4.1 The Derivative: Rates of Change, Velocity and Slope

Transcription

1 Chapter 4 The Derivative The earlier chapters are the analytical prelues to calculus. This chapter begins the stuy calculus proper, starting with the stuy of ifferential calculus, also known as the calculus of erivatives. We will evelop all of the funamental erivative computations in this chapter. Once we complete our evelopment of erivatives an see many of their most immeiate applications in this chapter an in Chapter 5, we will then look towars integral calculus, where roughly speaking we see how to reverse what we o here. Integral calculus begins with Chapter 6, with the avance computational techniques being introuce in Chapter 7. Subsequent chapters evelop several topics which are either offshoots of ifferential an integral calculus, or are greatly extene by these. As we will see, ifferential calculus aresses rates at which quantities change with resepct to each other, while integral calculus aresses how quantities (or the changes in quantities) accumulate. Once we lay the founations of both ifferential an integral calculus, we will further evelop an apply both in very iverse circumstances for the remainer of the text. 4. The Derivative: Rates of Change, Velocity an Slope Suppose that we are passengers in a car riving west to east along a highway. Further suppose that we cannot see the speeometer (measuring spee) but the highway is marke at regular intervals so we can measure our position accurately. Using a stopwatch, we see that we travele a total of 30 miles in hours for the whole trip. Then we woul say our average velocity (with positive measure in the eastwar irection) was 30 mi hr = 65 mi/hr. Now suppose that uring the trip we woul like to know our actual velocity at a particular time t. The average for the whole trip oes not usually reflect the velocity at any particular time t with acceptable accuracy, since we coul have been stoppe for a break at that particular time, or speeing up to pass a truck, or even riving in reverse (for a negative velocity). One way to attempt to approximate the velocity at time t is to begin our stopwatch at t, measure how far we travele in the next minute, an calculate the average velocity for the time interval t [t, t + minute]. Here we take a function an fin its erivative. Later we take the erivative an etermine what was the unerlying function. This reverse process is often a more formiable task, as we will see in later chapters. However a thorough unerstaning of the material in this chapter greatly simplifies the learning of integral calculus. Alternatively, we have a very accurate oometer or global positioning evice in plain view. 93

2 94 CHAPTER 4. THE DERIVATIVE At this point some notation will be useful. We will take the position at time t to be s(t). It is a function, which we will call the position function; its input is time t an its output is our position at time t. We will enote a change in t by t, rea elta t. 3 With t as the initial time in our experiment to approximate velocity, an t = t + t as the final time, we see the change is inee t t = t. The average velocity over any time interval [t, t ] is thus s(t ) s(t ) = s(t + t) s(t ). t t t If we take s = s(t + t) s(t ) to be the change in s(t) which results from the change in t from t to t + t = t, then we have the average velocity also equal to ( s)/( t), i.e., s(t ) s(t ) = s(t + t) s(t ) = s t t t t. This is akin to the ol-fashione rate equals istance ivie by time that is taught in grae school. 4 With this we can get back to the problem of attempting to fin the velocity at time t. If we let t be equal to one minute, then we look to see how far we travele in that one minute, an fin the average velocity for that minute. If the velocity i not change very much in that time interval, then the average velocity will closely approximate the actual velocity, which we will enote v(t ), where v(t) is the actual velocity at time t (what we woul have rea on the speeometer except for a possible sign ifference were it available): s t = s(t + minute) s(t ) v(t ). minute On the other han, many things can happen in a minute which can cause the velocity to change significantly. Perhaps we have a true velocity of 65 miles/hour at t, but then slow to a stop at a toll booth uring that minute, an thus unacceptably unerestimate v(t ) as approximate by the average velocity for the time interval [t, t ]. If possible, it woul likely be much better to measure how far we travele in the first secon after t, since most cars cannot change velocity as significantly in such a time interval except in catastrophic circumstances (e.g., collisions). Thus 5 v(t ) s(t + secon) s(t ). secon Following the same line of thinking, it seems reasonable that we can better approximate the actual value of v(t ) by taking the average velocity over an interval [t, t + t] with smaller an smaller values of t (such as one minute, one secon, 0.00 secons, etc.). For this reason we actually efine the velocity at time t by the following, 0/0-form limit: s(t + t) s(t ) s v(t ) = lim = lim t 0 t t 0 t. Recall that we have to consier t 0 as well as t 0 + in this calculation. This is not unreasonable, as we coul also approximate v(t ) by consiering how far we went in the minute, secon, 0.00 secon, etc., ening with t. Now we state the formal efinition of velocity. 3 Note that we take t as one quantity. It is not times t. One can rea t to be synonymous with the change in t. Occasionally we will write t = ( t) to remove ambiguity an reinforce that it is one quantity. (Here is the capital Greek letter elta.) 4 The grae school formula is lacking in that it always assumes velocity is constant, an oes not istinguish between istance an isplacement, or istance an position. (Distance only carries a nonnegative sign.) It is only mentione here because of its familiarity. 5 Of course we nee to convert units to be consistent, e.g., secon = (/3600) hour, an so on.

