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Andrés Arboleda Ap Precalculus...

Andrés Arboleda Ap Precalculus 1 Unit 01: Activity 1.1: Change in tandem 1. 2. 3. 4. 5. a. On what intervals f is decreasing and why? 1 1) C 2) D 3) C 4) B 5) a) f is decreasing on the intervals (- 1, 2); b) f is both negative and increasing on the intervals (- 5, - 3) c) f is both positive and decreasing on the intervals (- 1, 2) d) f is both positive and increasing on the intervals (2, 5)  Andrés Arboleda Ap Precalculus b. On what intervals is both negative and increasing and why? c. On what intervals is both positive and decreasing and why? d. On what intervals is both positive and increasing and why? 6. 2 a. On what intervals is g decreasing and the graph of g is concave up? b. On what intervals is the rate of change of g positive and decreasing? 7. 8. 9. 2 6a) g is decreasing and the graph of g is concave up on the interval (- 1, 1), 6b) the rate of change of g is positive and the decreasing on the interval (- 5, - 3), 9a) the function is concave down on the interval (2, 3), 9b) the function is concave up on the interval (0, 1)  Andrés Arboleda Ap Precalculus a. b. On what intervals is f concave up? c. 3On what intervals is f increasing? d. On what intervals is f decreasing? 10. 11. The tables describes the behavior of a function f for selected intervals of x a. On what intervals is the rate of change of f positive? b. On what intervals is the rate of change of f negative? c. On what intervals is the rate of change of f increasing? d. On what intervals is the rate of change of f decreasing? 12. 3 9c) f is increasing on the interval (1, 2), 9d) f is decreasing on the interval (3, 4), 10) A, 11a) (1, 2), 11b) (3, 4), 11c) (0, 1), 11d) (2, 3); 12a) (0, 4); 12b) ( -3, 0) U (4, 6), 12c) (0, 2), 12d) ( -3, 0)  Andrés Arboleda Ap Precalculus The graph of the function f is given for. f has a point of inflection at a. On what intervals is the rate of change of f positive? b. On what intervals is the rate of change of f negative? c. On what intervals is the rate of change of f positive and increasing? d. On what intervals is the rate of change of f negative and increasing? 4 Activity 1.2: Rates of change 1. The graph above shows. Complete the blanks below to correctly describe the graph of f and the rate of change of f 2. 3. Find the average rate of change for the following functions on the given intervals. Show all work a. d. b. e. 4 1) increasing, concave down, positive, decreasing, 2) positive, decreasing, negative, increasing, 3) a) - 4, b) ⅖ , c) ⅗ d) 4 , e) - 4/7, f) - 6/11  Andrés Arboleda Ap Precalculus c. f. 4. 5Selected values for the function f(x) are shown in the table below. Find the average rate of change (AROC) of f(x) from x = 1 to x = 8 5. Let The average rate of change of n(x) over the interval [c, 5] is equal to 3, where c is a constant. Find the value of c. 6. The table below list the annual budget, in thousands of dollars, for each of six different state programs in Kansas form 2007 to 2010 Which of the following best approximates the average rate of change in the annual budget for agriculture/natural resources in Kansas from 2008 to 2010? a. $50 000 000 per year c. 75 000 000 per year b. 65 000 000 per year d. 130 000 000 per year 7. Write the equation of the line with slope - 1.57 and passing through the point (21, 37). 8. The table gives the average rate of change of a function f over different intervals. On which of the intervals does the function increase the most? 5 4) Aroc= 5/7, 5) c = - 2, 6) b, 7) y - 37 = - 1.57 (x - 21), 7) b, 8) D 9) B  Andrés Arboleda Ap Precalculus 9. The graph of y = g(x) is given. On the following, on which intervals is the average rate of change of g the least? 6 10. The function f has a negative average rate of change on every interval of x in the interval. The function g has a negative average rate of change on every interval of x in the interval , and a positive average rate of change on every interval of x in the interval.Which of the following statements must be true about the function h, defined by , on the interval ? a. h is decreasing on b. h is decreasing on ; h is increasing on c. h is decreasing on ; h is neither increasing nor decreasing on d. h is decreasing on ; h can be increasing, decreasing, or both increasing and decreasing on (questions 11 and 12 refer to the following information) 11. The graph of h is shown below along with four points A, B, C, and D. a) sketch a line tangent to the graph of h, at the four points indicated on the graph, b) order the rates of change of the graph of h from least to greatest at the points A, B, C, and D. 12. For each of the following statements about the graph of h shown above, circle the correct answer and the correct reasoning. a. From point A to B, the rate of change of h is increasing/decreasing because the graph of h is concave up/concave down over the interval. b. From point B to C, the graph of h is increasing/decreasing because the rate of change of h is positive/increasing over the interval. 