Pre-Calculus Module 4

Pre-Calculus Module 4

4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational Functions and Their Asymptote (7 8) (RA) TEY A Graphic Organizer (9 16) Unit 10 Limits and Derivatives Average Rate of Change versus Instantaneous Rate of Change (17 0) Investigating Average Rate of Change (1 9) Slope of the Secant Line and Limits (0 ) Slopes of Curves (4 7)

Student Activity Rational Functions with Removable Discontinuities 1. a) Simplify the rational epression and state any values of where the epression is undefined. b) Using the simplified epression in part (a), predict the shape for the graph of the function f( ) =. Sketch your prediction. c) Graph the function f( ) = using the calculator window 10 10 and 10 y 10 Enter the function in your calculator as it is written in un-simplified form. Does the calculator graph appear the same as your graph in part (b)? d) Turn off the aes on your graphing calculator and graph the function again. What do you observe? Why does this happen? e) Draw an open circle on your graph in part (b) where the graph appears to have a hole.. a) Simplify the rational epression and state any values of where the epression is undefined. b) Sketch a graph for the function f( ) =. Do you think that the graph of the function will have a hole in it? Draw an open circle on the graph where you think the hole will occur. Turn off the aes on your graphing calculator and graph the function using a standard window. Compare the graph on the calculator to your graph in part (b). c) Use the trace or the table feature on your calculator to help you determine the value of where the function is discontinuous. Why is the function discontinuous at this value of? Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1

Student Activity d) Construct a table of values that shows values of f() when is close to zero. Use a graphing calculator to help you construct the table of values. < 0 f() > 0 f() 0.5 0.5 0.1 0.1 0.01 0.01 0.001 0.001 0.0005 0.0005 0.0001 0.0001 e) As approaches 0, what value does f() appear to be approaching?. a) Simplify the rational epression and state any values of where the epression is undefined. b) Sketch a graph for the function f( ) =. Do you think that the graph of the function will have a hole in it? Draw an open circle on the graph where you think the hole will occur. Turn off the aes on your graphing calculator and graph the function using a standard window. Compare the graph on the calculator to your graph in part (b). c) Construct a table of values that shows values of f() when is close to zero. Use a graphing calculator to help you construct the table of values. < 0 f() > 0 f() 0.5 0.5 0.1 0.1 0.01 0.01 0.001 0.001 Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 4

Student Activity d) As approaches 0, what value does f() appear to be approaching? e) What is the domain and range of f()? f) For what intervals of where f() is continuous? 4. a) Simplify the rational epression epression is undefined. + 4+ and state any values of where the + b) Sketch a graph for the function f( ) = discontinuous with an open circle. + 4+. Show any points where the graph is + c) Verify your graph in part (b) using a graphing calculator. d) Construct a table of values that shows values of f() when is close to -. Use a graphing calculator to help you construct the table of values, or use the simplified epression for f() you found in part (a). < - f() > - f() -.1 -.9 -.01 -.99 -.001 -.999 -.0001 -.9999 -.00001 -.99999 e) As gets closer to -, what value does f() appear to be approaching? f) What is the domain and the range of f()? g) For what intervals of is f() continuous? Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 5

Student Activity Note: This type of discontinuity is called removable because we could remove the discontinuity by redefining the function at just one number. For eample: The function f( ) = + 4+ is discontinuous at. + WINDOW min = 4 ma = scl = 1 ymin = 4 yma = yscl = 1 Y 1 X -4 - - -1 0 1-1 - - -4 By redefining f() at the number =, g() is a continuous function. The function + + 4 + if g ( ) = if = is continuous. Note: g() = + 1 for all. WINDOW min = 4 ma = scl = 1 ymin = 4 yma = yscl = 1 Y 1 X -4 - - -1 0 1-1 - - -4 Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 6 4

