Who invented Calculus Newton or Leibniz? Join me in this discussion on Sept. 5, 2017.

Transcription

1 Who invented Calculus Newton or Leibniz? Join me in this discussion on Sept. 5, 07. Sir Isaac Newton idology.wordpress.com Gottfried Wilhelm Leibniz et.fh-koeln.de Welcome to BC Calculus. I hope that you will enjoy the class as much as I do! I would like you to know a little bit about the course before we head off to enjoy our summers. Calculus is about how things change. That means that it can be applied to just about everything economics, medicine, engineering you name it. It is a highly creative endeavor, and it takes determination and commitment to the work to become proficient. Familiarity with basic functions in all their representations, computational accuracy and efficiency (AP tests are timed!), and use of your graphing calculator are essential to success. Pick up a TI-89 clinic packet from Laurie if you plan to buy a new TI-89 this summer. Mark Dickson prepared this wonderful self-paced clinic that will get you up-and-running in no time! This course is a rigorous college course. The pace of the course is such that four days per week is not enough class time to present the entire curriculum. Therefore, 8:0 classes called for Wednesday mornings are mandatory. I will notify you well in advance so that you can be here! You should expect to spend, on average, one hour each night on homework, and you will need to set aside additional time before tests. Tests will be cumulative. I suggest that you begin work on the attached summer assignments two or three weeks before school starts. That way, they will be fresh when you return to school. I recommend that you work with friends. You will need a study group throughout the year why not start now? Help each other with the mathematics, but please make sure that you understand everything that you put on your paper! And do your own written explanations. I will hold one help session the week before school begins. I will post the details on the HBW homepage once I have my schedule. You are also welcome to me with questions over the summer. I will be checking school periodically. The summer assignments explore limits, applications of the derivative, an approximation of a definite integral, and series. Please do your work on separate paper for each assignment and write neatly. Be thorough in your explanations, and show all work. The summer assignments will be graded. They are due the Friday of the first week of school, and there will be no late credit no exceptions. My hope is that you will enjoy this preview of the course. Please feel free to contact me with any questions or concerns. I wish you an exciting, yet restful summer, and I look forward to our class next year. Sincerely, Kris Kappmeyer kristin.kappmeyer@apsva.us

2 0 0 Summer Assignment #: Limits & L Hôpital s Rule You have studied limits and how they apply to the definition of the derivative of a function. (They also apply to the definition of an integral, as we will soon learn in BC Calculus.) Derivative: f '( a) = lim h 0 f ( a + h) h f ( a). In a few sentences, explain what the above expression conveys. Explain in such a way that a non-calculus student would understand. You can use a graph or table to support your case. Limits Numerically x 3 9x + 6x 4. Graph on your calculator and sketch on paper: y = x 4 You should see a parabola with a hole. Make sure your sketch reveals that. 3. What value is f approaching as x approaches 4? Use your calculator s table with tiny increments. Limits Algebraically Let s now evaluate that same limit more formally: x lim 3 9x + 6x 4 x 4 x 4 4. First, try direct substitution. Did you get 00? That s an indeterminate form. ( x 5x + 6)( x 4) 5. Factor the numerator (synthetic division works best). Did you get: lim? x 4 x 4 Now cancel & re-substitute x = 4. Does the result match your earlier work? Great! L Hôpital s Rule Using algebra to evaluate limits is fine, but what happens when quotients don t factor? The French mathematician L Hôpital (66-704) comes to our rescue. Here s his method:! First try direct substitution. If that works, then you re done.! If the result is indeterminate ( or ), then differentiate the numerator and the denominator and then re-substitute. There are other indeterminate forms,, 0 0, 0, and that we will explore later in the year. 3 x 9x + 6x 4 6. Try those steps with the quotient you just explored: lim x 4 x 4 First differentiate the numerator & denominator separately. There s no quotient rule here. Then re-substitute x = 4. Did it work? Terrific! sin5x 7. Try this example, a quotient that doesn t factor: lim x 0 3x First explore the limit graphically by placing the quotient in y. Do you see the hole at x = 0? Next, look at the calculator s table and predict the limit as x approaches zero. Finally, apply L Hôpital s Rule. Do all three results match? Great!

