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1 21 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? Ex: Can you approximate this line with another nearby? How would you get a better approximation? Eto KE e iee it :# :* :

2 Ex: A cardiac monitor is used to measure the heart rate of a patient after surgery It compiles the number of heartbeats after 5 minutes When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute t (minutes) Heartbeats The monitor estimates this value by calculating the slope of a secant line Use the data to estimate the patient s heart rate after 42 minutes using the secant line between the points with the given values of t a) t 36 to t 42 b) t 38 to t 42 c) t 40 to t 42 d) t 42 to t 44 slope 2* slope jfhztt slope A± At slope an 8,394 k 66 What are your conclusions? Notice in this example that we calculated slopes, which gave us " # # minutes that passed Change in total Heartbeats Heart rate

3 So, the slope told us about the rate of change of the graph Units Position: m, ft, in, miles, light year : Velocity: TD, Im% mph Acceleration: rysz or Y% Ex: If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y 10 t 186 t 2 a) Find l H Ei HIe± # " the average velocity over the given time intervals: i) [1, 2] 2 : ) 1256 m t 1 : Y ,14 m ii) [1, 15] iii) [1, 11] iroo5 ot#1 iv) [1, 101] v) [1, 1001] Avve6cit±s±'t#84%Y 42% b) Estimate the instantaneous velocity when t 1 va 628 ma See Next Page

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8 22 The Limit of a Function Ex: Graph this function:! " " (!&' " 2 1) What is f(1)? 2) What is f(3)? 3) What is f(2)? 4) What would you guess f(2) should be? 5) Fill in this chart: 's Kao 4% k fifa 23 1 : defined, we 4 don't,dne x f(x) x f(x)

9 The left side of the table tells us that lim!(") 4 0 ( 2 which is read as The lefthand limit of f(x) as x approaches 2 is equal to 4 or The limit of f(x) as x approaches 2 from the left is 4 The right side of the table tells us that lim!(") 4 0 ( 5 which is read as The righthand limit of f(x) as x approaches 2 is equal to 4 or The limit of f(x) as x approaches 2 from the right is 4 Note that x 3 means from the left and not necessarily negative numbers The same is true of x 3 8 In this case, the lefthand limit and righthand limit are equal, so we can say that lim!(") 4 0 ( In general, 678 ;(9) L if and only if 678 ;(9) L and 678 ;(9) L 9 : 9 : 9 : 5 Example 7, page 92: a) lim 0 ( < " d) lim < " 2 b) lim < " e) lim < " 0 ( c) lim < " f) lim < " 0 ( Not defined Does not exist 2 Note :gh1

10 Infinite Limits Ex: Find lim B 0 A 0 C if it exists Approach 1: Table of values (Why can we use ±?) x ± 1 ± 05 ± 02 ± 01 ± 001 ± ( I ,000 1,000,000 Approach 2: Sketch the graph # neex gain Hmmm it looks like f(x) keeps getting bigger and bigger as x approaches 0 from the left and the right In this case we say: 1 lim 0 A + ( Definition: Let f be a function defined on both sides of a, except possibly at a itself Then lim 5 + means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a, but not equal to a

11 Ex: Find lim 0 0 K 2 07K, lim 0 0 K 5 07K 0 and lim 07K tiny x ' lim : 4 ' YtnYi%n*o tjn 41 Fs, y a lim : positive 4+ Fiene Yjg+ 4

12 link 23 Calculating Limits Using the Limit Laws Limit Laws: Suppose that c is a constant and the limits lim '()) and lim +()) exist Then 1) lim [ ' ) + +())] lim '()) + lim +()) 2) lim [ '()) +())] lim '()) lim +()) 3) lim [1 ' ) ] 1 lim '()) 4) lim [ ' ) +())] lim '()) lim +()) 5) lim[ R 0 ] W X TUV R(0) S(0) TUV W X S(0) if lim +()) 0 Ex: Use Law #4 to show that: lim [ ' ) ]n [lim ' ) ] n " ' di ;na[fc D' ;zfc )] ' (6 Power Law) n2x±naac D : fixated H D lxiynafcx ) [ ti ) x a ;fc D '