Student Activity 5. Match each graph to the correct function. 4 f ( ) = g ( ) = h ( ) = + a) y b) y c) 4 + y For questions 6 8, follow the instructions listed below. a) Factor the numerator and denominator of the function and state the domain. b) Reduce any common factors and state any value(s) of where the function is undefined. c) State the range of the function. d) Graph the function using paper and pencil. Draw an open circle and label any removable discontinuities on the graph. e) Use your graphing calculator to check your answers. 6. f( ) = 10 + 9 4 7. f( ) = + 4 4 + 1 8. f( ) = 6 19 + 10 Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 7 5

Student Activity End Behavior of Rational Functions 1. Determine the horizontal asymptote of y = 4 by evaluating the limit as approaches + ± infinity. Does the function intersect its horizontal asymptote? If so, at what point(s)? 4. Determine the horizontal asymptote of y = by evaluating the limit as approaches + ± infinity. Does the function intersect its horizontal asymptote? If so, at what point(s)? +. Determine the horizontal asymptote of y = by evaluating the limit as approaches + 10 ± infinity. Does the function intersect its horizontal asymptote? If so, at what point(s)? + 4. The rational function, y =, has an oblique asymptote. Use long division to divide the + 10 numerator by the denominator. The line, y = quotient, is the oblique asymptote. Does the rational function intersect its oblique asymptote? If so, at what point(s)? + 5 6 5. Determine the oblique asymptote of y = 0 oblique asymptote? If so, at what point(s)?. Does the function intersect its 6. What is the rule that determines when a rational function has a horizontal asymptote and when it has an oblique asymptote? 7. When the degree of the numerator is two or more higher than the degree of the denominator, the end behavior of a rational function is modeled by a polynomial. Use long division to divide the numerator by the denominator. The polynomial is y = quotient. What is the polynomial that + 5 6 models the end behavior of the curve y = as approaches infinity? Will the 1 graph of the rational function intersect this polynomial? If so, at what point(s)? Copyright 008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 6

Mathematics Rational Functions and Their Asymptotes Part 1 1 1. Given the function y 1 a. What are the horizontal asymptote(s)? b. What are the vertical asymptote(s)? c. What are the - and y-intercepts? d. Graph the function. e. What is the maimum value of the function? Eplain how you found this value.. Do all rational functions have vertical asymptotes? Eplain your answer.. Given the function f() = a. What are the horizontal asymptote(s)? b. What are the vertical asymptote(s)? c. What are the - and y-intercepts? d. Determine f ( 1.5). What is interesting about this value? e. Complete the table of values. 0 0.5 1 1.5.5 4 5 6 100 f () f. Use your graphing calculator to confirm your findings. 4. Can the graph of a rational function cross its horizontal asymptote? Eplain your answer. 5. How can you determine if and where a function crosses its horizontal asymptote? Copyright 01 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 1 7

Student Activity Rational Functions and Their Asymptotes Part For questions 6 11, complete parts a f. a. What is the equation of each vertical asymptote? b. What is the equation of each horizontal asymptote? c. Determine all -intercepts. d. Determine the y-intercept. e. Does the function cross its horizontal asymptote? If so, where? Show the work that leads to your answer(s). f. Graph each function. 6. y 8 7. y 1 1 8. y 1 1 9. y 7 10. 6 9 y 4 11. y 1 Copyright 01 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 8