3 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 95 t = 0 v = 0 t = v = t = v = 4 t = 3 v = 6 s(t) t = t = t = 3 v = t = 4 v = 6 Figure 4.: Here we trace the one-imensional motion s(t) = t + as a position on a number line for times t = 3,,,0,,, 3 (note the progression of those positions in the figure). The velocities v = t are also given. The graph reflects how the particle comes in from the right for negative t, stops at t = 0 (s =, v = 0), an moves back out towars the right for positive t (faster an faster as t > 0 increases). Note that no t-axis appears explicitly. Definition 4.. Given a position function s(t), efine the velocity (or instantaneous velocity, to istinguish it from average velocity) at a time t to be the function given by the limit s v(t) = lim t 0 t = lim t 0 for each t for which the limit (4.) exists, an where we also efine s(t + t) s(t), (4.) t s = s(t + t) s(t). (4.) For now it is the secon part of (4.) that will be most useful. If we are lucky enough to know an algebraic formula for s(t) as a function, then we can use the limit to calculate v(t). This escribes a situation known to physicists as one-imensional motion. Note how (4.) allows for t 0 as well as t 0 +. Note also that for continuous s(t) a reasonable assumption in classical physics limits of form (4.) will be of 0/0 form. Example 4.. Suppose that position is given by s(t) = t +. We can use (4.) to calculate the velocity function for any fixe t as follows. As this limit will be a 0/0 form, we perform algebra to attempt to can cancel the t factor in the enominator. 6 s(t + t) s(t) v(t) = lim t 0 t ( (t + t) + ) ( t + ) = lim t 0 = lim t 0 = lim t 0 = lim t 0 = lim t 0 t (t + t t + ( t) + ) (t + ) t t + t t + ( t) + t t t t + ( t) t (t + t) = t. 6 Here we treat t as a constant in the calculation of v(t); t is fixe while t 0. We showe that s(t) = t + = v(t) = t. Some position an velocity ata are given for various times t in Figure 4.. Note that when s > 0 the position is to the right of s = 0 (as is always the case here), an when s < 0 position is to the left. Also, when v > 0 the motion is to the right, an when v < 0 the motion is to the left. For example, at time t = we have the position s() = + = 5, an velocity v() = () = 4. If s is measure in meters, an t in secons, then the

4 96 CHAPTER 4. THE DERIVATIVE units in the limit (4.) are meters/secon, as we woul hope. The ability to fin a nonconstant velocity function is a tremenous leap from the grae school notion of rate = istance/time. Having limits at our isposal mae it possible. Note that v(t) is really a limit of an expression of the form position change/time change, i.e., ( s)/( t) so it has some of the spirit of the grae school notion. Such limits are useful in more than just position = velocity problems; we will have use for them throughout the text in numerous contexts. Because they are ubiquitous we generalize the notation an call the functions which arise from these limits erivatives. (The following efinition shoul be commite to memory.) Definition 4.. Given any quantity Q which is a function of the variable x, i.e., Q = Q(x). The erivative of Q with respect to x is the function Q (x), rea Q-prime of x, efine by Q Q(x + x) Q(x) (x) = lim (4.3) x 0 x wherever that limit exists an is finite. If this limit oes not exist or is infinite at a given x 0, we say Q (x 0 ) oes not exist. If the limit oes exist as a finite number at x = x 0, we say Q(x) is ifferentiable at x 0. Q (x) is also calle the instantaneous rate of change of Q(x) with respect to x. 7 To efine the erivative Q (x) at a given value x, we require not only that the limit (4.3) exists, but also that it is finite (i.e., exists as a real number). We will make more use of the term ifferentiable in later sections where its justification is clearer. Note that in most cases we expect the limit (4.3) which efines the erivative to be of 0/0 form, requiring the usual techniques of algebraic simplification to compute. Definition 4..3 We also efine the average rate of change over an interval as before: if the initial value of x is x 0 (pronounce x-naught or x sub(script) zero ), an the final value is x f, then the average rate of change of Q(x) with respect to x for x [x 0, x f ] or x [x f, x 0 ] (epening upon whether x 0 < x f or x 0 > x f ) is given by the ifference quotient Q(x f ) Q(x 0 ) = Q(x 0 + x) Q(x 0 ) = Q x f x 0 x x (4.4) where x = x f x 0, (4.5) Q = Q(x f ) Q(x 0 ) = Q(x 0 + x) Q(x 0 ). (4.6) So we see that the erivative (4.3) is just the limit of the average rate of change in Q given in (4.4) on an interval with enpoints x an x + x, assuming that limit as x 0 is finite. With this notation, we can rewrite the (instantaneous) velocity function for a given s(t) as: v(t) = s (t). (4.7) 7 Instantaneous rate of change means the rate of change at that instant, as oppose to an average rate of change of the output variable which occurs over an entire interval s length of values of the input variable.

5 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 97 Because there are so many contexts, there are many ifferent notations. They each have their places an all are worth knowing. 8 Example 4.. Suppose our car has a very accurate fuel gauge an a very accurate oometer. Let V (s) be the volume of fuel in the tank at a particular position s, as rea from the oometer. 9 Then V V (s) = lim s 0 s = lim s 0 V (s + s) V (s) s represents the instantaneous rate of fuel flow in terms of volume per unit length. If s is in units of miles an V is in units of gallons, this woul be a flow rate in gallons per mile. (If we prefer miles/gallon, we can take the reciprocal.) So the erivative can also represent flow of a flui. Notice that the fuel shoul be leaving the tank whenever the engine is running, so V shoul be ecreasing as we rive. This gives V < 0 when s > 0, giving V/ s < 0, an thus V (s) 0. However the fuel running through the engine is exactly the fuel leaving the tank, an so the actual flow rate we woul report woul be V (s), a positive quantity, for any particular position s. There are countless other applications of the erivative. All we nee is a quantity Q as a function of another quantity x, to measure the rate that Q changes as x changes. The average rate of change of Q with respect to x is again ( Q)/( x), an the instantaneous rate is the number we get when we take ( Q)/( x) an let x 0, giving the rate at that instant. 4.. Slope of a Curve All of these applications have their own interpretations. Interestingly enough, the analytic geometric interpretation of the erivative of a function unifies them all in one graphical setting. For the bulk of the remainer of this section, we will concentrate on the significance of the erivative f (x) to the graph of y = f(x). First we will consier a very simple case. Suppose where m, b R are fixe constants. Then f f(x + x) f(x) (x) = lim x 0 x = lim x 0 f(x) = mx + b, mx + m x + b mx b x [m(x + x) + b] [mx + b] = lim x 0 x = lim x 0 m x x = lim x 0 m = m. Thus, when y = f(x) is the line y = mx + b we get f (x) = m; if the function is a line then the erivative is its slope. Recall that the slope of a line measures how rapily that line rises or falls as we move along the line an to the right. In other wors, slope measures the rate of change in y with respect to x. That rate is constant on a line, but changes on most curves. Still, if we look closely at a point (a, f(a)) on the graph of y = f(x), we can often associate a slope with the curve there. 0 To 8 In a later section we will introuce the very powerful Leibniz notation for the erivative Q (x), which we will then write Q/ (notice the resemblance to Q/ x). 9 Technically oometers isplay total istance travele an not position, but here we can use the reaing as a position uner the assumption all travel is in the same, positive irection. 0 Just as a naive, an nearsighte, observation of the Earth s surface can lea us to believe the Earth is flat, if we were staning on a curve at (a, f(a)), an very focuse on the curve at an aroun that point, we might believe we are looking at a constant slope. The actual slope is what we approach when we focus more an more on that point, by letting x 0.