6 10) D; 11) D, A, C, and D; 12) a) increasing and concave up; b) increasing and positive, c) decreasing and concave down  Andrés Arboleda Ap Precalculus c. From point C to D, the rate of change of h is increasing/decreasing because the graph of h is decreasing/concave down over the interval. 13. 7After Mr. Sepulveda missed a day of school, a rumor began to spread that he had won the powerball lottery and moved to Japan. Initially, seven students knew about the rumor (they were the ones that started it). After two hours (t = 2), a total of 15 students had heard the rumor. After six hours (t = 6), 67 students had heard the rumor. The number of students that have heard can be modeled by the piecewise function R given by Where R(t) is the number of students that have heard the rumor at time t = hours since the rumor first began. a. Use the given data to find the average rate of change in the number of students that have heard the rumor, in students per hour, from t = 2 to t = 6 hours. Express your answer as a decimal approximation. Show the computations that lead to your answer. b. Interpret the meaning of your answer from (a) in the context of the problem c. Use the average rate of change found in (a) to estimate the number of students that have heard the rumor after t = 9 hours. Show the computations that lead to your answer 14. if a. Find the average rate of change of f(x) on the interval [3, 10]. Write your answer a decimal approximation. Aroc - 0.857 b. Use the average rate of change found in part (a) to write the equation of the secant line on the interval [3, 10] y - 17 = - 0.857 (x - 3) or y - 11 = - 0.857 (x - 10) 15. The table above lists the life expectancy of US females born in a given year. Find the average rate of change in the life expectancy of US females born from 1850 to 2000. Include units of measure. 7 13) a) AROC: 13 students per hour, b) the AROC of the number of student who have heard the rumor since the rumor first began is 13 students per hour from t = 2 to t = 6 hours since the rumor first began, c) R(9) approximate 106 students  Andrés Arboleda Ap Precalculus a) estimate the rate of change of f at x = - 4 using the slope of the line tangent to f at x = - 4. b) sketch the tangent lines to f at the five points indicated on the graph above. A developer begins building houses in a large neighborhood. After one month (t = 1), the developer had built six houses. At the end of month 8 (t = 8), the developer had built 15 total houses. The number of houses that have been built by the developer after t months can be modeled by the function H given by where h(t) is the total number of houses built at time t months. a. Use the given data to find the average rate of change in the number of houses that have been built, in houses, in house per month, from t = 1 to t = 8 months. Express your answer as a decimal approximation. Show the computations that lead to your answer. AROC: slope of secant line= 1.286 or (1.285) houses per month b. Use the average rate of change found in (a) to estimate the number of houses built at time t = 12 months. Show the computations that lead to your answer. h(12) approximate 20.143 houses or 20.142 On the AP Exam, FRQ 2 part B will look like these questions. In MAy 2011 (t = 0), 65% of US adults did not own a smartphone. In November 2016 (t = 5.5), only 23% of US adults did not own a smartphone. The percent of US adults that did not own a smartphone can be modeled by the function S given by where S(t) is the percent of US adults that did not own a smartphone t years since May 2021.  