Capter 2 Te Derivative Applied Calculus 80 Section 2: Te Derivative Definition of te Derivative Suppose we drop a tomato from te top of a 00 foot building and time its fall. Time (sec) Heigt (ft) 0.0 00 0.5 96.0 84.5 64 2.0 36 2.5 0 Some questions are easy to answer directly from te table: (a) How long did it take for te tomato n to drop 00 feet? (b) How far did te tomato fall during te first second? (c) How far did te tomato fall during te last second? (d) How far did te tomato fall between t =.5 and t =? (2.5 seconds) (00 84 = 6 feet) (64 0 = 64 feet) (96 84 = 2 feet) Some oter questions require a little calculation: (e) Wat was te average velocity of te tomato during its fall? Average velocity = distance fallen total time = position time = 00 ft 2.5 s = 40 ft/s. (f) Wat was te average velocity between t= and t=2 seconds? Average velocity = position time = 36 ft 84 ft 2 s s = 48 ft s = 48 ft/s. Some questions are more difficult. (g) How fast was te tomato falling second after it was dropped? Tis question is significantly different from te previous two questions about average velocity. Here we want te instantaneous velocity, te velocity at an instant in time. Unfortunately te tomato is not equipped wit a speedometer so we will ave to give an approximate answer. One crude approximation of te instantaneous velocity after second is simply te average velocity during te entire fall, 40 ft/s. But te tomato fell slowly at te beginning and rapidly near te end so te " 40 ft/s" estimate may or may not be a good answer. Tis capter is (c) 203. It was remixed by David Lippman from Sana Calaway's remix of Contemporary Calculus by Dale Hoffman. It is licensed under te Creative Commons Attribution license.
Capter 2 Te Derivative Applied Calculus 8 We can get a better approximation of te instantaneous velocity at t= by calculating te average velocities over a sort time interval near t =. Te average velocity between t = 0.5 2 feet and t = is 0.5 s = 24 ft/s, and te average velocity between t = and t =.5 is 20 feet.5 s = 40 ft/s so we can be reasonably sure tat te instantaneous velocity is between 24 ft/s and 40 ft/s. In general, te sorter te time interval over wic we calculate te average velocity, te better te average velocity will approximate te instantaneous velocity. Te average velocity over a time interval is position time, wic is te slope of te secant line troug two points on te grap of eigt versus time. Te instantaneous velocity at a particular time and eigt is te slope of te tangent line to te grap at te point given by tat time and eigt. Average velocity = position time = slope of te secant line troug 2 points. Instantaneous velocity = slope of te line tangent to te grap. GROWING BACTERIA Suppose we set up a macine to count te number of bacteria growing on a Petri plate. At first tere are few bacteria so te population grows slowly. Ten tere are more bacteria to divide so te population grows more quickly. Later, tere are more bacteria and less room and nutrients available for te expanding population, so te population grows slowly again. Finally, te bacteria ave used up most of te nutrients, and te population declines as bacteria die.
Capter 2 Te Derivative Applied Calculus 82 Te population grap can be used to answer a number of questions. (a) Wat is te bacteria population at time t = 3 days? From te grap, at t = 3, te population is about 0.5 tousand, or 500 bacteria. (b) Wat is te population increment from t = 3 to t =0 days? At t = 0, te population is about 4.5 tousand, so te increment is about 4000 bacteria (c) Wat is te rate of population growt from t = 3 to t = 0 days? Te rate of growt from t = 3 to t = 0 is te average cange in population during tat time: cange in population average cange in population = cange in time = population time 4000 bacteria = 7 days 570 bacteria/day. Tis is te slope of te secant line troug te two points (3, 500) and (0, 4500). (d) Wat is te rate of population growt on te tird day, at t = 3? Tis question is asking for te instantaneous rate of population cange, te slope of te line wic is tangent to te population curve at (3, 500). If we sketc a line approximately tangent to te curve at (3, 500) and pick two points near te ends of te tangent line segment, we can estimate tat instantaneous rate of population growt is approximately 320 bacteria/day.
