Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange. Tis activity will define procedures for computing instantaneous rates of cange exactly and, wen exact is impossible or impractical, approximately. To do tis requires a rigorous definition of slope of a curve. Te slope of a curve at a point is defined to be te slope of te line tangent to te curve at tat point (see figure 1a). But te slope formula requires two points to get a slope. Often, as in figure 1a, a tangent line passes troug only one point on a curve. So were do we get anoter point from wic to compute te slope? We look to any oter nearby point. [Wen tey exist, tangent lines can be understood intuitively by zooming in at te point of tangency and noticing tat, after enoug zooms, te tangent and te curve appear to be te same grap. Hence, te curve, being virtually linear in tat window, can be tougt of as aving a definite slope. Before proceeding, exploring tis penomenon via anoter activity migt be advisable.] Fig. 1a Fig. 1b f ( x) If f is any function, ten is te slope of te secant line (see figure 1b) x a troug (a, f (a)) and (x, f (x)). Because it is a quotient in wic bot te numerator and denominator are differences, tat expression is called a difference quotient. Any secant line goes troug (at least) two points on te grap of a function; ence te numerator will always be te difference of two function values. If we want to someow use te secant line to approximate te slope of te tangent, ten x sould be near a. Exercise 1: Te point (x, f (x)) is referred to in te numerator of te difference quotient above. In figure 1b, were is tat point? Is tat point near enoug to P to give a decent approximation of te slope of te tangent at P (sown in fig. 1a)? f Oter forms of difference quotients include (wic is very generic, but concise) and x f ( a + x). If te latter two difference quotients are going to give slopes tat are ( a + x) a close to te slope of te tangent at (a, f (a)), teir denominators are going to ave to be near 0. (Convince yourself of tat.)
f ( a + x) Exercise 2: Simplify te difference quotient. (After simplification, ( a + x) a does it still look like a difference of two function values over te difference of two x s?) For tis difference quotient to be a good approximation of te slope of te tangent line at (a, f (a)), x must be near. If x is positive, te point (a + x, f (a + x)) will be to te rigt of (a, f (a)), so te difference quotient in Exercise 2 is called te forward difference quotient. f ( a) f ( a ) Exercise 3: Te difference quotient is te slope of te secant line troug two points on te grap of f. Assume tat is positive and locate te oter point on te sketc below and draw te secant line. [Te function expressions in te numerator tell you wat x s to use in te denominator.] Wat are te coordinates of te two points represented in te difference quotient above? In order for tis secant line to be a good approximation to te slope of te tangent line at P, ten would ave to be near. If is positive, te point (a, f (a )) will be to te left of (a, f (a)), so te difference quotient in Exercise 3 is called te backward difference quotient. Exercise 4: On te grap below, eac tick mark represents one unit and P is te point (a, f (a)). Let be 0.5, wic is not particularly close to 0, and draw te secant line troug te points (a, f (a )) and (a +, f (a + )). Ten express its slope as a difference quotient in terms of f, a, and. Do you see wy tis difference quotient is called te symmetric difference quotient?
Using te grid dots on te grap screen, estimate te slope given by te symmetric difference quotient. Draw te tangent line at P. How well does te symmetric difference quotient approximate its slope? π x Exercise 5: Te function in te grap in Exercise 4 is f( x) = sin and P as x- 3 coordinate 1. Grap tat function, zoom in [wit equal zoom factors] at P until te grap looks linear, and estimate te slope. Compute te symmetric difference quotient wit = 0.5. For wat value of are te forward and backward difference quotients as accurate as te symmetric wit = 1? Conclusions? Exercise 6: Sow algebraically [feel free to use your 89] tat te symmetric difference quotient is te average of te forward and backward difference quotients. (For aid, refer to figure 2, in wic te forward difference quotient gives te slope of PR, wile te backward gives te slope of PQ, and te symmetric gives te slope ofqr. You would be wise to let (a, f (a)) represent P and use eiter or x (consider bot to be positive) to determine te coordinates of Q and R.) Fig. 2 Exercise 7: Figure 2 was produced via te following window and function definitions. Matc te tangent line and forward, backward, and symmetric difference quotient secant lines wit te appropriate function in y2 troug y5. Eac equation in y2 troug y5 is a line in point-slope form, but every function defintion is at least partly cut off at te rigt margin. Write te equations for y2 troug y5.