6 98 CHAPTER 4. THE DERIVATIVE measure this slope, again we woul if effect measure the way y = f(x) changes (instantaneously) with respect to x at x = a. With this motivation, we make the following efinition: Definition 4..4 Given a function f(x), the slope of the graph of y = f(x) any point (a, f(a)) on the graph is given by f (a), assuming this erivative exists there. It is important to note that a function an its erivative give two types of information about the graph of the function: f(x) gives the height of the graph for a particular x-value; f (x) gives the slope of the graph at that x-value. For instance, we saw in Example 4.., page 95 (using ifferent variables) that f(x) = x + = f (x) = x. When we graph y = x +, i.e., when we graph the function f(x) = x +, the function f(x) gives the height at each x, an the erivative f (x) = x gives the slope there. This is illustrate in Figure 4.. Also illustrate in that figure, an of geometric interest, is the tangent line to the graph of y = f(x) at a given point (a, f(a)) on the curve. This is just the line through (a, f(a)) with the same slope slope as the curve, i.e., with slope f (a). Definition 4..5 The line through (a, f(a)) with slope f (a), i.e., the same slope as the function at x = a, is the tangent line to the graph of y = f(x) through (a, f(a)). Three separate tangent lines are rawn in Figure 4.. A formula for the tangent line through (a, f(a)) presents itself immeiately, since we have a point (a, f(a)), an a slope f (a), the moifie point-slope form (.59), page 3 gives us: y = f(a) + f (a)(x a). (4.8) For the function f(x) = x + in Figure 4., (4.8) gives us At x = : (a, f(a)) = (, ), f (a) = f () =, so the tangent line is y = + (x ). At x = : (a, f(a)) = (, 5), f (a) = f ( ) = 4, so the tangent line is y = 5 4(x+). At x = 0: (a, f(a)) = (0, ), f (a) = f (0) = 0, so the tangent line is y = + 0(x 0), or simply y =. This tangent line form (4.8) appears repeately in this text; much of Section 5.4 is evote to it. Example 4..3 Consier the function f(x) = x +. Then, using a conjugate multiplication Recall f f(a + x) f(a) (a) exists means exactly that lim exists as a limit an is finite. x 0 x

7 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 99 f( ) = 5 f ( ) = 4 f(0) = f (0) = f() = f () = 3 3 Figure 4.: The graph of f(x) = x +, along with the tangent lines to the graph at x =,0,. A tangent line is a line through a point (a,f(a)) on the curve which has the same slope as the curve s at that point. The height at each x is given by f(x) = x +, while the slope is given by f (x) = x. 3 f ( ) oes not exist f (0) = f (4) = 3 f (6) = Figure 4.3: The graph of f(x) = x +, along with a few slopes. Notice the behavior of the slope, in particularly how f (x) = / x + as x / +, an how f (x) 0 + as x.

8 300 CHAPTER 4. THE DERIVATIVE (thir line below) we can compute: f f(x + x) f(x) (x) = lim x 0 x (x + x) + x + = lim x 0 x (x + x) + x + = lim x 0 x = lim x 0 = lim x 0 = lim x 0 = lim x 0 ((x + x) + ) (x + ) ( (x ) x + x) + + x + x + x + x ( (x ) x + x) + + x + x ( (x ) x + x) + + x + ( (x + x) + + x + ) = To summarize, f(x) = x + = f (x) = x +. (x + x) + + x + (x + x) + + x + x + = x +. We can make several observations about the form of this erivative. First, note that f( /) = 0 exists, but f ( /) oes not. Of course for x = / we cannot take x 0 or we woul be taking square roots of negative numbers. Still, it is interesting to notice that f (x) as x / +. Furthermore, as x, we have f (x) 0 +, so the function becomes less slope as we move x farther to the right. This is all reflecte in the graph, as illustrate in Figure 4.3.

9 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 30 f (/) = 4 f () = f () = /4 3 f ( ) = /4 f ( ) = 3 f ( /) = 4 Figure 4.4: Illustration of the graph of f(x) = /x, from Example 4..4 showing also ata from its erivative f (x) = /x. From the graph or from the formula for f (x), we can notice that f (x) < 0 for all x 0, an observe the behavior of f (x) as x, as x, an as x 0. Example 4..4 Consier the function f(x) = x. We will fin its slope everywhere that it is efine. The metho below involves multiplying the numerator an enominator by a factor which will remove the fractions in the numerator. f f(x + x) f(x) (x) = lim x 0 x = lim x 0 x+ x x x x = lim x 0 ( x)(x)(x + x) Summarizing, f(x) = x = f (x) = x. x(x + x) x(x + x) = lim x 0 x+ x = lim x x 0 x x (x + x) ( x)(x)(x + x) = lim x 0 x(x + x) = x. We see some interesting features of this erivative as well. For instance, it is always negative, so the graph is always sloping ownwars, where the erivative exists (an the function is efine). Furthermore, it is the same at x = a as x = a, which is a result of the symmetry. Finally, we note that f (x) 0 as x ±, an f (x) as x 0. This is inee reflecte in the graph in Figure Marginal Cost As we will see, the erivative has many applications. While geometrically it represents the slope of the curve of a function, we have alreay seen it can represent any instantaneous rate of change of a quantity Q(x) with respect to its input variable x. A very goo illustration is

