FastTrack MA 137 BioCalculus

Transcription

1 FastTrack MA 137 BioCalculus Functions (2): More Eamples and Alberto Corso Department of Mathematics Universit of Kentuck Goal: We continue with more eamples of basic functions We also stud how certain transformations ( shifting, reflecting, and stretching) of a function affect its graph This gives us a better understanding of how to graph functions ma137 Lecture #2 Monda 1/30

2 A quadratic function is a function f of the form f () = a 2 + b + c, where a, b, and c are real numbers and a 0 The graph of an quadratic function is a parabola; it can be obtained from the graph of f () = 2 b elementar transformations Indeed, b completing the square, a quadratic function f () = a 2 + b + c can be epressed in the standard form f () = a( h) 2 + k The graph of f is a parabola with verte (h, k); the parabola opens upward if a > 0, or downward if a < 0 k k Verte (h, k) (Minimum) h a > 0 (Maimum) Verte (h, k) h a < 0 2/30 ma137 Lecture #2 Monda

3 Epressing a quadratic function in standard form helps us sketch its graph and find its maimum or minimum value There is a formula for (h, k) that can be derived from the general quadratic function as follows: f () = a 2 + b + c = a ( 2 + ba ) + c Thus: = a ( 2 ba b a 2 ( = a + b 2a h = b 2a ma137 ) 2 + c b2 4a k = ) + c b2 4a 4ac b2 4a Lecture #2 Monda 3/30

4 Geometric Interpretation of Completing the Square This interpretation goes back to the Bablonian scribes, who full used the cut-and-paste geometr developed b the ancient surveors (ca 1700 BC) Here,, a, and b are positive as the represent lengths: b/a + b 2a + b 2a where is a square of side b/2a; thus its area is (b/2a) 2 ma137 Lecture #2 Monda 4/30

5 The Quadratic Formula The previous calculation actuall allows us to derive the general formula for the solution of the quadratic equation: The Quadratic Formula The roots 1 and 2 of the quadratic equation a 2 + b + c = 0, where a 0, are: 1,2 = b ± b 2 4ac 2a Note: The easiest method to solve a quadratic equation is b factoring it Use the quadratic formula onl when a factorization is not readil visible ma137 Lecture #2 Monda 5/30

6 Eample 1 (Torricelli s Law): A tank holds 50 gallons of water, which drains from a leak at the bottom, causing the tank to empt in 20 minutes The tank drains faster when it is nearl full because the pressure on the leak is greater Torricelli s Law gives the volume of the water remaining in the tank after t minutes as ( V (t) = 50 1 t 20 ) 2 0 t 20 (a) Find V (0) and V (20) (b) What do our answers to part (a) represent? (c) Make a table of values of V (t) for t = 0, 5, 10, 15, 20 ma137 Lecture #2 Monda 6/30

7 Eample 2: When a certain drug is taken orall, the concentration of the drug in the patient s bloodstream after t minutes is given b C(t) = 006t 00002t 2, where 0 t 240 and the concentration is measured in mg/l When is the maimum serum concentration reached? What is that maimum concentration? ma137 Lecture #2 Monda 7/30

8 A Chemical Reaction (Eample 5, Section 12, p 20) Consider the reaction rate of the chemical reaction A + B AB in which the molecular reactants A and B form the molecular product AB The rate at which this reaction proceeds depends on how often A and B molecules collide The law of mass action states that the rate at which this reaction proceeds is proportional to the product of the respective concentrations of the reactants (Here, concentration means the number of molecules per fied volume) Denote the reaction rate b R and the concentration of A and B b [A] and [B], respectivel The law of mass action sas that R [A] [B] Introduce the proportionalit factor k We obtain R = k[a] [B] ma137 Lecture #2 Monda 8/30

9 Note that k > 0, because [A], [B], and R are positive We assume now that the reaction occurs in a closed vessel; that is, we add specific amounts of A and B to the vessel at the beginning of the reaction and then let the reaction proceed without further additions We can epress the concentrations of the reactants A and B during the reaction in terms of their initial concentrations a and b and the concentration of the molecular product [AB] If = [AB], then [A]=a for 0 a and [B]=b for 0 b The concentration of AB cannot eceed either of the concentrations of A and B (For eample, suppose five A molecules and seven B molecules are allowed to react; then a maimum of five AB molecules can result, at which point all of the A molecules are used up and the reaction ceases The two B molecules left over have no A molecules to react with) 9/30 ma137 Lecture #2 Monda

10 Therefore, we get R() = k(a )(b ) for 0 a and 0 b The condition 0 a and 0 b can be written as 0 min(a, b) Epand the epression for R(), to see that R() is indeed a polnomial function (of degree 2) R() = k(ab a b + 2 ) = k 2 k(a + b) + kab for 0 min(a, b) A graph of R(), 0 a, is shown for the case a b (We chose k = 2, a = 2, and b = 5) ma137 Lecture #2 Monda 10/30

11 Notice that when = 0 (ie, when no AB molecules have et formed), the rate at which the reaction proceeds is at a maimum As more and more AB molecules form and, consequentl, the concentrations of the reactants decline, the reaction rate decreases This should also be intuitivel clear: As fewer and fewer A and B molecules are in the vessel, it becomes less and less likel that the will collide to form the molecular product AB When = a = min(a, b), the reaction rate R(a) = 0 This is the point at which all A molecules are ehausted and the reaction necessaril ceases ma137 Lecture #2 Monda 11/30