NATIONAL MATH + SCIENCE INITIATIVE Mathematics RATEY A Graphing Organizer RATEY is an organizing tool for graphing rational functions. To use this tool effectively, the numerator and the denominator should be factored completely over the real numbers before beginning the graph. RATEY is an acronym for the following: R is for Roots (zeros) and Removable discontinuities. Factor the numerator and denominator. Identify common factors and label them as removable discontinuities (holes in the graph). Simplify the epression, set the numerator equal to 0, and solve. Graph these point(s) on the -ais. A is for Asymptotes. (Vertical only) Using the simplified epression, set the denominator equal to 0 and solve. Sketch the vertical asymptotes as dotted lines. T is for Two and means Two things. Do any of the factors of R and A have an even eponent (a multiple of ) such as ( 1) or ( 1) 4? In the numerator, this means T angency at the root. In the denominator, this means T ogetherness about the asymptote. E is for End Behavior. Determine the limit of the function that models the end behavior as ±. Compare the degree of the numerator and the denominator. If the numerator and denominator have the same degree, divide the coefficient of the largest degree term in the numerator by the coefficient of the largest degree term in the denominator to determine the horizontal asymptote. If the degree of the denominator is larger than the degree of the numerator, the end behavior asymptote is y = 0. If the degree of the numerator is larger than the degree of the denominator, perform long division to determine the oblique asymptote. Sketch the asymptotes for the end behavior as dotted lines. Y is for Y-Intercept. Let equal zero and solve for y. Graph this point on the y-ais. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 9 1

Mathematics RATEY A Graphing Organizer For questions 1-8, identify the RATEY characteristics and sketch the graph: 1. f () = 4 +. f () = ( +1)( ) +. f () = + 6 ( +1) Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 10

Mathematics RATEY A Graphing Organizer 4. f () = ( 1) ( +1) 5. f () = ( 1) ( +5) 6. f () = 9 4 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 11

Mathematics RATEY A Graphing Organizer 7. f () = ( + 4 +8)( 4) 4( 4) 8. f () = 9 9. Describe the similarities and differences between the graphs of f () = 6 6 and g() = 6 6. 4 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 1

Mathematics RATEY A Graphing Organizer 10. Write a possible equation for each of the functions in these graphs. Compare a calculator s graph of your equation to the given graph. a. f 1 1 1 b. f 1 1 c. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 1 5

Mathematics RATEY A Graphing Organizer 11. Given f () = a + b b a : a. What is the root of f ()? b. What is the equation of the function s vertical asymptote? c. What is the end behavior of the function? d. What is its y-intercept? e. If a > b > 0, sketch f (). f. On what intervals does f () appear to be increasing? g. On what intervals does f () appear to be concave up? 6 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 14

Mathematics RATEY A Graphing Organizer Use a graphing calculator to assist in answering questions 1 16. For questions 15 16, set the calculator to radian mode. 1. If f () = and g() =, describe how the graph of the composition g( f ()) = + + differs from the graph of f (). Support your answer with mathematical reasoning. 1. Let f () = 4 4 +1. Does this function eist for all real numbers? If not, where does it fail to eist? Let g() = 4 4. Does the square root affect the domain of this function compared to the domain +1 of f ()? If so, how is it affected? State the domain of g() and support your answer with mathematical reasoning. 14. Why are the domains of f () = ln() and g() = ln( ) different? Sketch the graphs of f and g. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 15 7

Mathematics RATEY A Graphing Organizer 15. Determine the end behavior asymptote of f () = sin() crosses its horizontal asymptote infinitely many times. Eplain why this is true. and state the domain of f (). The function f 16. The functions f () = sin() and g() = cos() are both discontinuous when the value of is 0. Determine which has an asymptote and which has a hole and describe the similarities and differences of the graphs of f and g. Sketch each function on the aes provided. 8 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 16

Mathematics 1. Beginning Average Rate of Change versus Instantaneous Rate of Change 1 miles from home, Jonathan drove away from home at a constant rate for 0 minutes. If his constant rate is 5 miles per hour, how far is he from home at the end of the 0 minutes? Draw a graph to model his distance from home during the 0 minute time period. 0 d in miles 18 16 14 1 10 8 6 4 0 4 6 8 10 1 14 16 18 0 t in minutes. Susan, Jonathan s sister, also drove away from home beginning 1 miles from home and following the same path as Jonathan. Susan kept varying her velocity by frequently speeding up and slowing down. She arrived at the same location as Jonathan at the end of 0 minutes. To model Susan s distance from home during the 0 minute time period, draw a smooth curve without any sharp corners. 0 18 16 14 1 10 8 6 4 d in miles 0 4 6 8 10 1 14 16 18 0 t in minutes Copyright 01 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 1 17