10 30 CHAPTER 4. THE DERIVATIVE an object s velocity v(t) = s (t) when the function s(t) is its position an t is time, but another very ifferent example was V (s) measuring the change of volume V in a vehicle s fuel tank as its position s varies. Nearly any fiel which eals with numerical quantities has some use for the erivative, which measures how those quantities change with respect to each other. One such fiel is business economics. For example, in economics the precise efinition of marginal cost is the cost of the next item, or (x+)st item, after x items are alreay prouce an their costs for prouction alreay pai. In function notation, if C(x) is the total cost of proucing x items, then (upon reflection) it is clear that the marginal cost at that level that is, the cost of the (x + )st item woul be C(x + ) C(x). One might assume that with the infrastructure require to prouce the first x items alreay in place, this next item woul be relatively inexpensive though probably not free to prouce. However there are exceptions to this, where it coul be much more expensive than the previous items if more infrastructure is suenly neee to prouce that next item. For instance, if one jetliner can carry a maximum of 400 passengers, the 400th may be nearly free for the airline to seat an fly (after the previous 399 are provie for), but the 40st woul likely be very expensive since another plane is require (even if ultimately the 40 passengers woul be ivie more evenly between the planes). That woul be a case where C(40) C(400) is quite large. In practice, rather than computing the actual marginal cost C(x + ) C(x), a proxy (substitute) for this is often use, usually the instantaneous rate of change of total cost C(x) with respect to the number of units x, that is, C (x), where C(x) is a function whose formula seems to make computational sense for x on actual intervals (an not just at nonnegative integer values). Thus many textbooks instea make the following efinition (without the parenthetical): Definition 4..6 If C(x) is the total cost function for proucing x items, then the (proxy) marginal cost function is given by C (x). This assumes C (x) makes sense as a function, usually requiring some formula for C(x). Then C (x) will have imensions which are units of money per units of items prouce; if C is In many fiels there will be a precise efinition an a ifferent, working efinition which itself may change from context to context. For example, in statistics a numerical atum sai (as a working efinition) to be in the 75th percentile is often unerstoo to be one for which 75% of the other ata is below it. Elsewhere being in the 75th percentile is escribe as meaning 5% of the other ata is above it. However these cannot both be exact because if 75% is below an 5% is above, when we a in the atum in question, we woul have over 00% of our ata, which is impossible. If the ata set is very large, so that one atum is much less than % of the total observations, the iscrepancy is minor enough to ignore. For a much smaller sample the iscrepancy is large. (The thir highest of four ata coul be escribe as being in the 50th percentile by one efinition, an 75th in the other.) Often one measurement is use as a proxy for the measurement which matches the exact efinition, an if that proxy is use more an more, it is sometimes given as the efinition. This is inee the case with marginal cost; there are textbooks which give only the proxy, erivative efinition. The two efinitions are connecte by the approximation below, in which the precise efintion of marginal cost at level x is on the left-han sie, an its usual proxy which is often given as the efinition is on the right in the following: C(x + ) C(x) C (x). {z } {z } exact efinition proxy While this may be somewhat confusing (or simply unsatisfying), it is similar to the iea that, if s (t ) = 5 ft/sec, we expect to have gone approximately 5 ft in the next secon after t, though the more exact answer woul be s (t + sec) s(t ). In a factory proucing 0,000 cars it is reasonable to assume the approximation above using C (x) is quite goo, assuming a reasonable formula for C(x) which makes sense on intervals, while an aerospace firm proucing a small number of stealth fighter jets woul likely use the exact efinition C(x + ) C(x). With the erivative rules we will erive as this chapter unfols, we will see that it is often easier an more efficient to compute s (t ) than to compute the ifference between s(t)-values at the en an start of the interval t [t, t + sec]. That is not the case here, where we fin erivatives from the limit efinition.

11 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 303 in ollars an x is in items prouce, then C (x) will be in ollars per item. This is because the limit inherits the units from the quotient in the efinition C (x) = lim x 0 C(x + x) C(x). x In contrast, the units of the actual (non-proxy) marginal cost C(x + ) C(x) will be in ollars only, though since it is for one item, it can be still consiere a ollar amount per item. Example 4..5 Consier the cost function C(x) = 3x + 90x + 500, representing the cost (in hunres of ollars) of manufacturing x cases of a particular prouct. Assume this function is vali for 0 x 5. Fin the average rate of change of cost per case as x ranges from 0 to 5 cases. Also fin the instantaneous rate of change in cost when 0 cases are manufacture. Solution: The average rate of change of cost per case on [0, 5] will be given by the ifference quotient (C(5) C(0))/(5 0). (See Equation 4.4 on page 96.) Thus we compute: C(5) = 3(5) + 90(5) = 75, C(0) = 3(0) + 90(0) = 00, an so the average rate of change of cost per case for 0 x 5 is C(5) C(0) 5 0 = = 75 5 = 5. Since C is in hunres of ollars, an the quantities in the enominator above are in cases of prouct, it follows that the units of this final quantity will be in hunres of ollars per case. We coul report our answer as 5 hunre ollars per case, or $500/case. To compute the instantaneous rate at x = 0, we will compute C (x) in the abstract an input x = 0 (instea of computing the special case C (0) irectly, for which C(x + x) woul be C(0 + x), an C(x) woul be C(0)). C C(x + x) C(x) (x) = lim x 0 [ x 3(x + x) + 90(x + x) ] [ 3x + 90x ] = lim x 0 = lim x 0 = lim x 0 = lim x 0 = lim x 0 = lim x 0 = C (x) = 6x [ ( x 3 x + x x + ( x) ) + 90x + 90 x ] + 3x 90x 500 x 3x 6x x 3( x) + 90x + 90 x x 90x 500 x 6x x 3( x) + 90 x x ( x)[ 6x 3 x + 90] x [ 6x 3 x + 90] Thus C (0) = 6(0) + 90 = 30. Because of the form of the limit, since C(x + x) an C(x) are in hunres of ollars an x is in cases (of prouct), the quotients above are all in units of hunres of ollars/case, an so therefore is the limit C (x). Thus C (0) = 30 hunres of ollars/case, an so we can instea report that the instantaneous rate of change at x = 0 cases is $3000/case.