3 0 0 Proof To see why L Hôpital s Rule works, consider this general expression: f ( x) lim where f ( c) = 0, g( c) = 0 x c g ( x ) 8. Write the point-slope form of equation for a line tangent to f at x = c. Do the same for g. f ( c) + f '( c)( x c) You should get: lim x c g( c) + g '( c)( x c) 9. Simplify that quotient, remembering that f ( c) = 0, g( c) = 0 f '( c) Did you arrive at lim? That s precisely what L Hôpital says to do! x c g '( c) (Actually, L Hopital doesn t deserve full credit. He published a colleague s work!) Extend the Idea 0. Sometimes, after applying L Hôpital s Rule, you still get an indeterminate form ( What should you do? Apply his rule again! Try this example: e x x lim x 0 x Did you arrive at? Well done. or ).. L Hôpital s Rule also works as x approaches infinity (although using growth rates is easier). Evaluate these limits: 3x 3x + 5 lim = x 8x 7x + 9 Fatal Flaw x 3 lim x e x =. It is important that you do not use L Hôpital s rule unless you get an indeterminate form! x 4 Evaluate lim and then try to apply L Hôpital s Rule. Why don t you get the same answer? x x + Summary of How to Evaluate Limits! When evaluating limits algebraically on equations of functions: " First try direct substitution. 0 " If that yields 0 or, either factor or apply L Hôpital s rule, then re-substitute. " If x, apply growth rates, from slowest to fastest: logarithmic, power, exponential! When evaluating limits on graphs of functions: " At any point on the graph, consider the limit from the left and from the right. " As x, look for the horizontal asymptote.

4 Summer Assignment #: Anuraag s Pendulum This BC summer assignment is Anuraag s AP Physics summer assignment. Students who are taking AP Physics next year should do the whole assignment and turn it in to Anuraag and turn in a copy to me. You will not be graded on Part I of the exploration for my class, but I d like to see your work for Part I as well as Part II. (It sounds fun!) Students who are not taking AP Physics next year should just do Part II. Of course you are welcome to do Part I if you d like to, but I will not be grading that part of the problem.

5 Name: Date: Block: Feel free to swing by this summer! Describing the Motion of a Pendulum Whether you know it or not (and if you ve taken Physics, you hopefully do!), you re very familiar with pendulums. They show up in devices from grandfather clocks to swings on a playground to the metronomes musicians use to keep time. A pendulum is a mechanical system which consists of a massive bob which swings back and forth in a periodic motion attached to a string or rod. This is shown in Figure below. In order to simplify the mathematics of this system, we assume that the rod is massless and worry only about its length. Figure : Diagram of a basic pendulum In this exploration, you ll take a closer look at the motion of a pendulum and become more comfortable with describing that motion mathematically. This will be a great springboard into using calculus concepts to describe physical situations. Complete all written answers to the following questions on your graph paper, and attach that to this sheet! Part I: Let s start by building a pendulum. This should not be too complicated all you need are a piece of string, a fixed platform or hook to hang the string from, and something massive to use as a bob. In the space below, record the length of the string you decide to use. L = m Aside from your pendulum, have a timer, a protractor, a calculator and some graph paper ready to take some quick data that you can then represent as a graph.

6 .) Tie the bob to the string and secure the other end of the string to your platform. Displace the pendulum by 5 from its equilibrium (vertical) position, and let it swing back and forth at least 5 times. As the pendulum swings, use your timer to measure and record the amount of time between each complete oscillation (back and forth swing)..) On your graph paper, create an angular displacement vs. time graph that shows what the angular displacement of the bob is at each point in time. Again, show at least five full oscillations on your graph. Be sure to label your axes and include units! 3.) What does the shape of your graph look like? Write down an equation that could describe this graph of angular displacement as a function of time. 4.) Looking at your angular displacement vs. time graph, think about where in its swinging the pendulum was moving the slowest, and where it was moving the fastest. Now, create an angular velocity vs. time graph which shows at least five oscillations of the pendulum. Keep the scale of your horizontal (time) axis the same as you did for your position vs. time graph to make for an easy comparison. The scale on your vertical (angular velocity) axis can be approximate, but provide a written rationale for why you chose the values you did. 5.) What does the shape of this graph look like? Write down a possible equation that could describe this graph of angular velocity as a function of time. 6.) Looking at your angular velocity vs. time graph, think about where in its swinging the pendulum was speeding up, and where it was slowing down. Create an angular acceleration vs. time graph which shows at least five oscillations of the pendulum. Again, keep the scale of your horizontal (time) axis the same as for your other two graphs to make for an easy comparison. The scale of your vertical (angular acceleration) axis can be approximate, but provide a written rationale for why you chose the values you did. 7.) Conceptually, how is angular velocity related to angular position? How is angular acceleration related to angular velocity? Looking at your graphs, briefly discuss the relationships between these three quantities for periodic (back and forth) motion. Part II: (Only required for AP Calculus BC students) Perhaps the most famous application of pendulums is in grandfather clocks. The pendulum in a grandfather clock swings back and forth, turning gears which control the hands on the clock. The pendulum is ideally suited as a timekeeping device because its motion is both repetitive and regular. Once it is set in motion, a pendulum completes one back and forth swing in a set amount of time. This time interval, in which the pendulum completes one cycle of motion, is called the period of the pendulum. Once you know the period of a pendulum, you can use it to mark out equal time intervals and thereby measure time. The equation for the period of a pendulum is: T = π l g Here, T is the period in units of seconds, l is the length of the pendulum in meters, and g is the acceleration due to gravity on Earth, which is equal to 9.8!!!. As long as you know the length of your pendulum, you can plug it into this equation to calculate the period. But what if the pendulum s length is changing?