Capter 2 Te Derivative Applied Calculus 83 Tangent Lines Do tis! Te grap below is te grap of y = f ( x). We want to find te slope of te tangent line at te point (, 2). First, draw te secant line between (, 2) and (2, ) and compute its slope. Now draw te secant line between (, 2) and (.5, ) and compute its slope. Compare te two lines you ave drawn. Wic would be a better approximation of te tangent line to te curve at (, 2)? Now draw te secant line between (, 2) and (.3,.5) and compute its slope. Is tis line an even better approximation of te tangent line? Now draw your best guess for te tangent line and measure its slope. Do you see a pattern in te slopes? You sould ave noticed tat as te interval got smaller and smaller, te secant line got closer to te tangent line and its slope got closer to te slope of te tangent line. Tat s good news we know ow to find te slope of a secant line. In some applications, we need to know were te grap of a function f(x) as orizontal tangent lines (slopes = 0). Example At rigt is te grap of y = g(x). At wat values of x does te grap of y = g(x) below ave orizontal tangent lines? Te tangent lines to te grap of g(x) are orizontal (slope = 0) wen x,, 2.5, and 5.
Capter 2 Te Derivative Applied Calculus 84 Let's explore furter tis idea of finding te tangent slope based on te secant slope. Example 2 Find te slope of te line L in te grap below wic is tangent to f(x) = x 2 at te point (2,4). We could estimate te slope of L from te grap, but we won't. Instead, we will use te idea tat secant lines over tiny intervals approximate te tangent line. We can see tat te line troug (2,4) and (3,9) on te grap of f is an approximation of te slope of te tangent line, and we can calculate tat slope exactly: m = y/ x = (9 4)/(3 2) = 5. But m = 5 is only an estimate of te slope of te tangent line and not a very good estimate. It's too big. We can get a better estimate by picking a second point on te grap of f wic is closer to (2,4) te point (2,4) is fixed and it must be one of te points we use. From te second figure, we can see tat te slope of te line troug te points (2,4) and (2.5,6.25) is a better approximation of te slope of te tangent line at (2,4): y 6.25 4 2.25 m = = = = 4.5 x 2.5 2 0.5 a better estimate, but still an approximation. We can continue picking points closer and closer to (2,4) on te grap of f, and ten calculating te slopes of te lines troug eac of tese points and te point (2,4): Points to te left of (2,4) Points to te left of (2,4) x y = x 2 Slope x y = x 2 Slope.5 2.25 3.5 3 9 5.9 3.6 3.9 2.5 6.25 4.5.99 3.960 3.99 2.0 4.040 4.0 Te only ting special about te x values we picked is tat tey are numbers wic are close, and very close, to x = 2. Someone else migt ave picked oter nearby values for x. As te points we pick get closer and closer to te point (2,4) on te grap of y = x 2, te slopes of te lines troug te points and (2,4) are better approximations of te slope of te tangent line, and tese slopes are getting closer and closer to 4.
Capter 2 Te Derivative Applied Calculus 85 We can bypass muc of te calculating by not picking te points one at a time: let's look at a general point near (2,4). Define x = 2 + so is te increment from 2 to x. If is small, 2 2 +, f(2 + ) = 2 +,(2 + ) is close to (2,4). ten x = 2 + is close to 2 and te point ( ) ( ) 2 Te slope m of te line troug te points (2,4) and ( 2,(2 ) ) of te slope of te tangent line at te point (2,4): + + is a good approximation 2 ( ) 2 4 4 4 2 y (2 + ) 4 + + 4+ m= = = = = 4 + x (2 + ) 2 Te value m = 4 + is te slope of te secant line troug te two points (2,4) and 2,(2 ) 2 + +. As gets smaller and smaller, tis slope approaces te slope of te tangent ( ) line to te grap of f at (2,4). y More formally, we could write: Slope of te tangent line = lim = lim(4 + ) x We can easily evaluate tis limit using direct substitution, finding tat as te interval srinks towards 0, te secant slope approaces te tangent slope, 4. Te tangent line problem and te instantaneous velocity problem are te same problem. In eac problem we wanted to know ow rapidly someting was canging at an instant in time, and te answer turned out to be finding te slope of a tangent line, wic we approximated wit te slope of a secant line. Tis idea is te key to defining te slope of a curve.