12 304 CHAPTER 4. THE DERIVATIVE The erivative computation above may seem teious, but fortunately techniques of later sections will make the erivation C(x) = 3x + 90x = C (x) = 6x + 90 as simple as writing just that. In Exercise 8 of this section it is note that there is a general formula for the erivative of a function of the form f(x) = ax + bx + c, epening only upon the coefficients. That an other rules for erivatives will make computations such as the one above almost trivial. We shoul note that with the exact efinition, at x = 0 we woul compute the marginal cost to be (in hunres of ollars per unit) C() C(0) = [ 3() + 90() ] [ 3(0) + 90(0) ] = 7 00 = 7. This is well approximate by our computation of C (0) = 30 above. Economics being rather far from an exact science anyhow, using C (0) = 6(0) + 90 = 30 seems like a reasonable proxy for the computation of the exact marginal cost of C() C(0) = 7 given above. From time to time we will make some use out of the iea that for small x, f(x + x) f(x) x f (x), epening upon how small is x, an upon how quickly (f(x + x) f(x))/ x f (x) as x 0. While the notation will change, this iea is present in many contexts in this text. A special case of this, then, is with the marginal cost but with x = item: C(x + ) C(x) C (x). The computation above woul automatically contain the units of money/item, which is unerstoo when the question is aske, how much woul the next prouct cost? but flows better from the approximation above (by the presence of the enominator, even if it is simply ). It shoul also be pointe out that there are similar, relate quantities in economics, such as total revenue an marginal revenue, an total profit an marginal profit. Whether total or marginal, it is assume that profit is the ifference between revenue an cost. Hence one often sees the equations P = R C an P = R C, relating the total versions of profit, revenue an cost, an also the erivative proxies for the marginal versions of profit, revenue an cost Some Futher Applications In Definition 4.. page 96 we saw that we can consier Q (x) as the erivative of Q(x) with respect to x, whatever Q an x represent, an Q (x) is given by (4.3): Q (x) = lim x 0 Q(x + x) Q(x). x The real power of the efinition lies in the interpretation an intuition in the meaning of the erivative Q (x), as the instantaneous rate of change of Q(x) with respect to x. So far we have ha four applications: velocity v(t) = s (t), the instantaneous rate of change in position s(t) with respect to the time t,

13 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 305 the rate V (s) of fuel volume change with respect to position (or istance travele) s, the slope f (x) of the graph of y = f(x) at a given value of x, which also represents the change in the height of the function f(x) with respect to the horizontal position x, an the (proxy) marginal cost C (x), measuring the rate of change in cost with respect to the number of items prouce. Throughout this chapter an the next we will consier further contexts for the erivative. Inee, the erivative arguably offers a unique insight into each setting, though all of these insights are ultimately unifie mathematically in the limit formula for Q (x), as the instantaneous rate of change of Q(x) with respect to x. As with any application problem, some care must be taken to be sure that the natural quantities consiere are the same quantities which are usually consiere in the common conversation. For instance, recall that if the volume of fuel in a tank is V (s), the fuel flow is actually V (s). (See Example 4.., page 97.) Example 4..6 Suppose a liqui is store in an inverte conical container where the height of the cone is twice its raius. Fin a formula for the instantaneous rate of change of the volume V with respect to the height h of the liqui in the cone. Solution: What we seek here is V (h), an so we first nee to fin a formula for V (h). In general, the volume of a cone (such as that represente by the liqui in the tank) is given by V = 3 πr h, where r is the raius of the base of the cone an h is the height of the cone. For the whole tank, we woul have h = r, an so by similar triangles this will be the case regarless of the height of the flui in the cone. h r We can use this to substitute r = h. Putting these together we get V as a function of h, namely V = 3 πr h = 3 π(h/) h, or Now we compute V (h) in the usual way: V (h) = lim h 0 = lim h 0 = lim h 0 Thus V (h) = 4 πh. V (h) = πh3. V (h + h) V (h) = lim π(h + h)3 πh3 h h 0 h [ π h3 + 3h h + 3h( h) + ( h) 3 h 3 ] h [ π (3h + 3h h + ( h) )] = π (3h ) = π 4 h. [ π = lim h 0 ( h)(3h + 3h( h) + ( h) ] ) h As an example, the instantaneous rate of change of the volume V when h = 5cm is V (5 cm) = π 4 (5 cm) = 5π 4 cm 3 cm 9.6 cm3 cm. The units first appear as cm, but we write the units as cm 3 /cm because the rate shoul be in units of volume change per unit of height length change.

14 306 CHAPTER 4. THE DERIVATIVE Exercises For problems 6, use the efinition of the erivative, f (x) = lim x 0 f(x + x) f(x), x to fin f (x) for the given function.. f(x) = 5 x.. f(x) = f(x) = x f(x) = 3x 5x f(x) = x. 6. f(x) = 3 x f(x) = 9 5x. 8. f(x) = x. 9. f(x) = x. 0. f(x) = x 3/. (Hint: Rewrite as x 3.). f(x) = x 3.. f(x) = 3 x +. (Hint: We mae use of the ifference of two squares (see Section.) in Example Here you will nee to use the ifference of two cubes in a similar manner.) 3. f(x) = x f(x) = x. (Hint: easier if f(x) is x + rewritten using long ivision.) 5. f(x) = x + x. 6. f(x) = x /3. 7. Suppose s(t) = 6t + 5t + 0 escribes the height of a projectile in free fall. a. Fin the velocity function v(t), using (4.), page 95. b. What is the projectile s velocity when t = 0? t = 0? c. Fin t so that the projectile is stationary (i.e., v = 0).. How high is the projectile when it is stationary? 8. Consier the general quaratic function f(x) = ax + bx + c. a. Use the efinition of erivative to fin a formula for the erivative of the general quaratic function f(x) = ax + bx + c. b. Assuming a 0, this represents a parabola. Assuming also that the slope is zero at the vertex, fin a general formula for the x-coorinate of the vertex. c. Fin a general formula for the y- coorinate of the vertex, an thus a formula for the point (x, y) at the vertex. 9. The cost (in hunres of ollars) from manufacturing x cases of a prouct is C(x) = 8x 0.x. a. Fin the average rate of change of cost between 0 an 0 cases are manufacture. b. Fin the marginal cost when 5 cases are manufacture. c. Do the same for 8 cases. For each of the following, fin the tangent line to the graph at the given point x = a for the given function. (See (4.8), page 98 an the examples following.) 0. f(x) = x 9, a = 4.. f(x) = x 3, a =.. f(x) = x, a = f(x) = x, a = 0.

