Section 3: The Derivative Definition of the Derivative

Transcription

1 Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding 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): m = y/ x = (6.25 4)/(2.5 2) = 2.25/.5 = 4.5, 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 rigt of (2,4) x y = x 2 slope of line troug x y = x 2 slope of line troug (x,y) and (2,4) (x,y) and (2,4) 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. Tis capter is (c) 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.

2 Capter 2 Te Derivative Business Calculus 86 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, ten x = 2 + is close to 2 and te point (2+, f(2+) ) = (2+, (2+) 2 ) is close to (2,4). Te slope m of te line troug te points (2,4) and (2+, (2+) 2 ) is a good approximation of te slope of te tangent line at te point (2,4): m = y x = (2+) 2 4 (2+) 2 = { } 4 = = (4 + ) = 4 +. 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 + ) 0 x 0 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.

3 Capter 2 Te Derivative Business Calculus 87 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 dx dx 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): dx d dx ( f ( x) ) = df dx 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.

4 Capter 2 Te Derivative Business Calculus 88 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. Example 1 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) We can estimate te values of f (x) at some non-integer values of x, too: f (.5) 0.5 and f (1.3) 0.3. We can even tink about entire intervals. For example, if 0 < x < 1, 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). Example 2

5 Capter 2 Te Derivative Business Calculus 89 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).

6 Capter 2 Te Derivative Business Calculus Exercises 1. 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) 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) (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).

7 Capter 2 Te Derivative Business Calculus (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). 5. 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 6 11, 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) ) 6. f(x) = 3x 7 7. f(x) = 2 7x 8. f(x) = ax + b were a and b are constants 9. f(x) = x 2 + 3x 10. f(x) = 8 3x f(x) = ax 2 + bx + c were a, b and c are constants 12. Matc te graps of te tree functions below wit te graps of teir derivatives.

8 Capter 2 Te Derivative Business Calculus Below are six graps, tree of wic are derivatives of te oter tree. Matc te functions wit teir derivatives. 14. 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.)