Student Activity Average Rate of Change versus Instantaneous Rate of Change. Calculate the average velocity for both drivers by calculating the change in position divided by the change in time. These two calculations have the same value; eplain why this makes sense. Compare your answers to the rate given in question 1. 4. On the graph showing Susan s position, draw the line segment connecting the point at t = 0 and the point at t = 0. What is the slope of this line segment and what are the units for the slope? How does this slope compare to the slope of line that modeled Jonathan s distance from home? 5. If the speed limit over the entire path is 5 miles per hour, did Susan ever drive over the speed limit? Eplain your answer by referring to Susan s graph. 6. For non-linear position functions, the eact velocity at a particular time, called instantaneous rate of change or instantaneous velocity, cannot be calculated precisely without the tools of calculus. However, the velocity can be estimated by approimating the slope of a short line segment drawn tangent to the curve at the particular time. On the graph showing Susan s distance, locate at least one time when Susan s instantaneous velocity has the same value as the average velocity. (Position a straightedge on the graph so that it is parallel to the line segment drawn on the curve in question 4. Move the straightedge around on the graph keeping the slope of the straightedge fied. When the straightedge appears to be tangent to the curve, mark the point(s) and sketch a short segment tangent to the curve. At these point(s), the instantaneous velocity is the same as the average velocity.) Copyright 01 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 18

Student Activity Average Rate of Change versus Instantaneous Rate of Change 7. Using the function g() shown in the graph, draw a small tangent line segment at each labeled point. A small tangent line is drawn at point P as a sample. There is not enough information to draw the segments perfectly, so sketches may vary slightly. Match the slope at each labeled point on the curve with an approimate rate of change value in the table. The slope at each point is called the instantaneous rate of change at a point because it is the rate of change at that one instant in time. Hint: The slope may be the same at different places along the graph. 6 Rate of Change at Point Letter 5 5 E P L 4 B 0 H 1 4 W C J -4 - - -1 0 1 4 P -1 15 - Copyright 01 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 19

Student Activity Average Rate of Change versus Instantaneous Rate of Change 8. The graph represents the position (t) in inches of an object that is moving along a line etending perpendicularly from a wall at a given time, t, measured in seconds. The distance between the object and the wall is indicated on the vertical ais, while time is measured on the horizontal ais. Points t (t) B 0.1 7.5 G 0.45 0 Z 0.87 V 1. 1 D.0 19 K.0 0 A.5 C 4.1 70 a. What do the coordinates of B (0.1, 7.5) and D (, 19) represent in the contet of this situation? b. Mark small tangent line segments on each of the points that are named. Using these tangent segments, for which of the point(s) is the instantaneous rate of change negative? What do you know about the motion of the object if the instantaneous rate of change is negative? c. Observing the tangent segments, over which time intervals is the object moving away from the wall? What do the slopes of these line segments mean in the contet of the position function? d. At which point(s) has the object stopped moving? Describe the slope of the tangent line(s) at the point(s). e. Speed indicates how fast an object is moving without regard to direction. Order the speeds at the following points from least to greatest: D, G, Z. Eplain your reasoning. Copyright 01 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at www.ltftraining.org. 4 0