15 4.. DERIVATIVE: RATE OF CHANGE, VELOCITY, SLOPE 307 Inserte so that page numbers for Calculus stuents notes woul be correct.

16 308 CHAPTER 4. THE DERIVATIVE 4. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial function f(x), an for sinx an cosx. Along the way we will erive a few general (though not comprehensive) rules for erivatives. We will also introuce the very powerful Leibniz notation for erivatives, an show how knowing the erivative helps us to further analyze a function. One consequence is that we can more accurately graph a function s behavior by han. 4.. Positive Integer Power Rule We will often be intereste in fining erivatives of functions f(x) = x n. Fortunately there is a simple rule which covers all such functions. It is usually calle the power rule, as state below. (Recall N = {,, 3, 4, 5, 6, }.) Theorem 4.. (f(x) = x n ) (n N) = f (x) = n x n. Note that implicit in this theorem is that the erivative of x n exists for every x R i.e., exists everywhere since it is equal to nx n, efine everywhere. Proof: The proof we give here epens upon the binomial expansion (.36), page 99. It is important to remember that x is a fixe number in the limit, an x is the variable approaching zero as far as the limit is concerne. With that in min, f f(x + x) f(x) (x) = lim x 0 x (x + x) n x n = lim x 0 = lim x 0 = lim x 0 = lim x 0 ( x ) x n + nx n x + n(n )xn ( x) + + ( x) n x n x nx n x + (n)(n ) x n ( x) + + ( x) n ( nx n + = nx n = nx n, q.e.. x ) (n)(n ) x n ( x) + + ( x) n The only term which survives in the limit in the fifth line is the nx n term because the others have positive integer powers of x, which is approaching zero. We will see later that this power rule is actually much more general. In fact, it can be use for n R but we nee some more avance methos to prove such generality. For now we will apply it only to n N. Example 4.. Here we list the erivatives of some of the positive integer powers of x. The first case liste below (n = ) oes follow from the proof, though we woul be reaing the statement of the theorem for that case f(x) = x = f (x) = x 0 =. Again we o not really wish to say x 0 = regarless of x, for several technical reasons (though it is fine as long as x > 0), but we see how the formula naively gives us what we want for n =. The rest of the table is more straightforwar: f(x) x x x 3 x 4 x 00 f (x) x 3x 4x 3 00x 99

17 4.. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION Leibniz Notation We will fin that other erivative rules will be unwiely to write with our present notation. Thus we will introuce the very powerful Leibniz 3 notation an use it except in a few settings where our present (prime) notation is simpler. Definition 4.. f(x) = f (x). This is also written f(x), an sometimes shortene to f/ when it is clearly unerstoo that f is a function of x. The symbol is a ifferential operator which takes a function of x an returns the erivative with respect to x. (Thus takes a function as its input, an returns a function as its output.) When we are intereste in position an velocity, we can write s (t) = s(t) t, or when it is unerstoo that s = s(t), we might write v = s t. (4.9) Notice that the notation resembles ifference quotients, because, with our efinition of erivatives, we have f(x) = lim f(x) x 0 x, s t = lim t 0 s t. Similarly for any such relate quantities. With Leibniz notation our power rule becomes: (xn ) = nx n. (4.0) If we woul like to compute f (a), i.e., the erivative at a particular point, in the Leibniz notation we use a vertical line which is rea, evaluate at, as in f (a) = f(x). x=a So, for example, 5 = 5x4, an the slope at x = of the function f(x) = x 5 is given by 4 f () = 5 4 = 5, or 5 = 5x 4 x= = 5 4 = 5. x= ( Note how the Leibniz notation is often assume to act like a fraction: ) x 5 = 5. However the in the numerator, an (separately) the in the enominator are treate as inviolable; we o not ever break those terms up further. 3 Name for Gottfrie Wilhelm Leibniz (July, 646 November 4, 76), a German mathematician an philosopher. Most creit him an English philosopher, mathematician, physicist an theologian Sir Isaac Newton, (December 5,64 March 0, 77) with inepenently iscovering calculus. Much is written about rivalries between the Newton camp an the Leibniz camp, regaring who iscovere what first. Newton s notation for erivative use a ot above the function, as in s/t = ṡ, a notation still use in some physics textbooks. 4 Often the x = is omitte when the variable is obvious, as in 5 = 5x 4 = 5 4 = 5.

18 30 CHAPTER 4. THE DERIVATIVE Note the flexibility of the Leibniz notation in the following: 3 = 3x, u 3 u = 3u, t 3 t = 3t. These are actually the same rule (with ifferent variables): that the cube of a quantity changes with respect to that quantity at the (instantaneous) rate of 3 times the square of the quantity, be it x, u or t. Put another way, if the horizontal axis is given by t, an we graph the height t 3 on the vertical axis, then the slope is always 3t. The Leibniz notation also keeps us from making the mistake of trying to use the erivative rules (such as the power rule) to compute, for example, u3. Since the variables (u an x) o not match, the power rule cannot be use irectly. 5 With the power rule (4.0) an a few other results we can quickly calculate the erivatives of polynomials. Much of this chapter will be evote to calculating erivatives using known rules, which save an enormous amount of time when compare to calculating erivatives using limits of ifference quotients as in the previous section Sum an Constant Derivative Rules Theorem 4.. (Sum Rule) Suppose f(x) an g(x) exist. Then (f(x) + g(x)) = f(x) + g(x). (4.) In other wors the erivative of a sum is the sum of the respective erivatives. Before we give the proof, it is worth mentioning that some texts write this using the prime notation: (f + g) = f + g. Proof: Assume that f(x) an g(x) exist at a particular x. Then (f(x + x) + g(x + x)) (f(x) + g(x)) (f(x) + g(x)) = lim x 0 x f(x + x) f(x) + g(x + x) g(x) = lim x 0 ( x f(x + x) f(x) = lim + x 0 x ) g(x + x) g(x) x f(x + x) f(x) g(x + x) g(x) = lim + lim x 0 x x 0 x = f(x) + g(x), q.e.. The reason that we coul break this into two limits legitimately is because the two limits both existe an were finite by assumption. (See Theorem 3.9., page 66.) 5 Later in the text we will have the chain rule, which helps us get aroun the problem of computing u3, an similar erivatives where the variable in the numerator oes not match the variable of the enominator, i.e., the variable of the ifferential operator (here /). There we will see some of the true power of the Leibniz notation, as we compute for instance u 3 = u3 u u = 3u u. Notice how we apparently multiplie an ivie by u to achieve the secon expression.

