Announcements. Topics: Homework: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook!

Transcription

1 Announcements Topics: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)

2 Exercise - Equation Analysis C = A rw βt If C is a function and t is independent variable: (1) Which quantities are parameters (2) Replace all parameters by numbers do you recognize the equation? (3) Keep the parameters (do not give them numeric values); what is the graph of C (you will need to make assumptions, for instance β could be positive, negative, or zero) (4) If C is a function of A, what is its graph? (5) If C is a function of r, what is its graph? (6) If C is a function of W, what is its graph?

3 Working With Functions Review addition, subtraction, multiplication, division, and composition of functions on your own Review transformations of graphs and inverse functions (we ll do a brief review here)

4 Inverse Functions The function f 1 is the inverse of f if f 1 ( f (x)) = x and f ( f 1 (x)) = x. f 1 Each of and f undoes the action of the other. Some simple examples:

5 What Functions Have Inverses? A function has an inverse if and only if it is a one-to-one function. A function f is one-to-one if for every y-value in the range of f, there is exactly one x-value in the domain of f such that y=f(x).

6 Horizontal Line Test If every horizontal line intersects the graph of a function in at most one point, then the graph represents a one-to-one function.

7 Finding the Inverse of a Function Algorithm: 1. Write the equation y=f(x). 2. Solve for x in terms of y. 3. Replace x by (x) and y by x. f 1 Note: The domain and range are interchanged Example:.

8 Temperature Conversion The relationship between degrees Celsius (C) and degrees Fahrenheit (F) is linear. We know that corresponds to 32 o F and corresponds to 212 o F. 0 o C 100 o C

9 Temperature Conversion (a) Find the function that converts o C to o F. Note: input is o C and output is o F Data Points: (0 o C, 32 o F) and (0 o C, 32 o F) slope = change in output change in input = ΔF ΔC = =1.8 Function: F(C) =1.8C + 32

10 Temperature Conversion (b) Find the function that converts o F to o C. Note: input is o F and output is o C One approach: Find the INVERSE of F(c): Note: we do not interchange variables at the end since F and C have a physical meaning F =1.8C + 32 F 32 =1.8C F = C C(F) = F

11 Exponential Functions An exponential function is a function of the form f (x) = a x where a base and Domain: Range: is a positive real number called the x is a variable called the exponent. x R y > 0 *Note: Please review EXPONENT LAWS on your own!

12 Graphs of Exponential Functions y y x x f (x) = 3 x f (x) = ( 1 ) x 2 When a>1, the function is increasing. When a<1, the function is decreasing. y=0 is a horizontal asymptote

13 Transformation of an Exponential Graph f (x) = e 2x + 3. Function y Recall: e is a special irrational number between 2 and 3 that is commonly used in calculus x Approximation: e 2.718

14 Logarithmic Functions The inverse of an exponential function is a logarithmic function, i.e. If f (x) = a x, then f 1 (x) = log a x. Cancellation equations: In general: For exponentials & logarithms: f ( f 1 (x)) = x f 1 ( f (x)) = x a log a x = x log a a x = x e ln x = x lne x = x

15 Graphs of Logarithmic Functions Recall: For inverse functions, the domain and range are interchanged and their graphs are reflections in the line y = x. Example: Graph f (x) = ln x.

16 Graphs of Logarithmic Functions e 2.7 y f 1 (x) = e x (1,e) ( 1,e 1 ) (0,1) x

17 Laws of Logs For x,y>0 and p any real number: ln(xy) = ln x + ln y ln(x / y) = ln x ln y ln(x p ) = pln x

18 Semilog Graphs Definition: A semilog graph plots the logarithm of the output against the input. The semilog graph of a function has a reduced range making the key features of certain functions easier to distinguish.

19 Example: Semilog Graphs

20 Semilog Graphs Example: Sketch the semilog graph of f (x) =10e 4 x.

21 Double-Log Graphs Definition: A double-log graph plots the logarithm of the output against the logarithm of the input.

22 Semilog and Double-Log Graphs Example: Blood Circulation Time in Mammals Sketch the semilog and double-log graphs for the model T(B) =17.73B B

23 Exponential Models When the change in a measurement is proportional to its size, we can describe the measurement as a function of time by the formula S(t) = S(0)e α t where S(t) is the value of the measurement at time t S(0) is the initial value of the measurement, and α is a parameter which describes the rate at which the measurement changes

24 Doubling Time. Example: A bacterial culture starts with 100 bacteria and after 3 hours the population is 450 bacteria. Assuming that the rate of growth of the population is proportional to its size, find the time it takes for the population to double.

25 Half-Lives of Drugs Example: Thinking in Half-Lives # of half-lives amount left in body % amount left in body 0 M(0) ** Many drugs are not effective when less than 5% of their original level remains in the body.

26 Trigonometric Functions Trigonometric functions are used to model quantities that oscillate.

27 Trigonometric Models Example: Seasonal Growth A population of river sharks in New Zealand changes periodically with a period of 12 months. In January, the population reaches a maximum of 14, 000, and in July, it reaches a minimum of 6, 000. Using a trigonometric function, find a formula which describes how the population of river sharks changes with time.

28 Example: Trigonometric Models

29 Inverse Trigonometric Functions Since the 3 main trigonometric functions are not one-to-one on their natural domains we must first restrict their domains in order to define inverses.

30 Inverse of Sine Restrict the domain of f (x) = sin x to [ π, π ]. 2 2 Now the function is one-to-one on this interval so we can define an inverse.

31 Inverse of Sine The inverse of the restricted sine function is denoted by f 1 (x) = sin 1 x or f 1 (x) = arcsin x. Cancellation equations: arcsin(sin x) = x sin(arcsin x) = x x [ π, π ] 2 2 x [ 1,1] (domain of sin x) (domain of arcsin x) Calculate: arcsin( 1) 2 sin(arcsin( 5 )) 7 arcsin(sin(π))

32 Graphs of Sine and Arcsine y = sin x y = arcsin x domain: x [ π 2, π 2 ] domain: x [ 1,1] range: y [ 1,1] range: y [ π 2, π 2 ]

33 Inverse of Tangent Restrict the domain of f (x) = tan x to ( π, π ). 2 2 This portion of tangent passes the HLT so tangent is one-to-one here

34 Inverse of Tangent The inverse of the restricted tangent function is denoted by f 1 (x) = tan 1 x or f 1 (x) = arctan x. Cancellation equations: arctan(tan x) = x tan(arctan x) = x x ( π 2, π 2 ) x (, ) (restricted domain of tan x) (domain of arctan x) Calculate: arctan(1) tan(arctan10) arctan( 3)

35 Graphs of Tangent and Arctangent y = tan x y = cos x y = arctan x y = arccos x domain: range: x ( π 2, π 2 ) y (, ) domain: x (, ) range: y ( π 2, π 2 )