Introduction. Math Calculus 1 section 2.1 and 2.2. Julian Chan. Department of Mathematics Weber State University

Transcription

1 Math 1210 Calculus 1 section 2.1 and 2.2 Julian Chan Department of Mathematics Weber State University 2013

2 Objectives Objectives: to tangent lines to limits What is velocity and how to obtain it from the position function One sided limits, infinite limits, and asymptotes.

3 Ideas of calculus 1 One of the biggest themes of calculus is to answer the question what is a rate?

4 Ideas of calculus 1 One of the biggest themes of calculus is to answer the question what is a rate? 2 You can recall that the equation: D = rt is a relationship between distace, rate (or velocity), and time. We could have also written D = Vt. 3 Why is this not the end of the story?

5 Velocity Consider the situation in which John is traveling from Ogden to Beaver county. John travels 75 MPH from Ogden to SLC for 1 hour, and 60 MPH from SLC to Provo for 1.5 hours, then 80 for 2 hours arriving at Beaver. Write an equation describing the distance he travels?

6 Velocity Consider the situation in which John is traveling from Ogden to Beaver county. John travels 75 MPH from Ogden to SLC for 1 hour, and 60 MPH from SLC to Provo for 1.5 hours, then 80 for 2 hours arriving at Beaver. Write an equation describing the distance he travels? 75t 0 < t 1 D(t) = t 1 < t t 2.5 < t 5.5 What rate did he travel for the whole trip?

7 Velocity Consider the situation in which John is traveling from Ogden to Beaver county. John travels 75 MPH from Ogden to SLC for 1 hour, and 60 MPH from SLC to Provo for 1.5 hours, then 80 for 2 hours arriving at Beaver. Write an equation describing the distance he travels? 75t 0 < t 1 D(t) = t 1 < t t 2.5 < t 5.5 What rate did he travel for the whole trip? r = D/t = 325/5.5 = 59 miles per hour!!!

8 Velocity Consider the situation in which John is traveling from Ogden to Beaver county. John travels 75 MPH from Ogden to SLC for 1 hour, and 60 MPH from SLC to Provo for 1.5 hours, then 80 for 2 hours arriving at Beaver. Write an equation describing the distance he travels? 75t 0 < t 1 D(t) = t 1 < t t 2.5 < t 5.5 What rate did he travel for the whole trip? r = D/t = 325/5.5 = 59 miles per hour!!! What is going on?????

9 Example Consider a ball that is dropped from rest at a height of 500 meters. It s position above the ground is given by P(t) = t 2 /2 where t is the time that has passes after the ball was dropped! What is the velocity 2 seconds after the ball is dropped? This is a hard question to answer (why)?

10 Example Instead try to calculate the rate (velocity) of the ball on the time interval [2, 10.1] (note 10.1 seconds is about the time the ball hits the ground). Instead try to calculate the rate (velocity) of the ball on the time interval [2, 4] Instead try to calculate the rate (velocity) of the ball on the time interval [2, 3] Instead try to calculate the rate (velocity) of the ball on the time interval [2, 2.5] Instead try to calculate the rate (velocity) of the ball on the time interval [2, 2.1] What would guess of the estimates calculated for velocity best respresent the rate of chane or (instantaneous) velocity?

11 Slopes The equation r = D/t can be written as: m = f (x + t) f (x) t = y f y i x f x i The latter formula is just the equation of the slope of a line!!! When we use average velocity we have the slope of the secant line, and the equation of the line is just the line that passes though the given point with the slope of the secant. The tangent line the the equation of the line that passes though the point with slope equal to the true rate of change. The tangent line is the LIMIT of the secant lines!!!

12 Example A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heart beat after t minutes. When the data in the table are graphed the slope of the tangent line represents the heart rate in beats per minute. Table: time to die t heart beats The monitor estimates the heart rate by calculating the slope of s secant line. Use the data to estimate the patients heart rate after 42 minutes.

13 Limits Now that the study of velocity and tangents has been discussed we see it is necessary to study limits! There are 3 ways to study limits: Using a table (these are best for conceptual understanding) Using a graph (these are best for conceptual understanding) Using the rules of limits (this is probably the most imporant method to know).

14 Idea of a limit What is a limit i.e. what does lim x a f (x) mean? A limit is the value f (x) approaches or gets really close to as x gets close to the value a

15 Example Let s compute lim x 1x by making a table. Table: limits x f(x) We see that as x gets close a 1 the values of f (x) get close to 1 therefore lin x 1x = 1. Draw this on a graph.

16 Example Let s try the same thing but for the following functions: lim x 1 1 x 2 +x. lim x 0sin(x)/x. lim x 1 x 1 x 2 1. Table: limits x f(x)

17 One sided limits 1 The one sided limit lim x a +f (x) is the value f (x) approaches as x gets close to a but for only values of x larger than a. 2 The one sided limit lim x a f (x) is the value f (x) approaches as x gets close to a but for only values of x smaller than a. 3 A one sided limit is just a restriction of the values of x to either being larger than a or smaller than a (but remember they can be as close to a as they like!)

18 Example Let s try the same thing but for the following functions: lim x 0 +sin(x). Table: limits x f(x)

19 Important fact 1 Fact: lim x af (x) = L if and only if lim f (x) = L x a and lim f (x) = L x a + 2 If this condition does not hold then lim x a f (x) DOES NOT EXIST (in the regular sense (but the one sided limit may exist!)).

20 Infinite limits 1 The infinite limit lim x af (x) = if as the value of x get close to a the output value f (x) get larger and larger without bound.

21 Example Let s try the same thing but for the following functions: lim x 0 +1/x 4. It may be illistrative to graph the function. Table: limits x f(x)

22 Infinite limits 1 The infinite limit lim x af (x) = if as the value of x get close to a the output value f (x) get larger and larger (in the negative direction) without bound.

23 Asymptote 1 The line x = a is a verticle asymptote of the curve y = f (x) is at least one of the following statements are true: 2 lim x af (x) = 3 lim x af (x) = 4 lim x a +f (x) = 5 lim x a +f (x) = 6 lim x a f (x) = 7 lim x a f (x) =

24 limits with graphs 1 Let s compute the limits of the graph (below). 2 lim x 1f (x) = 3 lim x 1f (x) = 4 lim x 1 +f (x) = 5 lim x 2 +f (x) = 6 lim x 1 f (x) =

25 limits

26 limits with graphs 1 Let s compute the limits of the graph (below). 2 lim x 4f (x) = 3 lim x 4 +f (x) = 4 lim x 4 f (x) =

27 limits

28 limits with graphs 1 Let s compute the limits of the graph (below). 2 lim x 2f (x) = 3 lim x 4 f (x) = 4 lim x 0 +f (x) = 5 lim x 2 +f (x) = 6 lim x 2 f (x) =

29 limits

30 Homework Homework

31 Homework Homework 2.1: 2,6,8 2.2: 6, 7, 13, 14, 25, 40