19 4.. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 3 Example 4.. ( x 3 + x + x ) = = 3x + x +. With practice, one learns to skip the first step in the example above. Note how =, as one might hope. This reflects the fact that x an x change at the same rate (i.e., the ratio of their rates of change is always ). Put another way, the slope of the line y = x is always. The next theorem is usually given separately for emphasis. Theorem 4..3 The erivative of a constant is zero; if a function is efine by f(x) = C for all x R, where C is some fixe constant, then f (x) = 0 for all x R. Written two ifferent ways, we thus have: f(x) = C = f (x) = 0; (4.) C = 0. (4.3) There are several ways to see this. From the ifference quotient limit efinition (4.3) (page 96), regarless of x 0 the ifference quotient [f(x + x) f(x)]/ x = [C C]/ x = 0/ x = 0, so it remains zero in the limit. From another perspective regaring what we know about lines we have that f(x) = C is a line of slope m = 0. From a qualitative stanpoint this theorem is reasonable since constants have rate of change zero (hence the term constant) with respect to x. Some texts write 6 (C) = 0. With this theorem we can write, for example, ( x ) = ( x 3 ) + ( ) 8 = 3x + 0 = 3x. ( With little or no practice one learns to write simply x ) = 3x. We nee just one more result before we can fin erivatives of arbitrary polynomials. This answers the question of what to o with the coefficients of a polynomial, an multiplicative constants in general. Theorem 4..4 Multiplicative constants are preserve in the erivative. In other wors, (C f(x)) = C f(x). (4.4) The proof is left as an exercise. It follows from the fact that multiplicative constants go along for the rie in limits as well. (See again Theorem 3.9., page 66.) For a simple example, we have ( 5x 7 ) = 5 ( x 7 ) = 5 7x 6 = 35x 6. Again, with very little practice one learns to compute such a erivative in one step. Note how the erivative operator treats aitive constants (which o not survive) ifferently from 6 One weakness of Taylor s prime notation is that we o not know what variable we are taking the erivative with respect to. For instance, in an earlier example we have fuel volume V as a function of position s, an so V /s measure the flow rate of fuel per mile. However, since s = s(t), we have V = V (s(t)), so ultimately V = V (t), i.e., V can be written as a (algebraically ifferent) function of t instea, in which case we can calculate V /t, measuring the flow rate with respect to time. So when aske to calculate V, or even V (5), there is this ambiguity which is not present in the Leibniz notation. If one wrote V (s), it woul probably be unerstoo to be V/s an not V/t. Similarly, V (5 secons) woul be unerstoo to mean V/t evaluate at t = 5 secons.

20 3 CHAPTER 4. THE DERIVATIVE multiplicative constants (which o survive). We can combine the power rule (4.0), (4.3), an (4.4) to quickly compute the erivative of any given polynomial (where n N): ( an x n + a n x n + + a x ) + a x + a 0 = a n nx n + a n (n )x n + + a x + a. (4.5) To be clear on the logic, note that we first use the sum rule to break this into a sum of erivatives of the a k x k, k =,, n an a 0, calculating the erivatives of the a k x k terms in turn, each time using the fact that the ( multiplicative constants a k are along for the rie in what are otherwise simple power rules: ak x k) = a k kx k. The final term a 0 is an aitive constant with erivative zero an thus oes not appear on the right han sie of (4.5). Example 4..3 To see how (4.5) can be carrie out quickly, we list a couple of brief examples: ( 5x 4 + 9x + 3x + 47 ) = 5 4x x = 0x 3 + 8x + 3. ( 9 6x + 5x ) = 0 + ( 6) + 5 x 0 = x 0. One learns quickly to think but not necessarily write the first computational step in such problems. Note that the negative sign also goes along for the rie, since it is just a factor of. In fact we coul list a ifference rule, (f(x) g(x)) = f(x) g(x), but that woul be reunant given the sum rule, an how a multiplicative constant, even if negative, is preserve in the erivative. We nee to also point out that to use (4.5), we nee to have the function written in the form of the left han sie of that equation, i.e., expane an not left factore. Consier for instance the following example. Example 4..4 [ (x + ) ] = [ x 4 + x + ] = 4x 3 + 4x. In the above we neee to multiply out the polynomial. Thus [(x + ) ] (x + ), since we are taking the erivative with respect to x an not (x + ). It is also important to note that erivatives o not allow variable quantities to go along for [ the rie. Thus ] [ x x 3 x ] x 3. Inee, x x3 = x 3x = 3x 3, while (x x3 ) = x4 = 4x 3. Finally we point out again that it shoul be clear from the previous examples that using (4.5) for such computations is much simpler than using the original efinition of the erivative (letting x 0 in a limit of ifference quotients as in (4.3), page 96) to calculate erivatives of polynomials Increasing an Decreasing Functions; Graphing Polynomials Recall that while f(x) gives the height of the graph y = f(x) at a particular value of x, the erivative f (x) gives the slope there. If the slope is positive the graph is sloping upwars; if negative the graph is sloping ownwars. Another way to speak of such things is to iscuss functions which are increasing or ecreasing on an interval, say (a, b).

