Columbus School Activities & Homework PDF

Andrés Arboleda Ap Precalculus Activities & Homework Andrés Arboleda Ap Precalculus  Andrés Arboleda...

Andrés Arboleda Ap Precalculus Activities & Homework Andrés Arboleda Ap Precalculus  Andrés Arboleda Ap Precalculus How to use your TI-83 or TI-84 During the entire AP Precalculus course, your calculator should always be in radians. Press [Mode] and make sure that the third line has the word [Radians] highlighted. The AP Calculus exam only requires you to know how to do four things on your calculator: 1. Plot the graph of a function within an arbitrary viewing window 2. Find the zeros of functions and the point of intersection between two functions 3. Numerically calculate the derivative of a function 4. Numerically calculate the value of a definite integral This guide will show you how to do each one of these four things. 1. Plotting the graph of a function within an arbitrary viewing window Press [Y =], the button on the upper left corner to get to the equation screen. Type the desired function(s) in any location (Y1 is the most common). Press [Graph], the last button on the top. To adjust the viewing window: a. Adjusting the viewing window manually. Press [Window]. Now you can specify the minimum and maximum values for the x-axis and y-axis. You can also set the axis scaling increment between tics. b. Adjusting the viewing window automatically. Press [Zoom] and choose one of the following options: 2: Zoom In enables you to define the center point of the zoom in location. 3: Zoom Out enables you to define the center point of the zoom out location. 6: ZStandard automatically sets x-min, x-max, y-min, and y-max to center the origin. 0: ZoomFit. recalculates y-min and y-max to include the minimum and maximum y values of all functions between the current x-min and x-max. 2. Finding the zeros of functions (and points of intersection) a. Finding zeros of functions. To find the roots (or zeros) of a function, graph the function in an appropriate window. Press [2 nd ][TRACE][2: zero]. Move the cursor to an x value that is to the left of the x-intercept and press [Enter]. Move the cursor to an x value that is the right of the x-intercept and press [Enter]. Move the cursor as close as you can to the x-intercept (the point you choose must be between the two points you marked previously) and press [Enter]. The cursor will flash on the root and the coordinates (correct to 13 decimal places) will be displayed at the bottom of the screen. (Note: If you have several functions entered into the equation page, use the down arrow to select the proper function after pressing [2 nd ][TRACE][2: zero] and before choosing an x value to the left of the root). b. Finding points of intersection. There are two ways to find the values of x that will make f (x) = g(x): Move g(x) to the other side so that f (x) – g(x) = 0 and use the method explained in the previous point to find the zeroes. OR Graph both f (x) and g(x) as two separate functions in an appropriate window. Press [2 nd][TRACE][5: intersection]. Press [Enter] twice to select both graphs that intersect. Type in a reasonable value for the x-coordinate of the point of intersection and press [ENTER]. The coordinates of the point will be displayed at the bottom of the screen. 3. Numerically calculating the derivative of a function. From the Home screen press [MATH ][8: nDeriv]. Next, write the equation you want to differentiate, press [ ,], write the variable that you are differentiating, press [ , ], write the x value where you want the derivative, press [ ) ] and finally press [ENTER]. Example: nDeriv(-5x 2+ 6x – 1, x, 2). You should get an answer of -14. 4. Numerically calculating the value of a definite integral. From the Home screen press [MATH ][8: fnInt]. Next, write the equation you want to integrate, press [ ,], write the variable that you are integrating, press [ , ], write the lower limit of integration, press [ , ], write the upper limit of integration, press t[ ) ] and finally press [ENTER]. Example: fnInt(5x 2+ 6x – 1, x, 0, 2). You should get an answer of 23.3333333333.  Andrés Arboleda Ap Precalculus How to use your TI-Nspire CAS During the entire AP Precalculus course, you need to make sure that your calculator is always in radians and displays enough decimals. You need to change this in two different places: ★ From the Home screen, press [5 Settings], select [2:Document Settings], from the first option of “Display Digits” select [Float 8] to show 8 significant digits and from the second option “Angle” select [Radian]. Scroll down to the bottom of the screen and select [Make Default] and press [OK]. ★ From the Graph screen, press [Menu], select [8: Settings], from the first option of “Display Digits” select [Float 8] to show 8 significant digits and from the second option “Angle” select [Radian]. Scroll down to the bottom of the screen and select [Make Default] and press [OK]. The AP Precalculus exam only requires you to know how to do four things on your calculator: 1. Plot the graph of a function within an arbitrary viewing window 2. Find the zeros of functions and the point of intersection between two functions 3. Numerically calculate the derivative of a function 4. Numerically calculate the value of a definite integral This guide will show you how to do each one of these four things. 