Introducing Instantaneous Rate of Change

Transcription

1 Introducing Instantaneous Rate of Change The diagram shows a door with an automatic closer. At time t = 0 seconds someone pushes the door. It swings open, slows down, stops, starts closing, then closes quietl at time t = 7 seconds. As the door is in motion the number of degrees, d, it is from its closed position depends on t. Sketch a reasonable graph of d versus t. Compare our graph with a graph drawn b a neighbor. What characteristics are possessed b both graphs. How do our graphs differ? Suppose that d is given b the equation d = 00t ( -t ). Use our calculator to graph this function for 0#t#7. What common characteristics does this graph have to our graph? Does our range make sense?

2 Make a table of values of d for each second from t = 0 through t = 0. Round to the nearest 0.0 degree. Time Degrees Looking at onl the chart, does the door appear to be opening, closing, or standing still at t = second? Eplain our reasoning. Using the graph, does the door appear to be opening, closing, or standing still at t = second? Eplain our reasoning. During the time interval [0, ] seconds, what is the average rate of change of the door s angle? Create a table of values for the angle of the door s opening over the interval [0, ] seconds with steps of 0., what is the average rate of change of the angle of door. Time Degrees Time Degrees Rahn 00 Material developed from Paul Forester s Calculus

3 During the time interval [0.9, ] seconds, what is the average rate of change of the door s angle? How does this average rate of change compare with the average rate of change on the interval [0,]? Does this make sense? Are ou surprised b the amount of difference? Eplain our reasoning. Create a table of values for the angle of the door s opening over the interval [0.9, ] seconds with steps of 0.0, what is the average rate of change of the angle of door. Time Degrees During the time interval [0.99, ] seconds, what is the average rate of change of the door s angle? How does this average rate of change compare with the average rate of change on the interval [0.9,]? Does this make sense? Are ou surprised b the amount of difference? Eplain our reasoning. Can ou predict the rate of change of the opening of the door s angle at time t =? Rahn 00 Material developed from Paul Forester s Calculus

4 How could ou get a better approimation? Continue until ou can predict the door s rate of change of the angle to a hundredths of a second. Describe in our own words how ou can find the instantaneous rate of change of the door at eactl time second. Rahn 00 Material developed from Paul Forester s Calculus

5 Graphs of Functions For each of the following functions: Draw a sketch of the each graph without using the graphing calculator. Check our answer b using the graphing calculator. C Create a set of table values for each function in the neighborhood of : 0.8 # #.. C Describe whether the function is increasing, decreasing, or not changing when =. C If the function is increasing or decreasing at =, use the graph and/or table values to decide if the function is increasing or decreasing at a rate that is slow or fast sin Rahn 00 Material developed from Paul Forester s Calculus

6 sin( ) Rahn 00 Material developed from Paul Forester s Calculus

7 Learning to Work with Area As ou pull out on the highwa on our biccle and graduall increase our speed according the graph below. Then ou notice our speedometer approaching 65 ft per minute so ou tap hand brake to slow down our speed to a constant rate of 65 feet per minute. 00 Velocit (feet per minute) time (minutes) How far do ou travel between time t=60 minutes and t=00 minutes? Eplain how ou found this answer. How is this distance represented in the graph? Shade this region in on the graph. How man feet are represented b one shaded in square. Eplain our reasoning. If a square was shaded in as illustrated at the right, to the nearest 0., what portion of the square is shaded? How man feet would be represented b this shaded in portion? Find the distance traveled from t=0 to t = 60 minutes Find out how far our biccle traveled in the first 00 minutes of the trip. The distance traveled b our biccle is represented b the bounded area under the velocit graph and above the time ais. This area is called the definite integral of the velocit from time t=0 to t= 00 minutes. Rahn Sept. 0 Material developed from Paul Forester s Calculus

8 You have just found a geometric method to calculate a definite integral of the velocit from t=0 to t= 60 minutes. A right circular cone is placed on its circular base. The cone is sliced parallel to the base, at points distance from the base, to create cross sections that are circles. The area of each circular cross section is graphed at the below as a function of its distance from the base. Find the volume of the right circular cone b determining the area bounded b the graph and the ais. Area (sq. cm.) Centimeters What is the largest area of a cross sectional area? What is the area of the circle created.75 cm from the base? At what distance from the base was a square centimeter circle cut? What are the units for the definite integral of the area with respect to in this problem? What does the definite integral represent in this problem? Estimate the definite integral of, the area, with respect to for 0##5. Show work that leads to our answer. Rahn Sept. 0 Material developed from Paul Forester s Calculus

9 Finding Definite Integrals Another Wa The velocit of a particle, in centimeters per second, is represented b the graph at the right. You are interested in determining how far the particle has traveled during the time interval 0 seconds # t # 5 seconds. Subdivide the interval 0#t#5. into 5 equal widths. Draw trapezoids in each interval b connecting two consecutive function values. Find the height of the velocit graph at the endpoints of each interval. Record them in the given table. Find an approimation for the definite integral of v(t) with respect to t b finding the sum of the area of the five trapezoids. vt t t () What does this definite integral represent? t 0 5 v(t) Increase the number of intervals to 0. Find a second approimation for the definite integral of v(t) with respect to t b finding the sum of the area of the ten trapezoids. t v(t) What do ou notice about our two answers for the definite integral of this velocit from 0#t#5. How could ou improve our answer for the definite integral of the velocit from 0#t#5? Can ou approimate the particle s rate of change of velocit when t=5? Was the particle speeding up or slowing down at that time. Eplain our reasoning. At what time is the particle s rate of change in velocit stopped. Eplain our reasoning. Rahn 0 Material developed from Paul Forester s Calculus

10 Introduction to Limits 8 5 Create a graph of. Did ou epect this graph? Wh or wh not? Set our window so 0.7. Did this change our graph? Insert = into the equation. What is the value of? Eplain the value ou see. Is this graph just jumping over one point or man points? Use the trace ke and eplain what ou are seeing. Build a set of table values in the neighborhood around. X Y Are the table values jumping over just one value or man values? When between 0.9 and., between what two values do the values sta between? Let s move in a little closer on. Build a set of table values between 0.96 # #.0. X Y When 0.96 # #.0, between what two values do all the values sta? Let s move in a little closer on. Build a set of table values between # #.00. X Y Rahn 00 Material developed from Paul Forester s Calculus

11 When # #.00, between what two values do all the values sta? What value to ou believe the function is jumping over when ou are close to =? 8 5 We call this the limit of the function when is approaching. Stud our last set of table values. Tr to complete this statement: "If is within 0.00 units of (but not equal to ), then f() is within units of, Rahn 00 Material developed from Paul Forester s Calculus