Capter 2 Te Derivative Applied Calculus 86 Te Derivative: Te derivative of a function f at a point (x, f(x)) is te instantaneous rate of cange. Te derivative is te slope of te tangent line to te grap of f at te point (x, f(x)). Te derivative is te slope of te curve f(x) at te point (x, f(x)). A function is called differentiable at (x, f(x)) if its derivative exists at (x, f(x)). Notation for te Derivative: Te derivative of y = f(x) wit respect to x is written as f '( x) (read aloud as f prime of x ), or y ' ( y prime ) dy df or (read aloud as dee wy dee ex ), or Te notation tat resembles a fraction is called Leibniz notation. It displays not only te name of te function (f or y), but also te name of te variable (in tis case, x). It looks y like a fraction because te derivative is a slope. In fact, tis is simply written in Roman x letters instead of Greek letters. Verb forms: We find te derivative of a function, or take te derivative of a function, or differentiate a function. dy We use an adaptation of te notation to mean find te derivative of f(x): d ( f ( x) ) = df Formal Algebraic Definition: f ( x + ) f ( x) f '( x) = lim 0 Practical Definition: Te derivative can be approximated by looking at an average rate of cange, or te slope of a secant line, over a very tiny interval. Te tinier te interval, te closer tis is to te true instantaneous rate of cange, slope of te tangent line, or slope of te curve. Looking Aead: We will ave metods for computing exact values of derivatives from formulas soon. If te function is given to you as a table or grap, you will still need to approximate tis way. Tis is te foundation for te rest of tis capter. It s remarkable tat suc a simple idea (te slope of a tangent line) and suc a simple definition (for te derivative f ' ) will lead to so many important ideas and applications.
Capter 2 Te Derivative Applied Calculus 87 Example 3 Find te slope of te tangent line to f( x) = wen x = 3. x Te slope of te tangent line is te value of te derivative f (3). f (3) =, and 3 f(3 + ) = 3 + Using te formal limit definition of te derivative, f(3 + ) f(3) f (3) = lim = lim 3+ 3 We can simplify by giving te fractions a common denominator. 3 3+ lim 3 3 3 + 3 + 3 3+ lim 9 3 = + 9 + 3 = lim 9 + 3 = lim 9 + 3 = lim 9 + 3 We can evaluate tis limit by direct substitution: lim = 9 + 3 9 Te slope of te tangent line to f( x) = at x = 3 is x 9 Te Derivative as a Function We now know ow to find (or at least approximate) te derivative of a function for any x-value; tis means we can tink of te derivative as a function, too. Te inputs are te same x s; te output is te value of te derivative at tat x value.
Capter 2 Te Derivative Applied Calculus 88 Example 4 Below is te grap of a function y = f ( x) table sowing values of f '( x) :. We can use te information in te grap to fill in a At various values of x, draw your best guess at te tangent line and measure its slope. You migt ave to extend your lines so you can read some points. In general, your estimate of te slope will be better if you coose points tat are easy to read and far away from eac oter. Here are my estimates for a few values of x (parts of te tangent lines I used are sown): x y = f ( x) f '( x) = te estimated SLOPE of te tangent line to te curve at te point ( x, y). 0 0 0 2 0 3 0 3.5 0 2 4 5 2 0.5 We can estimate te values of f (x) at some non-integer values of x, too: f (.5) 0.5 and f (.3) 0.3. We can even tink about entire intervals. For example, if 0 < x <, ten f(x) is increasing, all te slopes are positive, and so f (x) is positive. Te values of f (x) definitely depend on te values of x, and f (x) is a function of x. We can use te results in te table to elp sketc te grap of f (x).