7 Suppose you have a defective grandfather clock. The pendulum in the clock is measured to have a length of.5m. However, the string is slipping downwards as it swings back and forth, making the length of the string increase by m each second. We are going to investigate the effect of this increase on the period of the pendulum..) Using the information given above, write down an equation for the length of the pendulum as a function of time..) Fill out the table below to indicate the length of the string at different times as the pendulum is swinging. Assume that at t = 0, the pendulum is at its starting length of.5m. Time (s): Length of Pendulum (m): ) Now, for each of the lengths you entered in the table above, calculate the period of the pendulum at that length. Fill in this information in the table below. Time (s): Period of Pendulum (s):

8 4.) You have learned about derivatives as a tool to find the rate at which any given equation is changing. We have equations for the length of the string as a function of time, and the period of the pendulum as a function of length. a. What is an expression for the rate at which the length of the string changes with time (!"!" )? b. What is an expression for the rate at which the period of the pendulum changes with time (!"!" )? c. Let s compare the rate of change in the pendulum s period as expressed by the derivative!" with the average rate of change over each second interval from our!" data table. Using the work you have done so far, fill out the table below. Time Interval: Average Rate of Change of Period: Midpoint of Time Interval: Instantaneous Rate of Change dt Evaluated at dt Midpoint of Time Interval: 0s s 0.5s s s.5s s 3s.5s 3s 4s 3.5s 4s 5s 4.5s 5s 6s 5.5s 6s 7s 6.5s 7s 8s 7.5a 8s 9s 8.5s 9s 0s 9.5s

9 d. Compare the average rate of change over each time interval to the instantaneous rate of change at the midpoint of that time interval. i. Do the two calculations match up? If they don t, double check your derivative and make any changes you need to. Explain any changes you decided to make in the space below. ii. What do you notice with this comparison for later time intervals as opposed to earlier time intervals? iii. What does this say about the behavior of the equation for the defective pendulum s period (T = π!!! ) as more and more time goes by? e. A grandfather clock s pendulum increasing in length at a rate of m/s is not especially realistic if a clock were actually that defective, it would never keep oscillating for 0s! Why did we choose to give you such a striking rate of change for the length of the pendulum? What problems might you have encountered if we had picked a more realistic rate of change (say, 0.00 m/s)?

10 Summer Assignment #3: Driving Project This project involves taking a 5-minute car ride, collecting data, and analyzing results. You may collect data with another calculus student, but please submit your own write-up. Collect Data! Record your car s odometer reading before starting your car ride. DO NOT FORGET TO DO THIS!!!!! With another person driving, take a 5-minute car ride in a residential area. Find a route that allows you to change speeds throughout the trip.! During the trip, record your velocity every minute ( 5 th of an hour) even if you are at a stop. When done, you should have 6 pieces of data for t = 5 0, 5, 5 5,, hours. 5! Record the odometer reading after your trip. DO NOT FORGET TO DO THIS!!!! Analyze Results. Present your data in a table and plot velocity vs. time (in hours) on graph paper. Connect your data points with line segments to form your velocity graph. Clearly identify what each axis represents and include its measurement units. Also identify the increments on each axis. Use hours, not minutes!. Working from left to right, calculate the area under your velocity graph for each 5 time interval. You will calculate 5 areas. Most will be trapezoids, but some might be rectangles or triangles. Do you remember the area of a trapezoid? Area = b + b ) h (The bases are the parallel sides.) ( 3. Add your fifteen areas to find the total area under your velocity graph. 4. In a few well-written paragraphs, answer these questions:! How does the total area under your graph compare to the two odometer readings?! What does the area under your velocity graph represent? Address units. Also, divide the total area by ¼ hour, the total time you spent driving. What does that result represent? Include units.! What does the slope during each one-minute time interval represent? Include units. What does a negative slope imply about how you were moving?! Is your velocity data a complete picture of how you drove during your trip? If yes, explain why. If not, how could you make it more complete?! What other insights can you draw from your results and analysis?