Andrés Arboleda Ap Precalculus a. Use the given data to find the average rate of change in the percent of US adults that did not own a smartphone, in percent per year, from t = 0 to t = 5.5 years. Express your answer as a decimal approximation. Show the computations that lead to your answer. AROC: - 7.636 percent per year b. Une the average rate of change found in (a) to estimate the percent of us adults that did not own a smartphone for t= 10.2 years. Show the computations that lead to your answer. - 12.891 % - 12.890 8 Activity 1.3 Rates of change in linear and quadratic functions 1. The table gives you values of the function f for selected values of x. If the function f is linear, what is the value of f(13)? a. 4 c. 28/3 b. 29/4 d. 34/3 2. The function f is defined for all real values of x. For a constant a, the average rate of change of f from x = a to x = a + 1 is given by the expression 2a + 1. Which of the following statements is true? a. The average rate of change of f over consecutive equal - length input - value intervals is positive, so the graph of could be a line with a positive slope b. The average rate of change of f over consecutive equal - length input - value intervals is positive, so the graph of f could be a parabola that opens up. c. The average rate of change of f over consecutive equal - length input - value intervals is increasing at a constant rate, so the graph of f could be a line with a positive slope. d. The average rate of change of f over consecutive equal - length input - value intervals is increasing at a constant rate, so the graph of f could be a parabola that opens up. 3. An object is moving in a straight line from a starting point. The distance, in meters, from the starting point at selected times, in seconds, is given in the table. If the pattern is consistent, which of the following statements about the rate of change of the rates of change of distance over time is true? a. The rate of change of the rates of change is 0 meters per second, and the object is neither speeding up nor slowing down. 8 1) D, 2) D, 3) B  Andrés Arboleda Ap Precalculus b. The rate of change of the rates of change is 0 meters per second per second, and the object is neither speeding up nor slowing down. c. The rate of change of the rates of change is 4 meters per second, and the object is neither speeding up nor slowing down. d. The rate of change of the rates of change is 1 meters per second, and the object is speeding up. 4. 9 The table gives values of a function f for selected values of x. Which of the following conclusions with reason is consistent with the data in the table? a. F could be a linear function because the rates of change over consecutive equal - length intervals in the table can be described by y = 2x. b. F could be a linear function because the rates of change over consecutive equal - length intervals in the table can be described by y = 2x + 1. c. F could be a quadratic function because the rates of change over consecutive equal - length intervals in the table can be described by y = 2x. d. F could be a quadratic function because the rates of change over consecutive equal - length intervals in the table can be described by y = 2x + 1. 5. Directions: selected values for several functions are given in tables below. For each table of values, determine if the function could be linear, quadratic, or neither. 6. Directions: Selected values for several functions are given in the tables below. For each table of values, determine if the function could be concave up, concave down, or neither. 9 4) D; 5) 1. Quadratic; 2. Neither; 3. Linear; 5. Linear; 6. Quadratic, 6) 7. Concave Down; 8. Neither; 9. Neither; 7) 7. K = 8; k = -5; K = 6  Andrés Arboleda Ap Precalculus 7. Directions: For 7 - 9, the tables below give values of several quadratic functions at selected values of x. For each function, find the value of the constant K in the table. 10 Activity 1.4 Rates of change in linear and quadratic functions 1. Consider the graph of g(x) shown above. For each of the following intervals, determine if the rate of change of g(x) is positive and increasing, positive and decreasing, negative and increasing, or negative and decreasing. 2. For the polynomial function g, the rate of change of g is increasing for x < 2 and decreasing for x > 2. Which of the following must be true? A. The graph of g has a minimum at x = 2. B. The graph of g has a maximum at x = 2. C. The graph of g has a point of inflection at x = 2, is concave down for x < 2, and is concave up for x > 2. D. The graph of g has a point of inflection at x = 2, is concave up for x < 2, and is concave down for x > 2. 10 1) a) negative and increasing; b) positive and decreasing; c) negative and decreasing; d) positive and increasing; 2) Andrés Arboleda Ap Precalculus Andrés Arboleda Ap Precalculus