Lecture for Week 2 Secs 1 3 and 2 2 2 3 Functions and Limits 1 First let s review what a function is See Sec 1 of Review and Preview The best way to think of a function is as an imaginary machine or black box that takes in any of various objects labeled x whenever there s no reason to call it something else and processes it into a new object labeled y f x A function is not necessarily given by a formula 2 Usually both x and y are numbers and in that case we can easily think of a function as being the same thing as a graph A curve or any set of points in the x y plane defines a function provided that no vertical line intersects the set more than once y x 3 Sec 1 3 however deals with functions r t whose values are vectors or points not numbers The graph of such a function still exists but it lies in a more abstract space with three dimensions in this case one for t and two for r A point is not quite the same thing as a vector A point is represented by a vector r hx yi when we choose an origin of coordinates in space If you move the origin the numbers x and y will change but the numerical components of true vectors such as velocity and force will not change 4 Exercise 1 3 7 Sketch the curve represented by the parametric equations x 3 cos y 2 sin 0 2 and eliminate the parameter to find the Cartesian equation of the curve Note no actual vector notation here although one could have written r h3 cos 2 sin i 5 Well first I d plot the points for 0 2 4 2 The last point is the same as the first one Then I notice that x 2 y 2 cos2 sin2 1 3 2 so the curve is recognized as an ellipse It s the whole ellipse since we observed that the path closes 6 Fortunately my plotting software has an ellipse command so I didn t need to do the arithmetic for the first part y 7 x Exercise 1 3 27 Find a a vector equation b parametric equations and c a Cartesian equation for the line passing through the points 4 1 and 2 5 The exercise in the book doesn t ask for a Cartesian equation but the next group of exercises does so I ll do it here 8 First let s find the vector pointing from the first vector to the second v h 2 5i h4 1i h 6 6i If we add any multiple of v to any point on the line the result is a point on the line and you get all the points that way So an answer to a is r t h4 1i th 6 6i 4 6t i 1 6t j 9 r t h4 1i th 6 6i 4 6t i 1 6t j Notice that the question asked for a vector equation not the vector equation There are many other correct answers to a corresponding to different starting points on the line or different lengths and signs for v For b just write the components separately x t 4 6t y t 6t 1 10 To get a Cartesian equation we need to eliminate t In the present case that is easily done by adding the two parametric equations x 4 6t y 6t 1 x y 3 In more general situations you would need to solve one equation for t and substitute the result into the other equation 11 lus Limit is the fundamental concept of calcuEverything else is defined in terms of it continuity derivative integral sum of infinite series It took mathematicians 200 years of calculus history to arrive at a satisfactory definition of limit Not surprisingly the result is not easy for beginners to absorb 12 For that reason Sec 2 4 The Precise Definition of a Limit is not a required part of our syllabus That doesn t mean that you are forbidden to read it But you might find it more meaningful if you come back to it after gaining some experience with how limits are used and why they are important Two natural places in the textbook from which to loop back here are after infinite sequences and series Chap 10 when functions of several variables arise Sec 12 2 In that place Stewart simply states the multivariable generalization of the precise definition without fuss or apology 13 So we have to make do with intuitive ideas of a limit The graphical problems on p 89 are a good place to start However they don t lend themselves to this projector presentation because I have no good way to reproduce the graphs So we ll look at them in real time I ll come back to infinite limits and vertical asymptotes later 14 Let s go on to Sec 2 3 The key issue in that section is this When you are presented with a formula defining a function such as x2 1 f x x 3 usually the limit of the function at a point is just the value of the function at that point but not always 1 1 1 2 lim f x f 1 x 1 1 3 2 15 f x p x2 1 x 3 But limx 3 f x does not exist the function values get arbitrarily large near x 3 Even worse they are positive on one side negative on the other 16 So the big question is when can you get away with just sticking the number into the formula to find the limit lim f x f a x a Textbooks give you a list of limit laws that state conditions that guarantee that the limit can be taken in the obvious way Let s turn the question around and try to identify danger signs that label situations where the obvious way might go wrong 17 In practice the most common trouble is a zero of the denominator Notice that in the limit laws on p 91 the last one concerning division is the only one that needs a caveat if limx a g x 6 0 Now two things can happen 1 The limit of the numerator as x a is not 0 Then we probably have some kind of infinite limit and a vertical asymptote 18 2 The limit of the numerator as x a is 0 Then we have to look carefully to see which factor goes to zero faster the numerator or the denominator Typically they will vanish at the same rate so that the limit of the fraction is some finite nonzero number This is the situation that is fundamental to the definition of the derivative in calculus Here are some examples 19 Exercise 2 3 5 x 2 lim x 1 x2 4x 3 Exercise 2 3 23 1 h 4 1 lim h 0 h 20 x 2 x 1 x2 4x 3 …
View Full Document
Unlocking...