[FYI figure 3 below was produced by turning off y3 troug y5 and storing {.1,.075,.05,.25} into.] OK, fine. We can find a lot of different types of secant lines wose slope can be expressed as many different-looking difference quotients tat are related in an interesting way (Exercise 6). But ow do we find slopes of tangents? Refer to figure 3 to see several forward secant lines (for =.1,.075,.05,.025) tat approac, as a limit, te tangent line, wic is also included in figure 3. Fig. 3 Exercise 8: In figure 3, were is te point of tangency? Wic line is te tangent? If te point of tangency is (a, f (a)), wat would be te equation of te tangent tere? It looks like te slope of te tangent line in figure 3 is a positive number. Wen it exists [sometimes it doesn t], te slope of te line tangent to te function y = f (x) at (a, f (a)) is f ( a + ) defined as f ( a) = lim. Tis is called te derivative of f at a. 0 Exercise 9: Wic difference quotient is used in te definition of derivative? Could eiter or bot of te oter two also ave been used? Explain. If, for some reason, te limits cannot be computed, te derivative can be approximated by using any of te individual difference quotients, if a small enoug can be used. As figure 2 sows, te backward and forward difference quotients are not likely to very good at estimating te slope of te tangent line unless is very close to 0. But, as figure 2 also sows (at least for te function involved), te symmetric difference quotient can be very close to correct, even if is not so small, or if we ave only a grap to proceed from. Exercise 10: For te function defined grapically and numerically below, estimate its derivative at x = 2.45 using all 3 difference quotients.
Te data above came from te function f(x) = 2sin(3x) cos(x). Use te definition of derivative (a limit of a forward difference quotient) to calculate te exact value of te derivative. Your TI-89 can calculate many derivatives exactly. Define f(x)=2sin(3x) cos(x) and ten enter te command d(f(x),x) x=2.45. [d is above te 8 key.] Did you get te same result tat you did in te first bullet above? (Just cecking!) Wic difference quotient came closest? By ow muc did it miss? Wat is te relative error in te calculation? [Relative error is defined as actual value approximate value /actual value.] Wic missed by most? By ow muc? Wat is te relative error? Conclusions? Exercise 11: Consider te points (5,5), (10,25), and (15,10), plotted on 3 axes below. Te fact is tat any one of te 3 difference quotients could give te best estimate of te derivative. Draw a function for wic te FDQ would be te best estimate of te derivative at (10,25). Draw a function for wic te BDQ would be te best estimate of te derivative at (10,25). Draw a function for wic te SDQ would be te best estimate of te derivative at (10,25). Wic of te 3 functions you graped do you tink would be most likely to occur in a real-life problem? Wy? Exercise 12: Te symmetric difference quotient is so good tat it gives exact results for parabolas, even witout taking a limit! Prove tat tis is so. Exercise 13: Unlike te TI-89, some calculators, suc as te TI-83, cannot compute limits, since tey cannot do symbol manipulation. Still, tey ave te built-in capability to find derivatives approximately. Wy do you tink tey use te symmetric difference quotient to numerically approximate derivatives?
Exercise 14: Explain wy te symmetric difference quotient (and terefore te TI-83, for example) gives 0 as an approximate value of te derivative of x at 0. Wat do you get if you take te limit of te symmetric difference quotient for x at 0? (Hint: take te leftand rigt-and limits wen you compute te derivative from te definition.) Conclusions? Since te derivative is a limit, x can approac a from bot te rigt (x > a; see figure 1b) and te left (x < a), so it makes sense to ten talk about left- and rigt-and derivatives. If f ' (a) exists, te left- and rigt-and derivatives at a are equal, and all 3 derivatives equal te slope of te tangent line at (a, f (a). (See figure 1.) Sometimes x cannot approac a from bot sides. If ( ) 4 f( x) = x + x, te derivative only exists at 0 from te left. (Wy?) Oter times, te function may not even ave a derivative, aving different left- and rigt-and derivatives. (Can you tink of an example or two of suc a function?) And even if a function as a derivative for all x, if te function is piecewise-defined, it would be necessary to compute te left- and rigt-and derivatives separately at any point were te function definition canges from one piece to te next. Tus, te following definitions are necessary: Te rigt-and derivative of f at a, often designated as f ( a ) +, is defined as f( a+ ) f( a) lim. Similarly, te left-and derivative, f ( a), can be defined by eiter + 0 f( a+ ) f( a) f( a) f( a ) lim or lim. + 0 0 Exercise 15: Compute te left-and derivative for ( ) ( ) 4 f x = x + x at 0. ln( x),0 < x< 1 Exercise 16: For f( x) = 2, find te left- and rigt-and derivatives at 1 ( x 2) 1, x 1 and let tem tell you weter te derivative of f exists at 1. Draw te grap of te derivative. Describe its beavior at 1. Conclusions? Calculus Generic Scope and Sequence Topics: Derivatives NCTM Standards: Number and operations, Algebra, Geometry, Measurement, Connections, Communication, Representation