2.0 5-Minute Review: Polynomial Functions

Transcription

1 mat 3 day 3: intro to limits 5-Minute Review: Polynomial Functions You sould be familiar wit polynomials Tey are among te simplest of functions DEFINITION A polynomial is a function of te form y = p(x) = a n x n + a n x n + + a x + a, were n is a non-negative integer and eac a i is a real number (constant) If a n =, ten n is te degree of te polynomial (or igest power) Te domain of a polynomial is all real numbers, (, ) Note: Tese 5-minute reviews will be a feature of class for te first few weeks Tey cover material tat you sould already know Tey are meant to alert you to material tat tat is essential to te course and tat you sould review if it is not familiar to you Degree polynomials: Have te form y = a x + a and are just equations of lines Te more familiar form is y = mx + b (were m = a is te slope and b = a is te y-intercept) Degree polynomials: Have te form y = f (x) = a x + a x + a or y = ax + bx + c Tese are te familiar quadratic functions or parabolas Figure : Two parabolas (degree polynomials) Wen te coefficient of x is negative, te parabola is upsidedown y = x x + 3 y = x 3x Degree 3 polynomials: Have te form y = f (x) = a 3 x 3 + a x + a x + a Suc polynomials are also called cubics since te degree is 3 Suc polynomials can ave eiter,, or 3 roots Figure : Tree cubics (degree 3 polynomials) wit 3,, and roots, respectively YOU TRY IT Identify wic functions are polynomials and determine teir degree (a) p(x) = 4x 3 + x + (b) q(x) = 5x 7 x (c) r(t) = t + t (d) p(x) = sin(x + ) (e) s(x) = x x / + 7 (f ) q(t) = t 3 + t + (g) r(x) = () r(x) = 3 / x 4 x + π (i) f (x) = 3 x5 + 3x 4 + x (j) p(x) = 5x x /3 3 (k) g(x) = 6x + 4x + (l) q(x) = 3x 4x + x3 6 Answers to you try it Polynomials (degree): a (3), g (), (4), i (5), l (3)

2 Rates of Cange and te Slope Problem Slopes and Average Rates Goal: Recall tat our goal is to define te slope of a curve Key Observation: Te only slopes we know ow to calculate are slopes of lines We must someow use te slope of a line to determine te slope of a curve Find te line wit te same slope as te curve at te point in question Tis is called te tangent line 3 Set Up: Find te slope of y = f (x) at an arbitrary point (a, f (a)) on te curve f (a) f (x) f (a) Figure 3: Left: Te tangent line as te same slope as te curve at (a, f (a)) Rigt: Te secant line troug (a, f (a)) and (x, f (x)) a a x One of te ways we can find slopes is to use two points We ave one point (a, f (a)) We can get a second nearby point on te curve, call it (x, f (x)) Lines tat pass troug two distinct points of te curve are secant lines Te slope of te secant line troug (a, f (a)) and (x, f (x)) is given by te m sec = = difference quotient () Tis makes sense as long as x = a, ie, as long as te two points are distinct 4 Interpretation: If f represents position (distance) and x represents time, ten difference quotient = m sec = = position time = Average Velocity More generally, if f (x) represents any quantity and x represents time, ten te difference quotient represents te average rate of cange Tis interpretation is () Te term difference quotient as defined in () will be used repeated in tis course Make sure tat you are comfortable wit it

3 mat 3 day 3: intro to limits 3 wat makes calculus so useful in oter disciplines difference quotient = m sec = = Average Rate of Cange (3) Te difference quotient can be interpreted as average velocity, or average flow rate of a stream, average acceleration, etc Instantaneous Rates We ave identified average rates of cange as secant slopes of curves But our goal is to determine te slope of a curve rigt at a particular point (a, f (a)) on te curve, not an average slope between two different points (a, f (a)) and (x, f (x)) as in Figure 3 Figure 4 sows tat we can obtain a better approximation to te slope of a curve by letting te second point (x, f (x)) approac te point (a, f (a)) We indicate tis by using te symbol x a ( x approaces a ) We can interpret te figure as a time-lapse poto of x getting closer to a f (a) Figure 4: Left: Te secant lines troug (a, f (a)) and (x, f (x)) as x a better approximate te slope of te curve rigt at (a, f (a)) Rigt: Te tangent line tat we seek a x x x x a Tink of te curve as an icy road Te tangent slope rigt at te point (a, f a) is te direction tat a car would travel if it slammed on its brakes rigt at tat point and simply slid forward Te secant slopes of te lines in Figure 4 better approximate te tangent slope as x a So wat we want to do is let x a in te equation for te secant slope In oter words, as x a, m sec = slope of te curve at (a, f (a)) (4) If f (x) were position and x were time again, ten as x a, Average Velocity = Instantaneous Velocity at (a, f (a)) (5) But we can t just substitute x = a into eiter of tese expressions to get te slope (velocity) rigt at (a, f (a)) because we would end up wit wen x = a, f (a) f (a) a a = =???? (6) So wat does appen as x a? Te next example illustrates ow tis works out