11 Summer Assignment #4: Introduction to Series Series as Functions A transcendental function is a function which is not an algebraic (or polynomial) function. In other words, a function which transcends, i.e. cannot be expressed in terms of, algebra. (Wolfram MathWorld) The algebraic operations of addition, multiplication, and rooting do not result in the values of trigonometric, exponential, and logarithmic expressions. So how do our calculators kindly return values for expressions like sin? They are programmed to approximate them using series that are algebraic. Consider the rational function below and assume that it equals the polynomial series whose coefficients we wish to find. Function f(x): x Series P(x): = c 0 + c x + c x + c 3 x 3 + c 4 x 4 + c 5 x Let s start by considering the fact that we want the y-values of the function and the series to match at x = 0 so that they both pass through that point. Do your work on this paper, but write up your analysis on separate paper. at x: f (x) = P(x) x = c 0 + c x + c x + c 3 x 3 + c 4 x 4 + c 5 x Solve for c n at x = 0 : 0 = c 0 = We also want the slopes and the concavity of the function and series to match there. So we will take higher and higher derivatives of the function and of the series, equate them, and solve for the coefficients. It will be easier if you rewrite the function as x rule! ( ) and don t forget the chain at x: f '(x) = P'(x) Solve for c n at x = 0 : at x: f ''(x) = P''(x) at x = 0 : at x: f '''(x) = P'''(x) Solve for c n

12 at x = 0 :. at x: at x = 0 : at x: f ( 4) (x) = P ( 4) (x) f ( 5) (x) = P ( 5) (x) at x = 0 : Now that you have the first 5 coefficients of the polynomial series, you should be able to see a pattern. Write the 7 th degree polynomial for the function f (x) = x :. Graph the function as Y. Graph the series as Y, starting with the first degree polynomial P(x) =+ x. To distinguish the two curves, thicken Y. A good viewing window is x and 4 y 6. Copy your graph neatly, indicating which curve is the function and which is the series. 3. Now graph the function and the series, using the second degree polynomial P(x) =+ x + x. Copy your second graph neatly, indicating which curve is the function and which is the series. 4. Continue graphing the function and series, using the next higher degree polynomial each time until you reach the 5 th degree. Copy each graph neatly, indicating which curve is the function and which is the series. 5. Describe the effect of increasing the number of terms of the series - how does the graph of the series mimic the graph of the function? Why does it make sense that adding more terms to the series makes it more accurate? 6. Over what x-interval does the series best approximate the function? This is called the interval of convergence.

13 7. Using the 5 th degree polynomial series, make a table of values with your calculator for the function and the series. Start at x = and use Δx = 0.. Copy your tables down. Compare the values from the series (Y ) to those on the function (Y ). Describe how the accuracy of the series changes as you move away from x = 0. Be sure to look in both the positive & negative directions. Describe what happens to the accuracy when you move outside the interval < x < 8. Substitute x = into f (x) and into P(x), and simplify each expression. You should be able to find the sum of the infinite series! (This as a concept you learned in Algebra.) 9. Substitute x = into f (x) and into P(x), and simplify each expression. Do the results make sense? Why or why not? 0. Now replace each x in both the function and the series with x 3. Also replace x in the interval of convergence < x < with x 3 to determine the new interval of convergence. You should now have the series for and the interval on which it converges. 4 x. Graph both the function and its series, this time in Y 3 and Y 4. Go out to the 6 th degree. Turn off Y and Y. Also, extend your x-window to see Y 3 and Y 4 better. Does the resulting graph confirm your work in the previous step? It should! Sketch the graph and identify which curve is the function and which is the series. Trigonometric Series One famous series in mathematics is: x x + x x + 3! 5! 7! Notice that it has only odd powers and alternates sign. 3 Graph this series in Y 5. Go out to the thirteenth degree ( x ). 3! Turn off your other functions and use the window: 3 x 3, y Sketch the resulting graph.. Which trigonometric function does this series appear to model near the origin? Test your conjecture in Y 6. Were you correct? Terrific! 3. Look at your calculator s table of values for Y 5 and Y 6. Start at x = 3with Δx = 0. How does the accuracy of this series change as you move away from x = 0? 4. Add the next three terms to your series, so that it now goes out to the 9 th degree. Look again at the table of values. How did the accuracy change? Explain why that makes sense. 5. Evaluate sin() using your series. Is it a good approximation? Why or why not? Write a short summary about what you now know about series.