12 Eample 3: Find the scaling relation between the surface area S and the volume V of a sphere of radius R [More precisel, show that S = (36π) 1/3 V 2/3, that is, S V 2/3 ] ma137 Lecture #2 Monda 12/30

13 Eample 4: (Michaelis-Menten enzmatic reaction) According to the Michaelis-Menten equation (1913) when a chemical reaction involving a substrate S is catalzed b an enzime, the rate of reaction V = V ([S]) is given b the epression V = V ma[s] K m + [S], where [S] denotes substrate concentration (for eamples in moles per liter), and V ma and K m are constants V ma is the maimal velocit of the reaction and K m is the Michaelis constant K m is the substrate concentration at which the reaction achieves half of the maimum velocit Graph V assuming that V ma = 3 and K m = 2 That is, V = 3[S] 2 + [S] ma137 Lecture #2 Monda 13/30

14 Eample 5: (Lineweaver-Burk plot) The Lineweaver-Burk plot (1934) was widel used to determine important terms in enzme kinetics, such as K m and V ma, before the wide availabilit of powerful computers and non-linear regression software The Michaelis-Menten rate function V = V ma[s] traces out a K m + [S] hperbola The reciprocal of this epression is written 1 V = K m 1 V ma [S] + 1 V ma That is, the reciprocal epression is linear in = 1 [S] and = 1 V The slope of this line is K m /V ma ; the -intercept is 1/V ma and the -intercept is 1/K m The graph in the -plane is called the Lineweaver-Burk plot Eg: Given V = 3[S] 2 + [S], plot = ma137 Lecture #2 Monda 14/30

15 Let f be a function f is even if f ( ) = f () for all in the domain of f f is odd if f ( ) = f () for all in the domain of f f ( ) = 0 = f () = 0 f () = EVEN Graph smmetric wrt -ais f ( ) ODD Graph smmetric wrt (0, 0) ma137 Lecture #2 Monda 15/30

16 Eample 6: Determine whether the following functions are even or odd: f () = g() = ma137 Lecture #2 Monda 16/30

17 Curious/Amazing Fact! An function can be uniquel written as an even plus an odd function Eample: e = e + e }{{ 2 } cosh + e e 2 }{{} sinh = e = cosh = sinh /30 ma137 Lecture #2 Monda

18 Vertical Shifting: Suppose c > 0 Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching To graph = f () + c, shift the graph of = f () upward c units To graph = f () c, shift the graph of = f () downward c units c = f () + c = f () c = f () 0 0 = f () c 18/30 ma137 Lecture #2 Monda

19 Horizontal Shifting: Suppose c > 0 Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching To graph = f ( c), shift the graph of = f () to the right c units To graph = f ( + c), shift the graph of = f () to the left c units = f ( c) = f ( + c) = f () c c 0 0 = f () ma137 Lecture #2 Monda 19/30

20 Eample 7: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Use the graph of = to sketch the graphs of the following functions: = + 3 = 2 ma137 Lecture #2 Monda 20/30

21 Eample 8: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Use the graph of = to sketch the graphs of the following functions: = + 3 = ma137 Lecture #2 Monda 21/30

22 Eample 9: The graph of = f () is shown below Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Sketch the graph of = f ( 1) ma137 Lecture #2 Monda 22/30

23 Reflecting Graphs To graph = f (), reflect the graph of = f () in the -ais = f () Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching To graph = f ( ), reflect the graph of = f () in the -ais = f () = 0 = 0 = = = f () = f ( ) 23/30 ma137 Lecture #2 Monda

24 Eample 10: The graph of = f () is shown below 3 Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching /30 ma137 Lecture #2 Monda

25 Eample 10 (cont d): Sketch the graph of = f ( ) Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Sketch the graph of = f () Sketch the graph of = f ( ) ma137 Lecture #2 Monda 25/30

26 Vertical Stretching and Shrinking: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching To graph = cf (): If c > 1, STRETCH the graph of = f () verticall b a factor of c If 0 < c < 1, SHRINK the graph of = f () verticall b a factor of c = cf () = f () 0 = f () 0 = cf () c > 1 ma137 Lecture #2 Monda 0 < c < 1 26/30

27 Horizontal Shrinking and Stretching: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching To graph = f (c): If c >1, shrink the graph of = f () horizontall b a factor of 1/c If 0<c <1, stretch the graph of = f () horizontall b a factor of 1/c = f (c) = f (c) 0 = f () 0 = f () c > 1 ma137 Lecture #2 Monda 0 < c < 1 27/30

28 Eample 11: Sketch the graph of: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching = 2 2 = ma137 Lecture #2 Monda 28/30

29 Eample 12: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Use the graph of f () = 2 2 provided below to sketch the graph of f (2) /30 ma137 Lecture #2 Monda

30 Eample 13: Vertical Shifting Horizontal Shifting Reflecting Graphs Vertical Stretching and Shrinking Horizontal Shrinking and Stretching Use transformations to sketch the graph of = Use transformations to sketch the graph of = 3( + 2) ma137 Lecture #2 Monda 30/30