Capter 2 Te Derivative Applied Calculus 89 Example 5 Sown is te grap of te eigt (t) of a rocket at time t. Sketc te grap of te velocity of te rocket at time t. (Velocity is te derivative of te eigt function, so it is te slope of te tangent to te grap of position or eigt.) We can estimate te slope of te function at several points. Te lower grap below sows te velocity of te rocket. Tis is v(t) = (t). We can also find derivative functions algebraically using limits. Example 6 d 2 Find ( 2x 4x+ ) Setting up te derivative using a limit, f( x+ ) f( x) f ( x) = lim We will start by simplifying f( x+ ). 2 f( x+ ) = 2( x+ ) 4( x+ ) Expand
Capter 2 Te Derivative Applied Calculus 90 = x + x + x + = + + 2 2 2( 2 ) 4( ) 2 2 2x 4x 2 4x 4 Now finding te limit, f( x+ ) f( x) f ( x) = lim Substitute in te formulas ( 2x 2 + 4x + 2 2 4x 4 ) ( 2x 2 4x ) = lim Now simplify 0 2 2 2 2x + 4x+ 2 4x 4 2x + 4x+ = lim 2 4x + 2 4 = lim ( 4x+ 2 4) = lim Factor out te and simplify = lim 4x+ 2 4 ( ) We can find te limit of tis expression by direct substitution: f ( x) = lim 4x+ 2 4 = 4x 4 ( ) Notice tat te derivative depends on x, and tat tis formula will tell us te slope of te tangent line to f at any value x. For example, if we wanted to know te tangent slope of f at x = 3, we would simply evaluate: f (3) = 4 3 4 = 8. A formula for te derivative function is very powerful, but as you can see, calculating te derivative using te limit definition is very time consuming. In te next section, we will identify some patterns tat will allow us to start building a set of rules for finding derivatives witout needing te limit definition. Interpreting te Derivative So far we ave empasized te derivative as te slope of te line tangent to a grap. Tat interpretation is very visual and useful wen examining te grap of a function, and we will continue to use it. Derivatives, owever, are used in a wide variety of fields and applications, and some of tese fields use oter interpretations. Te following are a few interpretations of te derivative tat are commonly used.
Capter 2 Te Derivative Applied Calculus 9 General Rate of Cange: f '(x) is te rate of cange of te function at x. If te units for x are years and te units for f(x) are people, ten te units for df people are year, a rate of cange in population. Grapical Slope: f '(x) is te slope of te line tangent to te grap of f at te point ( x, f(x) ). Pysical Velocity: If f(x) is te position of an object at time x, ten f '(x) is te velocity of te object at time x. If te units for x are ours and f(x) is distance measured in miles, ten te units for f '(x) = df miles are our, miles per our, wic is a measure of velocity. Acceleration: If f(x) is te velocity of an object at time x, ten f '(x) is te acceleration of te object at time x. If te units are for x are ours and f(x) as te units miles our, ten te units for te acceleration f '(x) = df our per our. miles/our are our = miles our 2, miles per Business Marginal Cost, Marginal Revenue, and Marginal Profit: We'll explore tese terms in more dept later in te section. Basically, te marginal cost is approximately te additional cost of making one more object once we ave already made x objects. If te units for x are bicycles and te units for f(x) are dollars, ten te units for f '(x) = df are dollars, te cost per bicycle. bicycle In business contexts, te word "marginal" usually means te derivative or rate of cange of some quantity. Example 7 Suppose te demand curve for widgets was given by D( p) =, were D is te quantity of p items widgets, in tousands, at a price of p dollars. Interpret te derivative of D at p = $3. Note tat we calculated D (3) earlier to be D (3) = 0.. 9
Capter 2 Te Derivative Applied Calculus 92 Since D as units "tousands of widgets" and te units for p is dollars of price, te units for D tousands of widgets will be. In oter words, it sows ow te demand will cange as te dollar of price price increases. Specifically, D (3) 0. tells us tat wen te price is $3, te demand will decrease by about 0. tousand items for every dollar te price increases. 2.2 Exercises. Wat is te slope of te line troug (3,9) and (x, y) for y = x 2 and x = 2.97? x = 3.00? x = 3+? Wat appens to tis last slope wen is very small (close to 0)? Sketc te grap of y = x 2 for x near 3. 2. Wat is te slope of te line troug ( 2,4) and (x, y) for y = x 2 and x =.98? x = 2.03? x = 2+? Wat appens to tis last slope wen is very small (close to 0)? Sketc te grap of y = x 2 for x near 2. 3. Wat is te slope of te line troug (2,4) and (x, y) for y = x 2 + x 2 and x =.99? x = 2.004? x = 2+? Wat appens to tis last slope wen is very small? Sketc te grap of y = x 2 + x 2 for x near 2. 4. Wat is te slope of te line troug (, 2) and (x, y) for y = x 2 +x 2 and x =.98? x =.03? x = +? Wat appens to tis last slope wen is very small? Sketc te grap of y = x 2 + x 2 for x near. 5. Te grap to te rigt sows te temperature during a day in Ames. (a) Wat was te average cange in temperature from 9 am to pm? (b) Estimate ow fast te temperature was rising at 0 am and at 7 pm?
