Rational, Irrational & Absolute Value Functions Review

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Fredericton High School

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Unit 3 Review Rational Functions, Irrational Functions, Absolute Value Functions

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Calculus Application Problem #3 Name __________________________________________ The Deriving Dead! Due at the beginning of class ______________________ Introduction: Imagine that you are one of many people at a “party” and that, unknown to everyone else, one person was bitten by a zombie on the way to the party! How quickly will the “zombiepocalypse” spread, and what are the chances that you will leave the party as a zombie? The objective of this activity is to create a mathematical model that describes the spread of a disease (such as a zombie virus) in a closed environment, and then apply calculus concepts to this mathematical model. Collecting the Data: Let’s collect some data from an activity that will simulate the spread of a communicable disease over a period of time, divided into “stages”. The number of people in our “closed environment” is ________________ 1 1 131211109876543210 Number of Total Infected Number of Newly Infected Stage Number Applying Calculus to the Data: 1. Using the data from the chart, make a scatterplot of the "Stage Number" (in L1) vs. the "Number of Total Infected" (in L2). Sketch the scatterplot below. Connect the data points to create a continuous function for Y(t). 2. Using the data that was collected in the activity, answer the following questions about the derivative function Y’(t), which represents the instantaneous rate of change of the number of infected at any stage. Consider the domain to be [ 0 , 13 ]. a. When, if ever, is Y’(t) positive? ____________________________________ b. When, if ever, is Y’(t) negative? ___________________________________ c. When, if ever, is Y’(t) increasing? ____________________________________ d. When, if ever, is Y’(t) decreasing? ____________________________________ e. From your answers above, sketch a graph of Y’(t) below. f. The t-value where Y’(t) changes from increasing to decreasing is the inflection point on Y(t). According to the data in the chart, this occurs when t = _________, and the corresonding “y-value” is ________. (Note: We will check this later in the problem!) Finding a Logistic Function that Models the Data 3. Since the data (should) appear to be a model for a logistic function, we need to find a function in the form: Y(t ) = c 1 + a ⋅ e− b⋅t , where t represents the stage number and Y(t) represents the total number of infected people in stage t. Therefore, we need to find values for the three constants a, b, and c. The value of c should be easy. For our activity, c = _________ To find a, use the initial point ( 0 , 1 ). Substitute this ordered pair, with the value of c into our logistic model and solve for a. Show your work below. a = _______________ To find b, the last constant in the model, we need another ordered pair. Let’s use an ordered pair near the middle of the data, say during Stage #7. Record this ordered pair: ( 7 , _________) Substitut ...

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1. Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. (Enter NONE in any unused answer blanks.) Equation of horizontal asymptote: Equation of vertical asymptote: Value of y-intercept Value of x-intercept The function is 2. Use the graph of y = 2x to match the function with its graph. A B C D y = 2x – 4 y = 2x – 5 y = 2x + 4 y = 2–x 3. Use the graph of f to describe the transformation that yields the graph of g. Then sketch the graphs of f and g by hand. f(x) = −2x, g(x) = 5 − 2x The graph of g(x) = 5 − 2x is a vertical shift five units downward of f(x) = −2x. The graph of g(x) = 5 − 2x is a horizontal shift five units to the left of f(x) = −2x. The graph of g(x) = 5 − 2x is a vertical shift five units upward of f(x) = −2x. The graph of g(x) = 5 − 2x is a horizontal shift five units to the right of f(x) = −2x. Sketch the graphs of f and g. 4. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.) x f(x) = 5x − 3 -1 0 1 2 3 Sketch the graph of the function. Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.) vertical asymptote x = horizontal asymptote y = 5. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.) x g(x) = 4 − e−3x -4 -3 -2 -1 0 Sketch the graph of the function. Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.) vertical asymptote x = horizontal asymptote y = 6. Fill in the blank. If x = ey, then y = . 7. For what value of x is ln x = ln 9? x = 8. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25. log2 512 = 9 = 9. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25. log 1 100,000,000 = -8 = 10. Write the exponential equation in logarithmic form. For example, the logarithmic form of 23 = 8 is log2(8) = 3. 43/2 = 8 log = 11. Use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places. f(x) = log10(x) x = 4/5 12. Solve the equation for x. log10(102) = x 13. Write the logarithmic equation in exponential form. For example, the exponential form of ln(5) = 1.6094... is e1.6094... = 5. (Do not use ... in your answer.) = 14. Write the exponential equation in logarithmic form. For example, the logarithmic form of e2 = 7.3890 is ln 7.3890 = 2. (Do not use ... in your answer.) e2.2 = 9.0250 ln = 15. Use the properties of natural logarithms to rewrite the expression. 5 ln(e5) 16. Use the properties of logarithms to rewrite and simplify the logarithmic expression. ln 9 e9 17. Use the properties of logarithms to expand the expression as a ...

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