Math = Language Calculus = study of change Precalculus = study of functions Functions = data, graphs, equations Limits...Derivatives...

Transcription

1 Math = Language Calculus = study of change Precalculus = study of functions Functions = data, graphs, equations Limits...Derivatives...AntiDerivatives Rates of Change Average = slope between two points Instantaneous = slope at one point =Derivative=slope of the tangent line =lim f(x+h) f(x) h >0 h

2 Limits for the definitionof derivative epsilon delta formal definition using algebra to find limits using a graph or calculator to find limits one sided limits, two sided limits limits at infinity = end behavior Derivatives From the definition Quick and Dirty Rules Power Product Quotient Chain Transcendentals= Trig, logs, exponents,hyper Second Derivative=Acceleration or Deceleration

3 Definitions Epsilon Delta of Limits Continuity Derivative Therorems Squeeze certain limits /with sine Intermediate Value Theorem from Continuity Mean Value Theorem two rates equal

4 Skill work vs. Group Work On Connect Final Exam from tests Tests from Quizzes Quizzes from Homework If its not on a quiz, its not on a test. Help on videos On Blackboard Work with real data Apply to real life work with ugly numbers work with Technology Opportunity to Speak math Collaborate Groupwork and Project Join Group Divide Tasks Discuss work to post

5 Posting Groupwork Leader creates thread Writer writes math Speaker writes English 8 topics for individual project Before Starting: Give your data some life by assigning units x= cents y = cups of lemonade sold writer: r(25)=12 speaker: According to data, a price of 25 cents sells 12 cups of lemonade

6 Rates of Change Need a regression equation in y1 Need two endpoints a= and b= and an inner point c= V ave= ( y1(b) y1(a) ) / b a V instantaneous= y1' (c) = nderiv(y1,x,c) Average and Instantaneous according to the regression, between values x= a and x=b, the average rate of change Limits According to the split regression, as we look backwards to 2014, we expect a value of 7.8million dollars

7 according to the coninuous regression, we expect a value of 6.9 million in 2014 According to this continuous regression, as time goes on, there is no way of knowing what the value will be. But as we look back, the value drops.

8 Derivatives of Polynomials and Trascendentals Group Work Drivatives Objective: Make predictions from data that describes the value expected y(a), the rate at which the value is growing y'(a), and wheather or not the values are accelerating y''(a) using the calculator and properties of derivatives.

9 Part 1: Polynomials 1. Perform a cubic regression with your data 2. Place that regression into y1 and y2 on calculator 3. Edit y2 so it reflects the deriative 4. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 5. Use the table to make a prediction at a point of interest. 1. Perform a cubic regression with your data 2. Place that regression into y1 and y2 on calculator 3. Edit y2 so it reflects the deriative 4. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 5. Use the table to make a prediction at a point of interest.

10 Example: Writer: r(x) = ax^3+bx^2+cx+d r'(x)= 3ax^2+2bx+c r''(x)=6ax+2b r(poa)=50 r'(poa)=75 r''(poa)=100 Speaker: According to the cubic regression, at one dollar we expect to sell 4.3 cups of lemonade with the rate of loss of.0007 cups per dollar and is accelerating

11 Part 2: Transcendentals 1. Perform a exponential or logarithmic regression with your data whichever makes more sense. 2. Place that regression into y1 and y2 on calculator 3. Edit y2 so it reflects the deriative 4. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 5. Use the table to make a prediction at a point of interest. 1. Perform a exponential or logarithmic regression with your data whichever makes more sense. 2. Place that regression into y1 and y2 on calculator 3. Edit y2 so it reflects the deriative 4. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 5. Use the table to make a prediction at a point of interest.

12 Example: Writer: r(x) = a*b^x or a + blnx r'(x)= a*b^x*lnb or b/x r''(x)=6ax+2ba*b^x*(lnb)^2 or b/x^2 r(poa)=50 r'(poa)=75 r''(poa)=100 Speaker: According to the exponential regression, if we charge one dollar for lemonade, we anticipate selling 4.6 cups, and we would be losing.05 cups of sales for every penny more we charge, and this loss is accelerating.

13 Part 3: Sine Regression 1. Determine a period for your function. If it is constantly increasing/decreasing, find the difference between your largest and smalles X values and double that 2. Perform a sine regression with your data by changing the iterations to 1 and adding a period as found above. Use the command: sinreg 1,L1,L2,Period 3. Place that regression into y1 and y2 on calculator 4. Edit y2 so it reflects the deriative 5. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 6. Use the table to make a prediction at a point of interest. 1. Determine a period for your function. If it is constantly increasing/decreasing, find the difference between your largest and smalles X values and double that 2. Perform a sine regression with your data by changing the iterations to 1 and adding a period as found above. Use the command: sinreg 1,L1,L2,Period 3. Place that regression into y1 and y2 on calculator 4. Edit y2 so it reflects the deriative 5. Into y3, use the derivative feature of the calculator (math 8:nderiv) to find the second derivative 6. Use the table to make a prediction at a point of interest.

14 Example: Writer: r(x) = asin(bx+c)+d r'(x)= acos(bx+c)*b r''(x)= asin(bx+c)*b^2 r(poa)=50 r'(poa)=75 r''(poa)=100 Speaker: According to the sine regression with a period of 450, at 100 we expect to sell 4.77 cups of lemonade with a rate of loss of.09 cups per penny and is accelerating

15 Mean Value Theroem and Newton's Method 1. Find the cubic regression put in y1 2. Put the derivative of y1 into y2 3. Find the average rate of change between two values and put in y3 4. Adjust the window xmin and xmax 5. Turn off y1 and Do zoom 0: Fit 6. Calculate the intersection using Calc 5:intersect 1. Find the cubic regression put in y1 2. Put the derivative of y1 into y2 3. Find the average rate of change between two values and put in y3 4. Adjust the window xmin and xmax 5. Turn off y1 and Do zoom 0: Fit 6. Calculate the intersection using Calc 5:intersect

16 writer f(x) =ax^3+bx^2+cx+d f'(x)=3ax^2+2bx+c average rate between a= and b= (f(b) f(a))/(b a) = =f'(c) c = Speaker According to the cubic regression, the average rate between 25 and 250 cents is best represented at the instance of 127 cents

17 Newtons Method as a check 1. Let y4=y3 y2 2. Guess x=7 by putting 7 sto x 3. issue the command x y4/nderiv(y4,x,x) sto x Writer:

18 Error 1.Find exponential regression and put in y1 2. Determine the dx value 3. Put the derivative of y1 * dx in y2 4. Put y2/y1*100 in y3 dx = +/.5 if x values change by 1 dx = +/ 5 if x values change by 10 dx = +/.05 if x values chage by.1 Speaker: According to the exponential regression, we expect to sell 4.6 cups of lemonade at a price of 1.00 with an error of.1 which represents a 2.8% error Or dx=+/ 2.5 Writer: r(100)=4.59 error(100)=+/.1204 %error(100)=+/ 2.844

19 Analysis of Function Area and Average Value

20 Approximating Area and FTC