change in position change in time

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1 CHAPTER. THE DERIVATIVE Comments. The problems in this section are at the heart of Calculus and lead directly to the main idea in Calculus, limits. But even more important than the problems themselves is the technique we use to solve them. Generalizing this technique leads directly to limits and derivatives. Example. The height of a thrown ball is given by the following function: p(t).9t +.t + where t is in seconds and p is in meters. (This is the real equation for a moving object thrown from an initial height of meters, with an initial velocity of. m /s. Find an approximation of the velocity at t.. Solution. We start with the definition of velocity: Velocity change in position change in time. To calculate change in position we need to use two values for time. We have to make up the second value for t. In other words, we need something like this Velocity p(.) p(.).. Notice how useful the function notation is here: it makes it clear what number is getting plugged in, and what it s getting plugged into. Velocity p(.) p(.).. 0. (.79) m/s. To make our answer more accurate, we can repeat the calculation, but this time using t., instead of., because. is closer to.. Here are the same calculations done with.. (Note: we will show below how to do these calculation in the calculator, in a table.) Velocity p(.) p(.) (.79) m/s. We ll repeat the above calculations one last time (before using the table) with.: Velocity p(.) p(.)...8 (.79) m/s.

2 CHAPTER. THE DERIVATIVE 7 So, right now we can say that the velocity looks like it s around 7m/s. Now I ll show you a way to repeat these calculations but to do them faster, with more accuracy, and letting the calculator take care of more of the details. First, we enter the original formula in Y: Y.9x +.x + Now, we create a function to do the calculations like p(.) p(.9). The second..9 number can change, so we use x for this number: Y (Y(.) Y(x))/(. x) To enter Y in this formula go to VARS, Y-VARS, :Function, :Y. Now you could use Y directly to repeat some of the above calculations, and that would speed things up a lot. I mean you could enter things like Y(.9), etc. But I ll show you how to make a table that does it more quickly, especially if you want to do it for three or four or five numbers. First, go to TBLSET (that s ND, WINDOW ). Make sure you have Indpnt: Auto Ask selected (this means that the calculator will ask you for the x-value, the independent variable). Then go to TABLE (that s ND, GRAPH ) and enter some of the above x-values, and maybe some other ones too. You should get something like this: x Y Y ERR : Make sure you remember that Y is calculating things like p(.) p(.9). The..9 calculator gives an error in the fourth line because this formula would give division by 0 if x.. Also, if you want more accuracy than what is shown in the table, you can put your cursor on the number and the calculator will show you all the digits. Anyway, from the above table of numbers, the two best approximations are 7.09 and 7.0. We don t know for sure yet what the exacty velocity is, but our approximation gives us bounds, and I suppose our best guess would be something like 7.0. Example. The quantity of some drug in a person s blood is given by Q 00(0.9) t,withq in mg and t in hours. Estimate the rate of change of the quantity of drug at time t., and give a simple interpretation of your answer. Solution. Recall that rate of change means Q 0 (.) and is found by calculating average rates of change using shorter and shorter intervals that contain.. For convenience, we will fix one our t-values at., and change the other t-value As opposed to using two t-values that squeeze in on., such as. and.

3 CHAPTER. THE DERIVATIVE 8 Here s one calculation using. and.: Q 0 (.) rate of change from. to. Q(.) Q(.).9. mg/hr. Now we repeat, to get more accuracy, with a table of numbers. First, we enter the original formula in Y: Y x Now, we create a function to do the calculations like Q(.) Q(.). Y (Y(.) Y(x))/(. x) (Recall that you can enter Y by going to VARS, Y-VARS, :Function, :Y.) We go to TABLE (that s ND, GRAPH ) and enter some of the above x-values, and maybe some other ones too. You should get something like this: x Y Y ERR : Make sure you remember that Y is calculating things like Q(.) Q(.). Based on this calculation our best guess for the answer is.98 mg/hr. Our interperation should be something like the following: At t., the quantity of drug in the blood is decreasing by.98 mg each hour. Comments. There are two really hugely important conclusions we will draw from this example: () Sometimes we can make a sequence of approximations that appear to be getting closer and closer to the correct answer; () A di erence quotient (i.e. a fraction with a subtraction on top and on the bottom) may be an important example of a thing that we can approximate like this. In fact, both of these observations are extremely profound and important. The first one becomes, in its final from, the concept of a mathematical limit. The second one becomes the derivative. Finally, note that even the partial solution we have of this problem is a monumental step forward in the history of science. The very idea that you could quantify physical processes, and do mathematics with these quantities, was incomprehensible to almost everyone, until Galileo. After Galileo it took another 00 years before people understood that you could start with one quantified process, position in this case, and do something to it to calculate its rate of change.

4 CHAPTER. THE DERIVATIVE 9 Definition. The instantaneous velocity is the limit of average velocities over shorter and shorter time intervals. Comments. This definition merely records what we did in the previous example. However, we now want to extend what we have done, by applying the same idea, but to other functions besides position. Definition. Given a function f, and a number a in the domain of f, wedefinethe number f 0 (a) as follows: f 0 (a) the number that the fraction f(x) f(a) approaches as x x a gets close to a. the di erence quo- We call f 0 (a) thederivative of f at a. We call f(x) x tient (of f(x) at a). f(a) a The derivative, f 0 (a) is interpretd as the instantaneous rate of change of f(x) at a. If f(t) equals position as a function of time, then f 0 (a) is the instantaneous velocity at t a. Fact. The number f 0 (a) equals the slope of the line that is tangent to the graph of f(x) at the point (a, f(a)). More briefly: f 0 (a) is the slope of the tangent line of f at a. Explanation. Recall that the rate of change f(b) f(a) represents the slope through a b two points on the graph of f(x) (i.e. the slope of a secant line). Thus, the derivative represents the slope of a line that we get by moving the two points closer and closer together. If you move b closer and closer to a, in the picture below, the secant line (in red) moves closer and closer to the tangent line (in green): m f 0 (a) m f(b) b f(a) a f(x) a b

5 CHAPTER. THE DERIVATIVE 0 People frequently say the slope of f at a instead of the slope of the tangent line of f at a and there is nothing wrong with this provided one remembers the role of the tangent line. Example. Using the graph of f(x) shown below, estimate f 0 ( ) and f 0 (). (Hint: the best way to do this is to draw a tangent line at the point, extend it either as long as possible, or until it hits a point on the grid that has a clear value. Calculate the slope of the tangent line using either points at/near the end of the line, or points with a clear value on the grid.) Solution. We start by drawing the tangent lines, first for f 0 ( ). This means that we want a line that just barely touches the curve, and does so at one point only, namely the point (, ). The line should have the same slope at as does the curve: in fact, if you zoom in really close, they should look almost identical to each other.

6 CHAPTER. THE DERIVATIVE Now, we should pick a couple of points on the line which are () somewhat easy to estimate values for, and () somewhat far apart. The easiest point is (, ) itself: it s on both the curve and the line. A second point that s pretty good is (, 0) which is on the line. We could also use (.7, 7). Using any two of these points we should get (approximately) the same slope: f 0 ( ) m 7.7 ( ) ( ) Now we repeat the same sort of steps as above, but briefer this time:

7 CHAPTER. THE DERIVATIVE The points that are marked are (, ), (0., ), (, ). Using any two of these points we should get (approximately) the same slope: f 0 () m 0 0. This is where we ended on Wednesday, February.