1) Plotting the graph of a function within an arbitrary viewing window Select [B Graph] from the Home screen. Type the desired equation after the f1(x) =. (The x is at the bottom of the calculator). Press [Enter]. To edit your function, press [Tab]. To graph additional functions press the [Tab], scroll down to f2(x)= and input each additional function. To adjust the viewing window, press [Menu][4: Window/Zoom]. Choose one of the following useful options: 1: Window Settings lets you specify the minimum and maximum values for the x-axis and y-axis. Use this option to select a very specific viewing window. You can also set the axis scaling increment between tics. 3: Zoom - In enables you to define the center point of the zoom in location. 4: Zoom - Out enables you to define the center point of the zoom out location. 5: Zoom - Standard automatically sets x-min, x-max, y-min, and y-max to center the origin. A: Zoom-Fit recalculates y-min and y-max to include the minimum and maximum y values of all functions between the current x-min and x-max. 2) Finding the zeros of functions (and points of intersection) a. Finding zeros of functions. To find the roots (or zeros) of a function, graph the function in an appropriate window. Press the [Menu][6: Analyze Graph][1: Zero]. Move the mouse and click to select a lower bound (an x value that is before the zero) and an upper bound (an x value that is after the zero). The coordinates of the x-intercept will appear in the lower right corner of the screen. b. Finding points of intersection. There are two ways to find the values of x that will make f (x) = g(x): ★ g points of intersection. There are two ways to find the values of x that will make f (x) = g(x): Move g(x) to the other side so that f (x) – g(x) = 0 and use the method explained in the previous point (2a) to find the zeroes. OR ★ Graph both f (x) and g(x) as two separate functions in an appropriate window. Press [Menu][6:Analize Graph][4:Intersection]. Move the mouse and click to select a lower bound (an x value that is before the point of intersection) and an upper bound (an x value that is after the point of intersection). The coordinates of the points of intersection will appear in the lower right corner of the screen. c. A note about the solver. I do NOT recommend using the solver in the Calculate part of the calculator because the algorithms it uses sometimes are unable to detect some of the solutions. 3) Numerically calculating the derivative of a function. Select [A Calculate] from the Home screen. Press [Menu] [4:Calculus][2:Derivative at a Point]. Input the variable you are using and the x value where you want to find the derivative. Input the function exactly as you would by hand. 4) Numerically calculating the value of a definite integral. Select [A Calculate] from the Home screen. Press [Menu] [4:Calculus][3:Integral]. Input the function, the limits of integration and the variable exactly as you would by hand.  Andrés Arboleda Ap Precalculus 1 Activity 0: Finding linear equations graphically and analytically In exercises 1 - 3, estimate the slope and the equation (slope - intercept form) of the line from its graph. 1. 2. 3. In exercises 4 and 5, determine the equation of the line (point - slope form) and sketch the graph of the lines through the given point with the indicated slope. Point Slopes 4) (2, 3) 𝑎) 𝑚 = 0 𝑏) 𝑚 = 1 5) (− 4, 1) 𝑎) 𝑚 = 3 𝑏) 𝑚 = 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 A line passes through the points, for every case find an equation for the line in all three forms for linear equations 6) (− 1, − 1) (3, 4) 7) (− 1, 3) (1, 5) 8) Find the intercepts with the axes of the following lines 1 a. 𝑥 + 2𝑦 = −5 b. 𝑦 = 3 𝑥 +2 c. 3𝑥 + 9𝑦 = 9 In exercises 9 - 12, determine if the lines 𝑙1 𝑎𝑛𝑑 𝑙2 passing through the pairs of points are parallel, perpendicular, or neither (graph 9 and 11 only). 9) 𝑙1 : (0, − 1), (5, 9) 𝑙2 : (0, 3), (4, 1) 10) 𝑙1 : (− 2, − 1), (1, 5) 𝑙2 : (1, 3), (5, − 5) 7 1 11) 𝑙1 : (3, 6), (− 6, 0) 𝑙2 : (0, − 1), (5, 3 ) 12) 𝑙1 : (4, 8), (− 4, 2) 𝑙2 : (3, − 5), (− 1, 3 ) Applications of linear functions 13) A dam is built to create a reservoir. The water level W in the reservoir is given by the equation 𝑊 = 4. 5 𝑡 + 28, where t is the number of years since the dam was constructed, and W is measured in feet. a) Sketch a graph of this situation, b) What do the slope and w - intercept represent? (expressed in words), c) state the domain and range of this situation. 