13 for hl ( previous page ) 2 Assume for nk lxismacfcxdkfisnafcxdk@provefornkt1lxismaecxdhllinead 'kfh] : Bids :L ptvn lxignfftxiskghinafh ) by #4 [dxah DktiaKd G nah D" *

14 Additional Laws: 7) lim ' c 8) lim ( a 9) lim ( Y + Y he?xsf* [ [ 10) lim ( + [ (As long as + is defined) [ 11) lim (() [ lim (() (As long as [ lim (() is defined) Ex: Evaluate these limits, justifying each step % 1) lim 0 \ 2( ( 3( + 4 ;G t3dtxiz4 By Law # I High ' ;}x lix ' thing } ' Hing 3xttiy4 By kw#2 light 4 Law '±7 zomjnx +4 Law#3 2) lim (0C 7\08K 0 \ (07\

15 Given: 1) Any letter can change into an A 2) Two Cs in a row can turn into a B 3) Any word can be doubled Ex: C C C #3 CCCC #3 ACAC #1 ACAC Ex: CAC CAAAC Ex: C BACABC

16 lim 0 \ (2)( 3) + 4): 1) lim 0 \ (2) ( 3) + 4 ) lim 0 \ (2) ( 3)) + lim 0 \ 4 Law #1 2) lim 0 \ (2) ( 3)) + lim 0 \ 4 lim 0 \ 2) ( lim 0 \ (3)) + lim 0 \ 4 Law #2 ii ' ' 3) lim 2) ( lim (3)) + lim 4 lim 2) ( lim 3 lim ) + lim 4 Law #4 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 4) lim 2) ( lim 3 lim ) + lim 4 2 lim ) ( lim 3 lim ) + lim 4 Law #3 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 5) 2 lim 0 \ ) ( lim 0 \ 3 lim 0 \ ) + lim 0 \ 4 2 [lim 0 \ )] ( lim 0 \ 3 lim 0 \ ) + lim 0 \ 4 Law #6 6) 2 [lim 0 \ )] ( lim 0 \ 3 lim 0 \ ) + lim 0 \ 4 2(3) 2 3* Law #7, Law #8, Arithmetic 2) ( 3) + 4 lim 0 \ 2) 3 1) lim (0C 7\08K 0 \ (07\ TUV W a ((0C 7\08K) TUV W a ((07\) Law #5, plus lim 0 \ (2) 3) 0 W a 2) TUV((0C 7\08K) TUV W a ((07\) B\ TUV W a ((07\) (Problem above) 3) 4) B\ B\ TUV W a ((07\) TUV (0 7TUV W a B\ B\ TUV (0 7TUV W a W a \ ( TUV W a 0 7TUV W a \ Law #2 W a \ Law #3 5) B\ B\ B\ ( TUV W a 0 7TUV W a \ ( \7\ \ Law #7, Law #8, Arithmetic

17 * 0 Ex: Find lim C 7K 0 7( 08( time lxisndx ' 2) ± 4 \8b Ex: lim C 7c b A b tisane liz* Dat4s] : kjo[6#[h ] thing 6h i h lingo K6ntIhigd6+h ) tin [ 6th ] h O Ex: lim 0 B ) + 1 EF ) 1 G EF ) 1 2 It OneSided Analysis Remember that: IJK N(L) L if and only if IJK N(L) L and IJK N(L) L L M L M 2 L M 5 Ex: Show that lim 0 A ) 0 fg) disnotfc xzo x< 0 'Dtih+ O 1 10 king o king HD knot

