The Rule of Four Promotes multiple representations. Each concept and function is represented:

Transcription

1 The Rule of Four Promotes multiple representations. Each concept and function is represented: 1) 2) 3) 4)

2 Symbolically Numerically Graphically Verbally

3 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally What Is a Function? A function is a rule which takes certain values as inputs and assigns to each input value exactly one output value. The output is a function of the input. The inputs and outputs are also called variables.

4 Representing Functions Words Tables Graphs Formulas

5 Oecanthus Fultoni The Snowy Tree Cricket Nature s Thermometer" Page N/A 5

6 By counting the number of times a snowy tree cricket chirps in 15 seconds & adding We can estimate the temperature (in degrees Fahrenheit)!!! Page 2 6

7 What is the independent variable? What is the dependent variable? Chirps Temperature

8 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Representing Functions: Formulas Solution: As a Formula A formula is the equation giving T (temperature) in terms of R (chirps*). Dividing the chirp rate (in minutes) by four and adding forty gives the estimated temperature, so:

9 FORMULA (in minutes) Estimated temp (in F)= 0 1 Chirp rate (in chirps/min.)+40 4 T R 1 T R 40 4 Page 3 9

10 By assigning more substitutions into the formula, we can create a table: Page 3 10

11 R, chirp rate (chirps/minute) T, predicted temperature ( F) Page 3 11

12 the independent variable is Chirp rate Label: (chirps / minute) and the dependent variable is Temperature Label: (Fahrenheit)

13 Vertical : dependent variable Meaning? Horizontal : independent variable Page 3

14 When we use a function to describe an actual situation, the function is referred to as a mathematical model. 1 T R 40 4 This the mathematical model of the relationship between the temperature and the cricket's chirp rate. Page 3 14

15 1 T R 40 4 If the temperature is 30 degrees, what is R? Page 4 15

16 SHOW ALL WORK 1 30 R R R 40 Interpret the result - WHAT DOES THIS MEAN? IMPOSSIBLE!!!! Negative chirp rate

17 Therefore state the: Domain: 40 R (for the independent variable) Range: 0 T (for the dependent variable)

18 WRONG Reverse the values

19 1 T R 40 4 Is T a function of R, or R a function of T? Page 4 19

20 1 T R 40 4 T is a function of R. Page 4 20

21 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally What Is a Function? A function is a rule which assigns each independent variable x, of the domain to one and only one dependent variable y, in the range. The output is a function of the input.

22 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Mathematical Models & Functional Notation

23 Functional Notation Q is a function of the value, t Or: Q is a function of t We say: Q equals f of t We write: Q = f (t) or Q(t) Page 4 23

24 Q = f (t) means: applying the rule f to the input value, t, gives the output value, f(t). Q = dependent variable (unknown, depends on t) t = independent variable (known) Page 4 24

25 Q = f(t). In other words: Output = f(input) Or: Dependent = f(independent) Page 4 25

26 Generate functional notation given the two variables: hours of study final grade h=f(g) or g=f(h) Interpret Verbal meaning?

27 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2.

28 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Write the functional notation: n = f (A) Find a formula for f n f ( A) A 250

29 Explain in words (interpret) what the statement f(10,000) = 40 tells us about painting houses. Solution: An area of A = 10,000 ft 2 requires n = 40 gallons of paint. n f ( A) Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

30 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Functions Don t Have to Be Defined by Formulas The average monthly rainfall, R, at Chicago s O Hare airport is given in the Table, where time, t, is in months and t = 1 is January, t = 2 is February, and so on. The rainfall is a function of the month, so we write R = f (t). However there is no equation that gives R when t is known. a) Evaluate f (1) and f (11). b) Explain your answers. Month, t Rainfall, R (inches)

31 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Solution Month, t Rainfall, R (inches) The value of f (1) is the average rainfall in inches at Chicago s O Hare airport in a typical January. From the table, f (1) = 1.8 inches. Similarly, f (11) = 2.4 means that in a typical November, there are 2.4 inches of rain at O Hare.

32 NOT A FORMULA Semester Average Final Grade A B B C C D D Below 60 F

33 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally When Is a Relationship Not a Function? Exercise 38 (b) A person leaves home and walks due west for a time and then walks due north. (b) Suppose that x is the distance that she walks in total and D represents her (variable) distance from home at the end of her walk. Is D a function of x? Why or why not? Solution (b) D is NOT a function of x. Suppose the total distance walked is x = 10. By the Pythagorean Theorem, consider two scenarios: walk west 9 and north 1, then walk west 5 and north 5, then D(10) D(10)

34 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally How to Tell if a Graph Represents a Function: Vertical Line Test Visualizing the Vertical Line Test y vertical line x No matter where we draw the vertical line, it will intersect the red graph at only one point, so the red graph represents a function. But the vertical line intersects the blue graph twice, so the blue graph does not represent a function.

35 Quick Questions

36 Quick Questions

37 No calculator 10 f ( x) 1 x 2 x f(x)

38 The eyewall of a hurricane is the band of clouds that surrounds the eye of the storm. The eyewall wind speed v (in mph) is a function of the height above the ground s (in meters).

39 Hurricane cross-section

40 In the examples, use Table 1.6, which gives values of v (s), the eyewall wind profile of a typical hurricane.

41 Table 1.6 S V S V Interpret then evaluate v(300).

42 Table 1.6 S V S V At what altitudes does the eyewall wind speed appear to equal or exceed 116 mph?

43 Table 1.6 S V S V At what height is the eyewall wind speed greatest?