Capter 2 Te Derivative Applied Calculus 93 6. Te grap sows te distance of a car from a measuring position located on te edge of a straigt road. (a) Wat was te average velocity of te car from t = 0 to t = 30 seconds? (b) Wat was te average velocity of te car from t = 0 to t = 30 seconds? (c) About ow fast was te car traveling at t = 0 seconds? at t = 20 s? at t = 30 s? (d) Wat does te orizontal part of te grap between t = 5 and t = 20 seconds mean? (e) Wat does te negative velocity at t = 25 represent? 7. Te grap sows te distance of a car from a measuring position located on te edge of a straigt road. (a) Wat was te average velocity of te car from t = 0 to t = 20 seconds? (b) Wat was te average velocity from t = 0 to t = 30 seconds? (c) About ow fast was te car traveling at t = 0 seconds? at t = 20 s? at t = 30 s? 8. Te grap sows te composite developmental skill level of cessmasters at different ages as determined by teir performance against oter cessmasters. (From "Rating Systems for Human Abilities", by W.H. Batcelder and R.S. Simpson, 988. UMAP Module 698.) (a) (b) (c) (d) At wat age is te "typical" cessmaster playing te best cess? At approximately wat age is te cessmaster's skill level increasing most rapidly? Describe te development of te "typical" cessmaster's skill in words. Sketc graps wic you tink would reasonably describe te performance levels versus age for an atlete, a classical pianist, a rock singer, a matematician, and a professional in your major field.
Capter 2 Te Derivative Applied Calculus 94 0. Use te function in te grap to fill in te table and ten grap m(x). x y = f(x) m(x) = te estimated slope of te tangent line to y=f(x) at te point (x,y) 0 0.5.0.5 2.0 2.5 3.0 3.5 4.0. Use te function in te grap to fill in te table and ten grap m(x). x y = g(x) m(x) = te estimated slope of te tangent line to y=g(x) at te point (x,y) 0 0.5.0.5 2.0 2.5 3.0 3.5 4.0 2. (a) At wat values of x does te grap of f in te grap ave a orizontal tangent line? (b) At wat value(s) of x is te value of f te largest? smallest? (c) Sketc te grap of m(x) = te slope of te line tangent to te grap of f at te point (x,y).
Capter 2 Te Derivative Applied Calculus 95 3. (a) At wat values of x does te grap of g ave a orizontal tangent line? (b) At wat value(s) of x is te value of g te largest? smallest? (c) Sketc te grap of m(x) = te slope of te line tangent to te grap of g at te point (x,y). 4. Matc te situation descriptions wit te corresponding time velocity grap. (a) A car quickly leaving from a stop sign. (b) A car sedately leaving from a stop sign. (c) A student bouncing on a trampoline. (d) A ball trown straigt up. (e) A student confidently striding across campus to take a calculus test. (f) An unprepared student walking across campus to take a calculus test. For eac function f(x) in problems 5 20, perform steps (a) (d): f( x+ ) f( x) (a) calculate m sec = and simplify (b) determine m tan = lim m 0 sec (c) evaluate m tan at x = 2, (d) find te equation of te line tangent to te grap of f at (2, f(2) ) 5. f(x) = 3x 7 6. f(x) = 2 7x 7. f(x) = ax + b were a and b are constants 8. f(x) = x 2 + 3x 9. f(x) = 8 3x 2 20. f(x) = ax 2 + bx + c were a, b and c are constants 2. Matc te graps of te tree functions below wit te graps of teir derivatives.
Capter 2 Te Derivative Applied Calculus 96 22. Below are six graps, tree of wic are derivatives of te oter tree. Matc te functions wit teir derivatives. 23. Te grap below sows te temperature during a summer day in Cicago. Sketc te grap of te rate at wic te temperature is canging. (Tis is just te grap of te slopes of te lines wic are tangent to te temperature grap.) 24. Fill in te table wit te appropriate units for f '(x). units for x units for f(x) units for f '(x) ours miles people automobiles dollars pancakes days trout seconds miles per second seconds gallons study ours test points 25. If C(x) is te total cost, in millions, of producing x tousand items, interpret C (4) = 2. 26. Suppose P(t) is te number of individuals infected by a disease t days after it was first detected. Interpret P (50) = 200.