14) a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) in terms of the height h (in kilometers). (Assume that the relationship between T and h is linear), b) Draw the graph of the linear equation, c) What is the temperature at a height of 2.5 km?, d) state the domain and range of this situation 1 Answers: 1) m = 1 2) m = - 1 3) m = 0 4) a) y - 3 = 0 or y= 3 b) y - 3 = x - 2 5) a) y - 1 = 3( x + 4) or y - 1 = 3x + 12 6) a) y + 1 = 5/4( x + 1) or y - 4 = 5/4(x - 3); y = 5/4 x + ¼, 5x - y = -1 7) y - 3 = x + 1 or y - 5 = x - 1, y = x + 4, x - y = -4 8) a) Xint : (-5, 0) Yint : (0, -5/2) b) Xint : (-6, 0) Yint : (0, 2) c) Xint : (3, 0) Yint : (0, 1) 9) Perpendicular 10) Neither 11) Parallel 12) Perpendicular 13) b) The slope is m = 4.5, it represents THE RATE OF CHANGE of water level with respect to time. This means that the water INCREASES 4.5 ft per year. The w - intercept is 28, and occurs when t = 0. So it represents the water level when the dam was constructed 14) a) because we are assuming a linear relationship between T and h, the equations must be of the form y = mh + b, b) T = -10h + 20 c) T = - 5 °C d) Domain: h > 0 Range T < 20  Andrés Arboleda Ap Precalculus 15) The water is drained out of a bathtub. The liters of water left in the bathtub as a function of time is shown in the graph to the right How long did it take to completely drain the bathtub? a. How long did it take to completely drain the bathtub? b. How much water was in the bathtub when it started to drain? c. Write a linear equation to model this situation and identify the domain and range. 16)2 A telephone company charges a base fee plus a constant fee per minute spent. The table below shows the total of minutes spent with the total cost of the phone plan contracted (in dollars). a) what is the fee per minute consumed?, b) What is the base fee before any minute is spent?, c) write a linear equation to model this situation and identify the domain and range. Minutes Spent 120 140 180 Total cost of the phone plan 174 188 216 2 15) a) all the water was gone when the water reach cero liters at 5 minutes, b) before any time elapsed (o minutes) The bathtub had 360 liters of water in it c) y = - 32x + 360 Domain: 0 < equal x < equal 5 minutes Range: 0 < equal y < equal 360 liters 16) a) m = $ 0.7 per minute (70 cents per minute to every spent) b) b = $ 90 is the base fee c) y = 0.7x + 90 Domain: x > 0, Range: y > 120  Andrés Arboleda Ap Precalculus Solving equations and inequalities 3 Solve for x the following equations and inequalities: 17) 18) 19) 20) 21) 5(𝑥 − 2) ≥ 3(2𝑥 + 6) 𝑥 𝑥 22) 2 − 3 >− 1 3𝑥+4 1+2𝑥 23) 2− 5 ≥ 4𝑥 − 3 𝑥+3 𝑥 3𝑥−1 24) 4 − 3 ≥1− 2 3(2−𝑥) 16 𝑥+1 25) 2 − 3 - 6 23) x e (inf-,23/59]; x < or equal to 23/59; x e (-inf, ,23/59]; x < or equal to 23/59 24) x e [9/17,+inf); x > or equal to 9/17 25) x e (-64/69,+inf); x >-64/69 26) x = -8 27) Area 01: 5n; Area 02: x (n + x)/2; Area 03: (x + 5) x; Total Area: x (n + x)/2 + 5n + (x + 5) x 28) a) A = (8x^2 + 14x +3) / 2; A = 441/2 cm^2 b) A = 4x^2 + 17x -27; A = 158 cm^2; c) A = 18x^2 - 4x; A = 430 cm^2  Andrés Arboleda Ap Precalculus 29) 4The following figure represents land available to the García family. If the expressions shown represent the lengths of their sides: a) write a polynomial expression that represents the perimeter of the land; b) write a polynomial expression that represents the area of the land, c) what would be the value of perimeter and area if x = 8 meters? 30) simplify the expressions: a) b) c) 4 29) Perimeter: 28x + 34; Perimeter: 258 meters; Area: 29x^2 + 83x + 57; Area: 2577 m^2 30) a) ⅖ a^3 - 7/6 a^2b + 11/9 ab^2 + 14/5 b^3 -12, b) 40x c) -x - 20  Andrés Arboleda Ap Precalculus 5 Unit 01: Activity 1.1: Change in tandem 1. 2. 3. 4. 5. a. On what intervals f is decreasing and why? 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? 5 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 6. 6 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. a. b. On what intervals is f concave up? 6 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 c. 7On 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. 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? 7 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 8 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. c. f. 8 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 4. 9Selected 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? 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? 9 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 10 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. 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. 10 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 13. 11After 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. 11 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. 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  Andrés Arboleda Ap Precalculus 12 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. 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. 12 1) D, 2) D, 3) B  Andrés Arboleda Ap Precalculus 4. 13 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. 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. 13 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 14 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. 14 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