Familiar Functions - 1 Transformation of Functions, Exponentials and Loga- Unit #1 : rithms Example: The graphs of e x, ln(x), x 2 and x 1 2 are shown below. Identify each function s graph. Goals: Review core function families and mathematical transformations. Textbook reading for Unit #1: Read Sections 1.1 1.4 y x Familiar Functions - 2 Familiar Functions - 3 Comment on the properties of the graphs of inverse functions - Knowing the graphs and properties of essential families of functions is crucial for effective mathematical modeling. Name other families of functions. exponentials - logarithms - powers of x -
Familiar Functions - 4 Give examples of members of each family, and state some of their common properties. Transforming Functions - 1 The core families of functions can be made even more versatile by being transformed. Example: Sketch the graph of y = x 2, over the interval x [ 4, 4]. On the same axes, sketch the graph of y = 4 1 2 (x + 1)2. Transforming Functions - 2 Transforming Functions - 3 Review the four common types of function transformations. Type Form Example Type Form Example
Modeling With Transformations - 1 Modeling With Transformations - 2 Example: Consider the data shown below, showing the concentration of a chemical produced in a reaction vessel, over time. Give a general mathematical form for the function, based on the shape of the graph. e.g. C(t) =... What family of functions would best describe this graph? Point out specific features of the graph that make the choice a reasonable one. Modeling With Transformations - 3 Modeling With Transformations - 4 Determine as many of the numerical values in the formula C(t) =... as you can, given the graph. Sketching related graphs along the way might be helpful. Looking closely at the graph, you see that after 30 hours, the concentration has reached almost exactly 12 ppm. Determine the value for the final missing parameter in your concentration function.
Logarithm Definition - 1 Logarithm Definition - 2 Logarithm Review Without using a calculator, find log 10 (10, 000), and log 10 (1/100). Most students are quite comfortable with exponential functions, but many find logarithms less familiar. To address this we will do a more comprehensive review of the logarithmic function and its use in transforming equations. Log/Exponential Equivalency Simplify log a (a 7 ). a c = x means log a x = c Simplify a log a(25). Logarithm Definition - 3 These problems suggest the following equations, which also follow from the fact that a x and log a (x) are inverse functions. log a (a x ) = x and a log a x = x Rules for Computing with Logarithms 1. log a (AB) = log a A + log a B 2. log a (A/B) = log a A log a B 3. log a (A P ) = P log a A Changing logarithmic bases Changing Log Bases - 1 The functions a x and log a are not provided on calculators unless a = 10 or a = e (see next section of these notes). For other values of a, a x and log a can be expressed in terms of 10 x and log 10. To calculate log a x, we use the following formula: Conversion of Log Bases log a x = log 10 x log 10 a or log e x log e a Prove the above formula, using the Rules for Computing Logarithms and the fact that log a x = c means x = a c.
Changing Log Bases - 2 Example: Without an exact calculation, determine which of log 10 1000 and log 2 1000 is the larger numeric value. Compute the numeric value of both the log values above, using your calculator if necessary. Graphs of Logarithmic Functions Graphs of Logarithmic Functions - 1 The graph of log a x may be obtained from the graph of its corresponding exponential function by reversing the axes (that is, by reflecting the graph in the line y = x). (If drawing the graph of inverse functions is unfamiliar, please read Section 1.3 in the text.) E.g. for y = log 10 x and y = 10 x, 10 x log 10(x) Note: since the logarithm in base 10 is commonly used in science, we define log x (no subscript) to mean log 10 x, for brevity. For the natural logarithm (base e), we use ln instead of log e. Graphs of Logarithmic Functions - 2 What is the domain of log x? What is the range of log x? Sketch the logarithm function for the bases e and 2. Continuous Growth With Exponentials - 1 Classic Applications of Exponentials and Logarithms Example: A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. Sketch the graph of caffeine level over time, after drinking one cup of coffee.
Continuous Growth With Exponentials - 2 There are two natural interpretations of the question statement which lead to two different formulae for A(t). Write down both formulae. Continuous Growth With Exponentials - 3 The key phrase continuous rate has a special meaning in mathematics and science, and it associated with the natural exponential form e rt. It is typically associated with processes like chemical reactions, population growth, and continuously compounded interest. Common alternative statements about percentage growth or decay, where the rate is assumed to be measured at the end of one time period (hour, day year), are usually of the form (1 ± r) t. Compare the predicted caffeine level after 10 hours, using each model. Based on those values, how similar are these two models in practice? Continuous Growth With Exponentials - 4 Write out an appropriate mathematical model for the following scenarios: Infant mortality is being reduced at a rate of 10% per year. Continuous Growth With Exponentials - 5 We now return to our earlier modeling problem. Example: A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. Write the formula for A(t). My $10,000 investment is growing at 5% per year. What is the caffeine level at t = 4 hours? A savings account offers daily compound interest, at a 4% annual rate. Bacteria are reproducing at a continuous rate of 125% per hour. At what time does the caffeine level reach A = 10 mg?
Find the half-life of caffeine in the body. Continuous Growth With Exponentials - 6