2.11 That s So Derivative

Transcription

1 2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point in terms of te average rate of cange of te function f over related intervals. In addition, tis quantity is called te derivative of f at a, wit tis value being represented by te sortand notation f (a). Definition 2.1A. Let f be a function and x = a, a value in te function s domain. We define te derivative of f wit respect to x evaluated at x = a, denoted f (a), by te formula, provided tis limit exists. ff (aa) = lim 0 ff(aa + ) ff(aa) Aloud, you may read te symbol f (a) as eiter f -prime at a or te derivative of f evaluated at x = a. Here are some important notes to keep in mind: Te derivative of f at te value x = a is defined as te limit of te average rate of cange of f on te interval [a, a + ] as 0. It is possible for tis limit not to exist, so not every function as a derivative at every point. One says tat a function tat as a derivative at x = a is differentiable at x = a. Te derivative is a generalization of te instantaneous velocity of a position function: wen y = s (t) is a position function of a moving body, s, a tells us te instantaneous velocity of te body at time t = a. Because te units on ff(aa+) ff(aa) are units of f per unit of x, te derivative as tese

2 very same units. For instance, if s measures position in feet and t measures time in seconds, te units on s (a) are feet per second. Because te quantity ff(aa+) ff(aa) represents te slope of te line troug (a, f (a)) and (a +, f(a + )), wen you compute te derivative you are taking te limit of a collection of slopes of lines, and tus te derivative itself represents te slope of a particularly important line. As you move from an average rate of cange to an instantaneous one, tink of one point as sliding towards anoter. Investigation 1: Investigate te GeoGebra app at ttps://ggbm.at/ndej8dkj 1a) In your own words, describe wat appens to te point (a +, f(a + ) as 0. In te figure above, notice te sequence of figures wit several different lines troug te points (a, f (a)) and (a +, f(a + )), tat are generated by different values of. Tese lines (sown in te first tree figures in magenta), are often called secant lines to te curve y = f (x). A secant line to a curve is simply a line tat passes troug two points tat lie on te curve. For eac suc line, te slope of te secant line ff(aa+) ff(aa), were te value of depends on te location of te point you coose. Return to te GeoGebra app and Click on Sow secant lines. b) Wen 0, wat appens to te slope of te secant line? c) How would tis compare to a tangent line at x=a? Differential Calculus Page 2

3 It is most important to note tat f (a), te instantaneous rate of cange of f wit respect to x at x = a, also measures te slope of te tangent line to te curve y = f (x) at (a, f (a)). Te following example demonstrates several key ideas involving te derivative of a function. Example 1. For te function given by f (x) = x- x 2, use te limit definition of te derivative to compute f (2). In addition, discuss te meaning of tis value and draw a labeled grap tat supports your explanation. Solution. From te limit definition, you know tat ff ff(2+) ff(2) (2) = lim 0 Now you use te rule for f, and observe tat f (2) = = -2 and f (2 + ) = (2 + ) - (2 + ) 2. Substituting tese values into te limit definition, you ave tat ff (2) = lim 0 (2 + ) (2 + ) 2 ( 2) Observe tat wit in te denominator and our desire to let 0, you ave to wait to take te limit (tat is, you wait to actually let approac 0). Tus, you do additional algebra. Expanding and distributing te numerator results in Combining like terms, you get ff (2) = lim ff (2) = lim Next, observe tat tere is a common factor of in bot te numerator and denominator, wic allows us to simplify and find tat ff (2) = lim 0 3 Finally, you are able to take te limit as 0, and tus conclude tat f (2) = -3. Now, you know tat f (2) represents te slope of te tangent line to te curve y = x - x 2 at te point (2, -2); f (2) is also te instantaneous rate of cange of f at te point (2,- 2) Differential Calculus Page 3

4 Graping bot te function and te line troug (2, -2) wit slope m = f (2) = 3, you indeed see tat by calculating te derivative, you ave found te slope of te tangent line at tis point, as sown in te figure below. Te following investigation will elp you explore a variety of key ideas related to derivatives. Investigation 2: Consider te function f wose formula is f (x) = 3-2x. 2a) Wat familiar type of function is f? Wat can you say about te slope of f at every value of x? b) Compute te average rate of cange of f on te intervals [1, 4], [3, 7], and [5, 5 + ] ; simplify eac result as muc as possible. Wat do you notice about tese quantities? c) Use te limit definition of te derivative to compute te exact instantaneous rate of cange of f wit respect to x at te value a = 1. Tat is, compute f (1) using te limit definition. Sow your work. Is your result surprising? Witout doing any additional computations, wat are te values of f (2), f (π), and f ( 2)? Wy? Differential Calculus Page 4