4 mat 3 day 3: intro to limits 4 EXAMPLE Consider te function f (x) = x x near a = 3 wic is graped below Suppose tat tis represents te position (in meters) of an object moving along a straigt line and tat x represents time Find te instantaneous velocity of te object at a = 3 wic is te same as finding te slope tan of tis curve rigt at a = 3 SOLUTION We do so by calculating some average velocities (secant slopes) on te interval [a, x] = [3, x] as x gets closer and closer to a = 3 Let s begin: First find m sec (average velocity) on te interval [3, 5] From () Average Velocity = m sec = f (5) f (3) 5 3 = 5 3 = 6 (m/s) (7) Te secant line troug (3, f (3)) and (5, f (5)) wit slope 6 is drawn in Figure 5 Tis slope is a roug approximation of te slope of te curve rigt at (3, f (3)) Let s try for a better approximation: Coose x closer to a = 3 Try x = 4 Tis time Average Velocity = m sec = f (4) f (3) 4 3 = 8 3 = 5 (m/s) (8) Plot tis secant line troug (3, f (3)) and (4, f (4)) Does tis line appear to ave a slope closer to te slope of te curve rigt at (3, f (3)) tan our first secant line? Take a second and try one more point: Coose x closer still to a = 3 Try x = 35 Finis te calculation and plot te corresponding secant line Fill te value in te table Wat do you conclude? Average Velocity = m sec = f (35) f (3) 35 3 = (9) x m sec = ave vel on [3, x] 5 f (5) f (3) 5 3 = m/s OK, we could continue in tis way coosing lots of x s closer still to a = 3 and determine te corresponding secant slope Instead, let s do it for a general value of x On te inteval [3, x] we find 9 5 Figure 5: Determine te slope of f (x) = x x at x = 3 = 6 m/s Ave Vel = m sec = f (x) f (3) x 3 = (x x) 3 x 3 = (x + )(x 3) x 3 = x + m/s ()

5 mat 3 day 3: intro to limits 5 So for example, wen x = 5, te secant slope would be x + = 5 + = 6 Wen x = 4, te secant slope would be 4 + = 5 Or if x = 35, te secant slope would be 45 Tese matc te earlier calculations above Tis general calculation makes it is easy to fill in te table of values for te secant slopes We no longer ave to compute eac difference quotient individually We ave a formula tat gives us te slopes of te secant lines passing troug (3, f (3)) and (x, f (x)), namely m sec = x + Wen x = 3, m sec = 4 Stop! Take a moment to fill in te rest of te table Notice tat te formula works even if x < 3 Draw a few more secant lines in Figure 5 It sould be apparent from your table of values tat as x 3, m sec 4 Draw te line of slope 4 tat passes troug te point (3, f (3)) in Figure 5 Tis line is called te tangent line to y = f (x) at x = 3 It just touces te curve once near x = 3 Ceck your drawing Anoter way to say tis is tat as x 3, te average velocity 4 m/s We conclude tat 4 m/s is te instantaneous velocity rigt at time 3 One last point We can t just let x = 3 in () for te slope of te secant line oterwise we d ave f (3) f (3) m sec = 3 3 wic is nonsensical since it involves division by More Examples EXAMPLE (Finding te general formula for a secant slope) Let s(t) = 3t + t + (a) Find te average velocity or m sec on te interval [, ] (b) Find te formula for te average velocity on te general interval [, t] Use te formula to estimate m sec on te interval [, ] (c) Make a conjecture about m tan or te instantaneous velocity rigt at t = SOLUTION For (a) For (b) Ave Vel = s(t) s() t Ave Vel = m sec = s() s() = 3t + t + ( ) t = 9 ( ) = 3t + t + t = = 8 m/s ( 3t )(t ) t So on te interval [, ], te average velocity is 3() = 53 m/s Looks like m tan migt be 5 Wy? = 3t m/s EXAMPLE 3 (Using a table) Fill in te average velocities (m sec ) for s(t) = sin t on te intervals listed in te table Ten estimate m tan at t = Is factoring possible tis time? Time interval [, t] [, ] [, 5] [, ] [, ] [, ] Ave Vel SOLUTION Make sure your calculator is set in radians We get y = sin(t) sin() sin() Ave Vel = = sin() 689 sin(5) sin() Ave Vel = = sin(5) sin() sin() Ave Vel = = sin() sin() sin() Ave Vel = = sin() sin() sin() Ave Vel = = sin() Looks like m tan = No factoring is possible Figure 6: Wat is te slope of te red tangent line to y = s sin(t) at t =?

