derivatives. Step 1:Find the domain of f(). Step : Solve for -values that make f () = 0 or make f () undefined. These are the potential minimums and maimums (critical values). Step 3: Plug the solutions from step 3 into f() find the ordered pairs. Eliminate solutions that are not in the domain of f(). Step 4: Plug the solutions from step 3 into f (). If the value is positive, then the ordered pair is a minimum and the graph is concave up at this point. If the value is negative, then the ordered pair is a maimum and the graph is concave down at this point. Step 5: Solve for -values that make f () = 0 or that make f () undefined. These are the potential inflection points. Step 6: Plug the solutions from step 5 into f() find the ordered pairs. Step 7: Choose test values an plug them into f () to determine where the graph is increasing and decreasing.

12 1 Eample: Sketch the graph of each function. List the coordinates of where etrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down. a) g( ) ( 1) 3

13 13 f( ) b) 1

14 14.3 II. Rational Functions r ( ) P( ) Q( ) an b d n d a b 0 0 m b R( ) Q( ) ( ( z1)( z )( 1 z)...( z )...( zn) z ) d E) ( )( 3) r( ) 3 3 ( 1)( )

15 15 Steps To Graph Rational Functions 1. X-intercept: Let y = 0, and solve for. Ordered pair is (, 0). Zeros of the numerator.. Y-intercept: Let = 0 and solve for y. Ordered pair is (0, y). 3. Locate asymptotes. A. Vertical Asymptotes lim f( ). c. lim f( ) c Definition: If, as approaches some number c, the values y approach infinity, then the line = c is a vertical asymptote. Runs parallel to the y-ais Rational functions WILL NOT cross a vertical asymptote.

16 16 To locate vertical asymptotes: Factor the numerator and the denominator. If there is not a common factor to the numerator and the denominator, then the factors that cause a zero in the denominator correspond to vertical asymptotes. Eample: 5 has a vertical asymptote at =. If there is a common factor in the numerator and denominator, then this point forms a hole on the graph. THERE IS NOT A VERTICAL ASYMPOTOTE FOR COMMON FACTORS. ( 5)(4 3) r( ) E: ( 5)( 3)

17 17 B. Horizontal Asymptotes lim f( ) lim f( ) Definition: If, as approaches positive or negative infinity, the values y approach a fied number L, then the line y = L is a horizontal asymptote. Horizontal asymptotes run parallel to the -ais. Rational functions MAY cross a horizontal asymptote.

18 18 Horiztonal Asymptotes n P( ) an... a0 r( ) d Q( ) b... b For d 0 1. Proper Form (n < d): y = 0 is the horizontal asymptote (the -ais). Eample: 5 has a horizontal asymptote at y = 0.. Improper Form. Case I: If n = d, then f() = 0+b = b, and y a b n d is the horizontal asymptote. Eample: 4 ( ). S CaseII: If n > d, then f() = m+b and m+b is an oblique (slant) asymptote. 3 Eample: 3 4

19 19 C. Oblique (Slant) Asymptotes: If the asymptote is not parallel to the -ais or the y-ais, then the asymptote is called oblique. lim f ( ) m b lim f ( ) m b Notice that if m = 0, then f() = m+b becomes f() = b. So the slant asymptote becomes a horizontal asymptote. D. There is a way to locate horizontal and slant asymptotes without comparing degrees of the numerator and denominator take the limit! E) Locate horizontal and slant asymptotes for the following using limits: a) 5 b) S( ) 4

20 0 E: Locate horizontal or oblique asymptotes for the rational function below. c) E: Solve lim 10 3

21 1 4. Determine the domain of f(). 5. Locate min/ma points. 6. Locate inflection points. 7. Sketch the graph. E) Graph 3 r( ) 1.

22 E: Graph s ( ) 1 7

23 3.4 and.5 Recall: RELATIVE MAXIMUM RELATIVE MINIMUM GLOBAL(ABSOLUTE) MAXIMUM GLOBAL (ABSOLUTE) MINIMUM Etreme-Value Theorem If a function f is: Then: 1. Continuous over. A closed interval, say [a, e] 1. The function must have an absolute (global) maimum. The function must have an absolute (global) minimum To find the global minimum and maimum: 1. Determine all critical values in the interval [a, e]. List the critical values along with the endpoints of the interval. 3. Evaluate f at each value listed in step (). 4. The largest output is the global maimum over [a, e]. 5. The smallest output is the global minimum over [a, e].

24 4 E: Find the absolute maimum and minimum values of each function over the indicated interval, and indicate the -values at which they occur.

25 5 E: Find the absolute maimum and minimum values of each function, if they eist, over the indicated interval. Also indicate the -value at which each etremum occurs f ( ) ; (0, 3)

26 6 Aren t there any shortcuts????!!!! If a function f is: Then: 1. Continuous over. A closed interval, say [a, e] 3. f () eists for every in *a, e+ 4. there is eactly ONE critical value in *a, e+ that comes from solving f ()=0. 1. f(c) is the absolute maimum over *a, e+ if f (c) < 0.. f(c) is the absolute minimum over *a, e+ if f (c) > 0. E: Find the absolute maimum and minimum values of each function over the indicated interval, and indicate the -values at which they occur.

27 7 Recall: = number of items Cost C() = cost to produce items C'( ) = rate at which the cost changes = MARGINAL COST FUNCTION. **To minimize cost, find the absolute (global) minimum Revenue R() = revenue from the sale of items R'( ) = rate at which the revenue changes = MARGINAL REVENUE FUNCTION. **To maimize revenue, find the absolute (global) maimum Profit P() = R(X) C(X) = profit from the sale of items P'( ) = rate at which the profit changes = MARGINAL PROFIT FUNCTION. **To maimize profit, find the absolute (global) maimum

28 8 Eample: Wrangler Jeans determines that in order to sell jeans, the price per jean must be p= Wrangler also determines that the total cost of producing jeans is given by C( ) a) Find the total revenue equation, R(). b) How many jeans must the company produce and sell to maimize revenue? c) How many jeans must the company produce and sell to minimize cost? d) Find the profit equation, P(). e) How many jeans must the company produce and sell to maimize profit? f) What is the maimum profit?

29 9 E: Of all numbers whose sum is 50, find the two that have the maimum product.