21 4.. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 33 Definition 4.. Consier a function f(x) with an interval (a, b) containe within its omain.. We say f(x) is increasing on (a, b) if an only if ( x, y (a, b))(x < y f(x) < f(y)).. We say f(x) is ecreasing on (a, b) if an only if ( x, y (a, b))(x < y f(x) > f(y)). (Note that it is possible that a function is not consistently increasing or consistently ecreasing on a given interval.) Clearly, for an increasing function on (a, b), the height increases as x increases through the interval. Similarly, for a ecreasing function on (a, b), the height ecreases as x increases through the interval. If we know exactly where a function is increasing, an where it is ecreasing, that information can be of great help in plotting or analyzing the function. To see what this has to o with erivatives we state the following theorem. Its proof relies on the Mean Value Theorem which will be introuce in a later section. However, it shoul alreay have the ring of truth given what we know of erivatives an slopes. Theorem 4..5 Suppose f(x) is efine for x (a, b), an f (x) exists for x (a, b). Then. ( x (a, b))(f (x) > 0) = f(x) is increasing on (a, b);. ( x (a, b))(f (x) < 0) = f(x) is ecreasing on (a, b). (Again, if f (x) changes sign on (a, b), then neither of these hol.) Example 4..5 To see how we might use this to graph polynomials, consier the graph of the function f(x) = x 3 3x. This function is continuous on all of R = (, ). Also notice that lim f(x) = lim [x ( 3 3x )] x x lim f(x) = lim [x ( 3 3x )] x x,. If we raw a sign chart for f(x), showing where the function is positive an where it is negative, we can get some iea of what the graph looks like. To construct a sign chart for any function we look at all the possible points where the function can change signs. Recall that the Intermeiate Value Theorem (Corollary 3.3., page 96) implies a function f(x) can only change signs, as we increase x, by either passing through zero height or having a iscontinuity. Since our particular f(x) here is continuous on all of R, we look to where f(x) = 0 to ivie R into intervals of constant sign. Now f(x) = x 3 3x = x(x 3) is zero for x = 0, ± 3. This gives us four intervals on which f(x) oes not change signs. We can test for the sign of f(x) at a single point in each interval to get the sign of f(x) on that interval. Doing so as we i in Section 3.3, we construct the sign chart for f(x): Function: f(x) = x(x 3) Test x = 0 Sign Factors: 0 Sign f(x): 3 0 3

22 34 CHAPTER 4. THE DERIVATIVE local maximum? 3 3 local minimum? Figure 4.5: Rough graph of f(x) = x 3 3x base upon its sign chart an behavior as x ±. In particular we o not know the exact locations of the local maximum(s) or minimum(s) without investigating the erivative of f(x). From the sign chart an the behavior as x ± we can get some iea of what the graph of f(x) looks like. That information is reflecte, however imprecisely, in Figure 4.5. A serious rawback to such a graph is that we know from the Extreme Value Theorem (Corollary 3.3., page 96) that there will be a value in [ 3, 0 ] which is a local maximum, an another in [ 0, 3 ] which is a local minimum, but we o not know exactly where these are from the sign chart of the function (we will formally efine the bolface terms shortly). However, a sign chart for the erivative of f(x) can possibly give us this information. Since f(x) = x 3 3x, it follows quickly that f (x) = 3x 3. Recall that on intervals where f > 0, the function f is increasing, while on those intervals on which f < 0, the function is ecreasing. Since f (x) is also an easily factore polynomial, constructing its sign chart is easy. Note f (x) = 3x 3 = 3(x ) = 3(x + )(x ) is zero exactly where x = ±. Test x = Sign f (x) = f (x) = 3(x + )(x ) 0 0 Sign f (x): Behavior of f(x): INC DEC INC ր ց ր Here we use INC to abbreviate increasing, which we also signifie by the arrow pointing upwars (ր), an we use DEC an (ց) to signify ecreasing. From this we euce that we get a local maximum at (, f( )) = (, ), an a local minimum at (, f()) = (, ). These two bits of information allow us to raw a more accurate sketch of the graph of f(x) = x 3 3x, as illustrate in Figure 4.6. That graph is computer-generate, but we can get a very accurate picture of the function s general behavior by plotting the information we have gathere: the sign

23 4.. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 35 (true) local maximum 3 3 (true) local minimum Figure 4.6: Partial graph of f(x) = x 3 3x showing the sign of f(x), the limiting behavior as x ±, an the sign of f (x) (which inicates also the locations of local extrema). The x- intercepts (where f(x) = 0), the local maximum an local minimum points are also illustrate, as are the facts that x = f(x), an x = f(x). of f, incluing the x-intercepts (where f(x) = 0), the limiting behavior of f(x) as x ±, an where f(x) is increasing/ecreasing, incluing any local maximum an minimum points. 7 It is important to istinguish the meanings of a sign chart for f(x), an one for f (x). The former just tells us where the function is below or above the x-axis; the latter tells us where the function is increasing an where the function is ecreasing. In the above we use the following terms, which we now efine: Definition 4..3 Given a function f(x).. We call a point x 0 a local maximum of f(x) if an only if ( (a, b) x 0 )( x (a, b))(f(x) f(x 0 )). (4.6). We call a point x 0 a local minimum of f(x) ( (a, b) x 0 )( x (a, b))(f(x) f(x 0 )). (4.7) In other wors, x 0 is a local maximum of f(x) if there is an open interval containing x 0 in which the function is never greater than x 0 on that interval. Local minimum is efine analogously. If f(x) is continuous in an open interval aroun x 0, an f exists in that interval, then a change of signs of f at x 0 inicates one of these local extrema. If, for instance, f > 0 to the left of x 0 an f < 0 to the right, then f increases before an ecreases after x 0, making x 0 a local maximum. This can be seen in the erivative sign chart an graph above for our example function f(x) = x 3 3x. Example 4..6 Use the erivative to etermine where the graph of f(x) = x 4 6x + 8x is increasing, an where it is ecreasing. Use this information to sketch a graph of y = f(x). 7 It is also worth noticing that f (x) = 3(x + )(x ) for both x an x, an so the slope of f(x) grows larger as x ±. This is not the case with all graphs (see Figure 4.3, page 99 for example), but it is a nice feature to notice when plotting a graph such as Figure 4.6 above.