6 mat 3 day 3: intro to limits 6 Recap and a Pilosopical Problem Let s recap te important ideas we ve encountered We can interpret wat we ave done in a couple of different ways In te example above, Geometry: As x 3, te secant line tangent line to te curve at x = 3 In oter words, as x 3, m sec m tan = te slope of f (x) at x = 3 Motion: As x 3, te average velocity instantaneous velocity at x = 3 More generally for any quantity f (x) varying wit time, as x a, te average rate of cange instantaneous rate at x = a An Analogy: Tese observations can be combined into an analogy secant slope : average velocity :: tangent slope : instantaneous velocity (rate) In particular, if f (x) = x x, ten as x 3, te tangent slope = 4 = slope of te curve at x = 3 or te instantaneous velocity is 4 m/s at x = 3 3 Problems Finding te general formula for a secant slope Let f (x) = x + x + 3 Te point P = (4, 3) lies on te curve Suppose tat Q = (x, x + x + 3) is any oter point on te grap of f Find te slope of te secant line troug P and Q Simplify your answer Finding te general formula for a secant slope Let f (x) = x Te point P = ( 3, 6) lies on te curve Suppose tat Q = (x, x ) is any oter point on te grap of f Find te slope of te secant line troug P and Q Simplify your answer 3 Finding te general formula for a secant slope Let f (x) = x + 3 Te point P = (9, 6) lies on te curve Suppose tat Q = (x, x + 3) is any oter point on te grap of f Find te slope of te secant line troug P and Q Simplify your answer Big int: Multiply bot te numerator and denominator by x + 3 Te term x + 3 is te conjugate of te original numerator in te difference quotient Answers: x + 5; 6 x ; 3 x+3 4 (a) A secant line to te grap of a function y = f (x) is a line tat passes troug two points (a, f (a)) and (b, f (b)) on te curve (see below, left) Wat is te general formula for te slope m sec of te secant line containing te points (a, f (a)) and (b, f (b)) Tis general formula works for all functions 4 (a, f (a)) (b, f (b)) (b) Suppose we want secant lines for f (x) = x Use tis particular function in your formula for m sec Simplify te expression by factoring You now ave a formula for te slope of te secant, m sec, between any two points (a, f (a)) and (b, f (b)) on te curve y = x (c) Still assume tat f (x) = x Use your formula in part (b) to easily calculate te value of te slope m sec if a = and b = Tis is te slope of te secant line troug (, f ( )) and (, f ()) Wat is te equation of te line? Draw tis line

7 mat 3 day 3: intro to limits 7 on te grap (above, rigt) Find m sec and te equation of te secant line troug (, f ( )) and (, f ()) (d) If te secant line troug (, f ()) and (b, f (b)) as slope 7, wat is b? (e) More interesting Repeat part (b) for te function f (x) = x Tis gives a formula for m sec for te curve y = x Caution: Watc your algebra Ten use your formula to find te slope of secant troug (, f ()) and (, f ()) 5 Average velocity is cange in position over cange in time Wen distance is given by a function y = f (x), were x is time, ten average velocity is just cange in y over cange in x wic is just te secant slope again Assume a ball is trown upward from te roof of Gulick Hall so tat its position above te ground after x seconds is given by y = f (x) = 5x + x + 5 meters over te time interval x 5 (see grap) (a) Let (x, f (x)) be any point on te curve Find te general formula for te average velocity between (3, f (3)) and (x, f (x)) Remember: Tis is te same ting as finding m sec between tese two points using 3 and x instead of a and b as in te previous problem Simplify your answer Hint: First factor out 5 in te numerator (b) Use te formula from part (a) to evaulate average velocities in te table tat follows Tis is very easy (c) Wy is *!*&*%**%$***" in te table above? Time interval [3, x] Average velocity on [3, x] [3, 3] [3, 3] [3, 3] [3, 3] [3, 3] *!*&*%**%$*** [9999, 3] [999, 3] [99, 3] [9, 3] (d) Wat is te average velocity approacing as x gets close to 3? (e) Determine te instantaneous velocity at x = 3 Wat is m tan at x = 3? 6 Slope is simply te cange in y over te cange in x Tis is easy to compute for a straigt line But in calculus we will want to determine te slope of a curve y = f (x) Suppose tat represents te cange in x, ie, x canges from x to x + (instead of a to b) Ten y = f (x) canges from f (x) to f (x + ) So te slope becomes y f (x + ) f (x) = = x (x + ) x f (x + ) f (x) Care must be taken in computing and simplifying f (x + ) f (x) For example, if f (x) = x + ten f (x + ) f (x) = [(x + ) + ] [x + ] = x + x + + [x + ] = x + Notice ow it simplifies You sould be comfortable wit tis type of calculation Calculate f (x+) f (x) if = x + (a) f (x) = 3x + (b) f (x) = x (c) f (x) = x x

8 mat 3 day 3: intro to limits 8 7 (a) Find te equation of te tangent line to te circle at te point indicated Hint: How is te tangent line related to te radius? (b) Circles don t ave slopes in te usual sense But ow migt you define" te slope" of te circle at te point (3, )? 8 A unit semi-circle is drawn to te rigt and a general point is marked on it (a) Determine a formula for slope" of te circle at te point indicated (You will first need to find te y-coordinate of te point on te circle in terms of x) (b) Use your formula to calculate te slope of te circle wen x = 9 and wen x = 4 (c) Wat is te domain of your slope" function? (d) Use your formula to determine wen te slope is Does tat make sense given te grap? (e) Use your formula to determine wen te slope is positive Negative Compare to te grap Express your answers using interval notation 4 (3, ) Figure 7: Te circle for Problem 7 x Figure 8: Te semi-circle for Problem 8