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Investigating Average Rate of Change y f( b) f( a) For y = f( ) on the interval [ a, b ], the average rate of change is =. This quotient is the b a slope of the secant line. In other words, this is the slope calculated between two points on the function f(). The instantaneous rate of change, the slope of the tangent line at one point, will be eplored in this lesson. 1. f( ) = + 1 y a. Calculate the average rate of change,, of the function over each of the given intervals. i. [ 5, 1] ii. [, 8] iii. Choose any different interval. iv. [0.9, 1] v. [0.999, 1] b. What is the instantaneous rate of change at = 1?. f( ) = + y a. Calculate the average rate of change,, of the function over each of the given intervals. i. [ 6, ] ii. [, 9] iii. Choose any different interval. iv. [0.9, 1] v. [0.999, 1] b. What is the instantaneous rate of change at = 1?. a. Eplain why the answers in question 1 are the same and why the answers in question are the same. b. Describe an easy method for determining the average rate of change of a linear function. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 1 1

Mathematics Investigating Average Rate of Change 4. f ( ) = + 1 a. Sketch the function by carefully plotting the points at integer values of. b. Draw a secant line for each of the following intervals and graphically determine the average rate of change of the function (slope of the secant line) over each interval. i. [, 1] ii. [ 1, 0] iii. [0, ] c. Are the secant lines in part (b) parallel? Do the secant lines in part (b) have the same slope? d. Based on the answers for part (b), are the average rates of change for a quadratic function constant? Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org.

Mathematics Investigating Average Rate of Change e. Using a colored pencil, draw a secant line for each interval given. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and determine the -coordinate of the midpoint of each segment. Record your information in the table provided in part (i). i. [ 1, ] ii. [0, 1] f. Are the secant lines in part (e) parallel? Do the secant lines in part (e) have the same slope? g. Using a colored pencil, draw a secant line for each of the following intervals. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and determine the -coordinate of the midpoint of each segment. Record your information in the table in part (i). i. [0.4, 0.6] ii. [0.49, 0.51] iii. [0.499, 0.501] h. Are the secant lines in part (g) parallel? Do the secant lines in part (g) have the same slope? Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org.

Mathematics Investigating Average Rate of Change i. Complete the table including your information from parts (e) and (g). First Point Second Point Δy Δ ( 1, ) (, ) (0, ) (1, ) (0.4, ) (0.6, ) (0.49, ) (0.51, ) (0.499, ) (0.501, ) Δy Δ -coordinate of the midpoint of the segment j. Do the coordinates in the table seem to approach a certain point? What is that point? k. Estimate the instantaneous rate of change (slope of the tangent line) at = 0.5. l. At what specific point of f( ) on [ 1, ] is the instantaneous rate of change of the function equal to the average rate of change of the function on the interval [ 1, ]? For what other intervals given in this question is this same relationship also true? m. Using your estimate for the instantaneous rate of change at = 0.5 found in part (k), write the equation of the tangent line through the point. Using a colored pencil, draw this line on your graph. 4 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 4

Mathematics Investigating Average Rate of Change 5. f 1 4 ( ) = ( + ) + 6 a. Sketch the function by carefully plotting the points at integer values of. b. Using a colored pencil, draw a secant line for each given interval. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and calculate the -coordinate of the midpoint of each segment. Record your information in the table in part (f). i. [, 5] ii. [, 4] iii. [0, ] c. Are the secant lines in part (b) parallel? Do the secant lines in part (b) have the same slope? Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 5 5

Mathematics Investigating Average Rate of Change d. Using a colored pencil, draw a secant line for each given interval, calculate the average rate of change of the function (slope of the secant line) over each interval, and record your answers in the table in part (f). i. [0.9, 1.1] ii. [0.99, 1.01] iii. [0.999, 1.001] e. Are the secant lines in part (d) parallel? Do the secant lines in part (d) have the same slope? f. Complete the table to include your information from parts (b) and (d). First Point Second Point Δy Δ (, ) (5, ) (, ) (4, ) (0, ) (, ) (0.9, ) (1.1, ) (0.99, ) (1.01, ) (0.999, ) (1.001, ) Δy Δ -coordinate of the midpoint of the segment g. Do the coordinates in the table seem to approach a certain point? What is that point? h. Estimate the instantaneous rate of change (slope of the tangent line) at = 1. 6 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 6