5 Investigation 3: A water balloon is tossed vertically in te air from a window. Te balloon s eigt in feet at time t in seconds after being launced is given by s(t) = 16t t Use tis function to respond to eac of te following questions. 3a) Sketc an accurate, labeled grap of s on te axes provided below. You sould be able to do tis witout using computing tecnology. b) Compute te average rate of cange of s on te time interval [1, 2]. Include units on your answer and write one sentence to explain te meaning of te value you found. c) Use te limit definition to compute te instantaneous rate of cange of s wit respect to time, t, at te instant a = 1. Sow your work using proper notation, include units on your answer, and write one sentence to explain te meaning of te value you found. d) On your grap in (a), sketc two lines: one wose slope represents te average rate of cange of s on 1, 2, te oter wose slope represents te instantaneous rate of cange of s at te instant a = 1. Label eac line clearly. e) For wat values of a do you expect s (a) to be positive? Wy? Answer te same questions wen positive is replaced by negative and zero. Investigation 4: A rapidly growing city in Arizona as its population P at time t, were t is te number of decades after te year 2010, modeled by te formula PP(tt) = 25000ee tt 5. Use Differential Calculus Page 5

6 tis function to respond to te following questions. 4a) Sketc an accurate grap of P for t = 0 to t = 5 on te axes provided below. Label te scale on te axes carefully b) Compute te average rate of cange of P between 2030 and Include units on your answer and write one sentence to explain te meaning (in everyday language) of te value you found. c) Use te limit definition to write an expression for te instantaneous rate of cange of P wit respect to time, t, at te instant a = 2. Explain wy tis limit is difficult to evaluate exactly. d) Estimate te limit in (c) for te instantaneous rate of cange of P at te instant a = 2 by using several small values. Once you ave determined an accurate estimate of P (2), include units on your answer, and write one sentence (using everyday language) to explain te meaning of te value you found. e) On your grap above, sketc two lines: one wose slope represents te average rate of cange of P on [2, 4], te oter wose slope represents te instantaneous rate of cange of P at te instant a = 2. f) In a carefully-worded sentence, describe te beavior of P (a) as a increases in value. Wat does tis reflect about te beavior of te given function P? Differential Calculus Page 6

7 II. Exercises 1. Consider te grap of y = f (x) provided at rigt. a) On te grap of y = f (x), sketc and label te following quantities: te secant line to y = f (x) on te interval [-3, - 1] and te secant line to y = f (x) on te interval [0, 2]. te tangent line to y = f (x) at x = -3 and te tangent line to y = f (x) at x=0. b) Wat is te approximate value of te average rate of cange of f on [-3, -1]? On [0, 2]? How are tese values related to your work in (a)? c) Wat is te approximate value of te instantaneous rate of cange of f at x = -3? At x = 0? How are tese values related to your work in (a)? 2. For eac of te following prompts, sketc a grap on te provided axes of a function tat as te stated properties. a) y = f (x) suc tat te average rate of cange of f on [-3, 0] is -2 and te average rate of cange of f on [1, 3] is 0.5, and te instantaneous rate of cange of f at x = 1 is 1 and te instantaneous rate of cange of f at x = 2 is Differential Calculus Page 7

8 b) y = g(x) suc tat gg(3) gg( 2) 5 = 0 and gg(1) gg( 1) 2 gg (2) = 1 and gg (2) = 1 = 1, and 3. Suppose tat te population, P, of Cina (in billions) can be approximated by te function P(t) = 1.15(1.014) t were t is te number of years since te start of a) According to te model, wat was te total cange in te population of Cina between January 1, 1993 and January 1, 2000? Wat will be te average rate of cange of te population over tis time period? Is tis average rate of cange greater or lesser tan te instantaneous rate of cange of te population on January 1, 2000? Explain and justify, being sure to include proper units on all your answers. b) According to te model, wat is te average rate of cange of te population of Cina in te ten-year period starting on January 1, 2012? c) Write an expression involving limits tat, if evaluated, would give te exact instantaneous rate of cange of te population on today s date. Ten estimate te value of tis limit (discuss ow you cose to do so) and explain te meaning (including units) of te value you ave found. d) Find an equation for te tangent line to te function y = P(t) at te point were te t- value is given by today s date. 4. Te goal of tis problem is to compute te value of te derivative at a point for several different functions, in tree different ways, and ten compare te results Differential Calculus Page 8

9 For eac of te following functions, use te limit definition of te derivative to compute te value of f (a) using tree different approaces: strive to use te algebraic approac first (to compute te limit exactly), ten test your result using numerical evidence (wit small values of ), and finally plot te grap of y = f (x) near (a, f(a)) along wit te appropriate tangent line to estimate te value of f, a visually. Compare your findings among all tree approaces; if you are unable to complete te algebraic approac, still work numerically and grapically. a) f (x) = x 2-3x, a = 2 b) f (x) = 1, a = 1 c) f (x) = xx a = 1 d) f (x) = 2 - x - 1, a = 1 e) f (x) = sin(x), a = ππ 2 III. Assessment Kan Academy 1. Complete te first 10 practice exercises from te Derivative Introduction unit in Kan Academy s AP Calculus AB course: ttps:// 2. Optional: Practice #11 (Callenge) Differential Calculus Page 9

10 Differential Calculus Page 10