From average to instantaneous rates of change (and a diversion on con4nuity and limits)
Extra prac4ce problems? Problems in the Book Problems at the end of my slides Math Exam Resource (MER): hcp://wiki.ubc.ca/science:math_exam_resources
Homework help: WW Assignment 2 Problem 11 Input the formula (f(x 2 )-f(x 1 ))/(x 2 -x 1 ) for the func4on in your problem!
Homework help: WW Assignment 2 Assignment2: Problem 11 You do the thinking. WW does the grinding. No need for calculator!
Remember that in WeBWorK you can use basic operators (+,-,*,/,\^{}, etc.) instead of compu4ng numerical values by hand or by calculator. Always pair your (), [], etc! See: WebWork tutorial
Last 4me Aphid-Ladybug problem: See OpenBook p 40 hcps://www.desmos.com/calculator/wtyyewqtjq What happens when =???
From average rates of change, (by refining the data) To instantaneous rates of change
Last 4me: falling object Distance fallen versus 4me: We computed average velocity Distance fallen 4me
Average velocity over the 4me interval is Ajer some algebra.. See P 57 Example 2.6 Be sure you can arrive at this result yourself!
Our goal Use what we learned about average velocity over a 4me interval to define an instantaneous velocity at a given 4me. Will do this by refining our measurements to get more data points.
Strobe of a Falling ball hcp://video.mit.edu/watch/strobe-of-a-falling-ball-3151/ Δt = h = 4me between flashes
Strobe of a Falling ball hcp://video.mit.edu/watch/strobe-of-a-falling-ball-3151/ Height of ball Δt Δt = h = 4me between flashes
Refine the measurements Make strobe flash faster.. Height of ball Δt Δt = h = 4me between flashes
Refine the measurements Even faster: Height of ball Δt Δt = h = 4me between flashes
1. As the measurements are refined (A) Δt à 1 (B) The 4me between flashes gets shorter (C) Δt à 0 (D) None of the above (E) More than one of the above is correct
As the measurements are refined The 4me between measurements gets smaller and smaller
2. What happens to the value of the average velocity as The average velocity v av = approaches (A) 2 c (B) c h (C) 2 c h (D) 2 c t 0 (E) not sure
Defn: Instantaneous velocity Let v av be the average velocity over the 4me interval Then the instantaneous velocity, v at 4me t 0 is the value of v av obtained by shrinking the 4me between measurements
Instantaneous velocity for falling object at 4me t 0
Deriva4ve Then the instantaneous velocity at 4me t 0 is We also call this the deriva4ve of the func4on y(t) at 4me t 0
Defn: the deriva4ve The deriva4ve of a func4on at is We denote this by the nota4on or
Remarks about the instantaneous velocity It depends on 4me! It is the velocity at the instant t=t 0. It is obtained as a (formal) process involving a limit
Geometric view The slope of the secant line is the average rate of change (e.g. average velocity) O
Geometric view The slope of the secant line (rise/run) is the average rate of change (e.g. average velocity) O
Geometric view Rise and run are obtained from the coordinates of two points on the graph O Rise= Run = h h
Geometric view Slope of secant line = Rise / run = O h
Geometric view Now shrink the run. O h h
As we refine the data (shrink h) the secant line approaches a tangent line h
Secant à tangent line See P 61
Geometric and Analy4c views Secant line Tangent line Slope=average rate of change slope= instantaneous rate of change h à 0
Spreadsheet intro: slope of secant line on the func4on f(x)=x 2
Fill in the cells
You can change the step size, h, and the spreadsheet will recompute all the values automa4cally.
Average and instantaneous rates of change computa4on Consider See P 67 prob 2.15 Examtype ques4on
Limits, Con4nuous func4ons and the deriva4ve
Con4nuous func4on Three things need to be true! 1. Func4on defined at the point 2. Limit exists as x approaches that point 3. limit is same as value of func4on
Places where a func4on is not con4nuous Break in graph Hole in graph Ver4cal asymptote:
Types of (dis)con4nui4es
First, some con4nuous func4ons All power func4ons and polynomials are con4nuous everywhere. Limit exists and equals the func4on at every point
Con4nuous func4on
Discon4nuous fn: Hole in graph
Hole in graph Func4on con4nuous except at x=a. At x=a, limit exists, but does not equal the func4on We can fill in the hole by defining f(a) the right way
Fill the hole? The wrong way : Value assigned to f(a) does not fit the hole
Fill the hole! The right way :
1. Hole in graph We can fill in the hole by defining f(a) to be (A) 0 (B) a (C) a 2 (D) a 3 (E) x 3 -ax
Plug the hole in the graph of Make the func4on con4nuous by assigning Define : Removable discon4nuity
Jump discon4nuity Approaching x=a from above (from the right):
Jump discon4nuity Approaching x=a from below (from the lej):
Jump discon4nuity Approaching x=a from above: Approaching x=a from below:
Jump discon4nuity We can define lej and right limits at x=a. But they are not equal. We say that the limit does not exist (DNE)
Ver4cal asymptote
Ver4cal asymptote Func4on not defined at x=a, f(x) à + or - as x à a. Limit does not exist at x=a.
Nice summary of (dis)con4nui4es: hcps://www.youtube.com/watch?v=lzzyonrkjgs
3. Recap: did you follow all that?? Which of the following limits DNE? (A) (B) (C) (D) (E) More than one of the above DNE
4. Limits at infinity Which of the following limits exists?
4. Limits at infinity Which of the following limits exists?
5. Limits at infinity What is the following limit?
6. Piecewise func4on Compute
7. Piecewise func4on Which of the following limits exist? (A) (D) Both A and B, not C (B) (E) None of them (C)
7. Piecewise func4on Which of the following limits exist? (A) (D) Both A and B, not C (B) (E) None of them (C) When in doubt, sketch! x=1
Piecewise func4on Which of the following limits exist? (A) (D) All of them (B) (E) None of them (C) x=1
So what else are limits good for? We have encountered limits in the defini4on of a deriva4ve, and now we have more skills in using them. We will need them to compute deriva4ves of func4ons.
Defn: the deriva4ve The deriva4ve of a func4on at is We denote this by the nota4on or
See P 67 prob 2.20 Examtype ques4on
8. The deriva4ve To compute the deriva4ve of this func4on, we need to find the following limit:
Solu4ons
Answers 1. E 2. D 3. B 4. C 5. D 6. B 7. D 8. E
Solu4ons to previous problems First try out the problems at the end of the last lecture slides. Only then should you peek at the answers.
Soln to Test problem 1 from last 4me MT1, 2008 SOLUTION:
Soln to Test problem 2 from last 4me MT1 2013 Why? The only interval [0,b] such that the average rate of change=1, should have b=1. (See red secant line whose slope is 1)
Quiz problem (2015) See next page for reasoning
Reasoning for previous page
Quiz 1 (2015)
Problems to test your skill A selec4on of problems from tests and exams to help you study concepts in this lecture
Secant and tangent lines
A secant line is (A) A line whose slope is instantaneous velocity (B) A line connec4ng two points on a graph (C) The same as an average velocity (D) The same all along the curve (E) Not sure.
Which of the below is an average velocity? (A) Av veloc= (v 1 +v 2 +v 3 +v 4 )/4 (B) Av veloc = (y 1 +y 2 +y 3 +y 4 )/4 (C) Av veloc = (y 1 +y 2 +y 3 +y 4 )/(t 1 +t 2 +t 3 +t 4 ) (D) Av veloc = (y 1 +y 2 ) /(t 1 +t 2 ) (E) Av veloc = (y 2 -y 1 )/(t 2 t 1 )