Mathematics Investigating Average Rate of Change i. At what specific point on [, 5] is the instantaneous rate of change of the function equal to the average rate of change of the function? j. Using your estimate for the instantaneous rate of change at = 1 found in part (h), write the equation of the tangent line through the point. Using a colored pencil, draw this line on your graph. 6. Fill in the blanks for each statement using the choices provided. Note: Some choices may be used more than once and some may not be used at all. constant different endpoint length midpoint slope the same zero a. The average rate of change between two points of a function is the of the secant line. b. Since the slope of a linear function is, the average rate of change is. c. For a constant function, the y-coordinate is for every pair of points selected, so the average rate of change always has a value equal to. d. The average rate of change for a quadratic function is not for every pair of points selected. e. For a quadratic function, the -value of the point where the average rate of change over a given interval equals the instantaneous rate of change of that interval is the of the interval. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 7 7

Mathematics Investigating Average Rate of Change 7. A car company is testing the speed and acceleration of one of its new sports cars. The table shows the distance the car travels when it accelerates from a standstill. Use a graphing calculator to answer the following questions. Elapsed time in seconds (t) Distance in meters (d) 0 0 10 5 14 75 18 550 0 650 a. Eplain why this data is not linear and justify your answer mathematically using the slopes of a pair of secant lines. d b. In the contet of the problem, what does t represent? What is the average rate of change, d t, on the interval [0, 0]? Indicate appropriate units of measure. c. Determine the quadratic regression function, Rt (), for the data and superimpose its graph on a scatterplot of the data. Copy the graph and the data from your calculator onto the grid provided. Quadratic Regression Equation 8 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 8

Mathematics Investigating Average Rate of Change d. What is R (0)? Eplain the meaning of this value in terms of the problem situation, and eplain why this value is different from the value in the table. e. According to R(t), what is the average rate of change over the 0-second time interval from 0 seconds to 0 seconds? Convert the answer to the nearest whole number in miles per hour and eplain its meaning in terms of the problem situation. (1 km = 0.614 miles) f. Since the regression function is quadratic, where should the average rate of change be equal to the instantaneous rate of change for the interval [0, 0]? g. The car company claims the car can accelerate from 0 to 60 mph in 6 seconds. This means that the instantaneous rate of change at 6 seconds must be 60 mph. Prove or disprove this claim by eamining the average rate of change over an interval for which 6 seconds is the midpoint. miles meters Note: 60 6.81. hour second Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 9 9

Student Activity Slope of the Secant Line and Limits For f() =, find the slope of the line containing the following points. 1. (, f()) and (, f()). (, f()) and (1, f(1)). ( + 1, f( + 1)) and ( 1, f( 1)) 4. (, f()) and ( + h, f( + h)) 5. (, f()) and ( +, f( + )) Find the following limits. Fill in answers in table. 6. 7. 8. lim lim 1 9 1 1 ( + 1) lim 1 ( 1) Problem answer 6 lim as 7 lim as 1 8 lim as 1 9 lim as h 0 9. ( + h) lim h 0 h Can you generalize about the slope of the secant line connecting two very, very, very close points on? Use the generalization to answer question 10. 10. ( + a) lim a 0 a Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 0

Student Activity The following points lie on f() =. Find the slope of the line containing the two points. Use the table function on the graphing calculator. For problems 1 through, let y1 = ( +.001) y = y = (y1 y)/.001 11. (, 9) and (.001, f(.001)) 1. (1, 1) and (1.001, f(1.001)) 1. (0, 0) and (0.001, f(0.001)) 14. (, f()) and ( +.001, f( +.001)) For 15 through 18, let y1 = ( +.0001) and y = (y1 y)/.0001 15. (, 9) and (.0001, f(.0001)) 16. (1, 1) and (1.0001, f(1.0001)) 17. (0, 0) and (0.0001, f(0.0001)) 18. (, f()) and ( +.001, f( +.001)) For 19 through, let y1 = ( +.00001) and y = (y1 y)/.00001 19. (, 9) and (.00001, f(.00001)) 0. (1, 1) and (1.00001, f(1.00001)) 1. (0, 0) and (0.00001, f(0.00001)). (, f()) and ( +.00001, f( +.00001)) Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1

Student Activity. Fill in the table below: sin( +.001) sin cos.001 0 π 6 π 4 π π What conclusion can you make from the table above? Use the given values and the function to answer the following. 4. f() = + 1 for = and =.001. a) Find the slope of the secant line. b) Draw a picture of the function and the secant line. c) Find the equation of the secant line. 5. f() = 1 + 1 for = 1 and = 1.001. a) Find the slope of the secant line. b) Draw a picture of the function and the secant line. c) Find the equation of the secant line. Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 4

Student Activity 6. f() = + for = and =.001. a) Find the slope of the secant line. b) Draw a picture of the function and the secant line. c) Find the equation of the secant line. 7. f() = e for = 1 and = 1.001. a) Find the slope of the secant line. b) Draw a picture of the function and the secant line. c) Find the equation of the secant line. 8. f() = cos for = 1.56 and = 1.57. a) Find the slope of the secant line. b) Draw a picture of the function and the secant line. c) Find the equation of the secant line. Copyright 009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 5

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Slopes of Curves 4 1. For y = 8+ 1, the slope function m for any point on the curve is given by a. Determine the points where the slope is zero. Show the algebraic steps. m = 4 4 8. b. Write the equation(s) of the horizontal tangent line(s) to the curve at the point(s). c. Determine if the curve has a maimum or minimum value or neither at the point(s). Eplain your answer using the values of m.. For y + 4y = 10, the slope function m for any point on the curve is given by a. Is this slope ever zero? Show why or why not. 1 m = y +. b. What does part (a) indicate is true about the graph? c. When would the slope be undefined? Show the process used to determine the answer. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 4 1

Mathematics Slopes of Curves d. What is when y =? e. Describe the tangent line to the curve where the slope is undefined. f. When would the slope be positive? Describe the curve when the slope is positive. g. On a TI-84, solve the original equation for in terms of y and then graph the curve using parametric equations. Let (t) = equation for in terms of t instead of y, and let yt () On the TI-Nspire, solve the original equation for in terms of y and then use the scratchpad to graph = f( y) List the equations used and sketch the graph. = t. h. Verify your answers to part (f) by eamining the graph.. Consider the curve y + 6y 1 + 6y = 1. The slope function m of the curve at all points is given by 4 y m = y + + 1. 1 1 a. Determine the slope of the curve at the point,. b. Is the curve increasing or decreasing at the point 1 1,? Eplain. Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 5

Mathematics Slopes of Curves c. Determine the values of and y where m = 0. d. Using your answers to part (c), determine the point(s) on the curve where the slope is zero. e. Write the equations of the horizontal tangent lines to the curve. f. On the TI-84, solve the equation of the curve for in terms of y then use parametric equations to graph the curve. Let t () = equation for in terms of t instead of y, and let yt () = t. On the TI-Nspire, solve the equation of the curve for in terms of y and then use the scratchpad to graph = f( y) List the equations used and sketch the graph. g. What is the horizontal asymptote of the curve? Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 6

Mathematics Slopes of Curves 4. Consider the curve given by y y y is m = y. y = 6. The slope function m of the curve at any point a. Determine all points on the curve whose -coordinate is 1. b. Write an equation for the tangent line at each of the points where the -coordinate is 1. c. Determine the -coordinate of each point on the curve where the tangent line is vertical. d. Use the quadratic formula to solve for y in terms of. e. On the TI-84, use parametric equations to graph the curve. On the TI-Nspire, use the scratchpad to graph the curve, using the equations from part (d). List the equations used and sketch the graph. 4 Copyright 014 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at www.nms.org. 7