18 0 Ex: Prove that lim 0 A 0 Ex: If f(x) 2 ) EF ) < 2 ) EF ) 2 does not exist 1 1 x± lifx >O T t ifx< 0 ' ti ' disnot limo 'x±t notd ', K% InE, determine whether lim F()) exists 0 ( not Exist lxiytfkkbiz,xz tinfoil int?ie#5y@t5 Greatest Integer Function [[x]] The largest integer that is less than or equal to x Ex: [[5]] 5 [[G]] 3 [[ 3]]? 17 [ [D]]1

19 Theorem: If f(x) < g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then lim F()) < lim T()) Sandwich Squeeze Theorem: If f(x) < g(x) < h(x) when x is near a (except possibly at a) and then lim F()) lim h()) L lim T()) L Ex: Show that lim 0 A ) ( VEW B 0 0 I x 2 < ) ( VEW B 0 < x2

20 Summary of Techniques So Far:!"# $ & '($) 1) When in doubt, look at the graph 2) If the function is defined at a, try substituting a for x 3) If the function is not defined at a, try expanding and/or factoring to cancel out problem factors 4) If you suspect that the limit might be infinite, look at a sign analysis/sign chart 5) If you suspect that the limit might not exist, consider a onesided analysis 6) If you can find g(x) > f(x) > h(x), use the Squeeze Theorem

21 24 The Precise Definition of a Limit (DeltaEpsilon Proofs) What does x gets really close to a actually mean? The distance between x and a gets smaller and smaller How do we express the distance between x and a? Ix " We use the Greek letter! ($%&'() to express an upper bound for this distance: x a <! What does f(x) approaches a limit actually mean? The distance between f(x) and the limit, L, gets smaller and smaller How do we express the distance between f(x) and L? f(x ) Ll We use the Greek letter * (epsilon) to express an upper bound for this distance: f(x) + < * In fact, if the limit actually exists, we should be able to get arbitrarily close to L as x gets closer to a So we start with a value for *, and then solve to find a! that works Translation: Step 1: Pick any value for * that you d like, or use one that someone else gives you (It has to be positive, and usually it s between 0 and 1 often very close to 0) Step 2: Set up the inequality L * < f(x) < L + * Step 3: Solve to find the interval of x values that will satisfy the inequality Step 4: Use this interval to come up with an appropriate!

22 Ex: f(x) x + 2, * 001 Ee tae l E t!e 6I obi Now it s possible that * 001 was a special case, so what we really want to do is to look at all possible values of *, not just specific ones Ex: f(x) x + 2, all values of * footsteps Ex: g(x) 3x, all values of *

23 Ex: For f(x) x + 2, a 1, and L 3 find a! that corresponds to " 001 limfcx ) 3 % c) I g L x a 1<8 / 's, horizontal fat y 's, Now it s possible that " 001 was a special case, so what we really want to do is to look at all possible values of ", not just specific ones Ex: f(x) x + 2, a 1, and L 3, find a! that corresponds to all values of " j m, 6+2 ) 3 t te, Ex: Prove that lim 0 ( 3* 6 (for all values of ") vertical ; f k i #a " We want : / far ) L / < E How about#? 1 11 < e * * 3 61 < { 13 ( x 41 < { 3 / xi I < { 1 21<5 of E 1 21 so 1 al < or

24 Infinite Limits Let f be a function defined on some open interval that contains the number a, except possibly at a itself Then lim * means that for every positive number M there is a positive number! such that if 0 < x a <! then f(x) > M (Note that an infinite limit technically means the limit doesn t exist, just in a particular way) Ex: Prove that lim (0 C Start 0 A B with x al < 8 0<1 01 < J 0 < 1 1<6 Such that ( ) > M 2< ZMT X? k > x? ZM ( bkx3o) : Etym 1 > ZM xzzra 12 ( bk M >0 )

25 So, if I want fcx ) > M, I need 2 M How do I ensure x n? < an a * m

26 25 Continuity A function f is continuous at a number a if lim ' ( '(+) This means three things: 1) f(a) is defined 2) lim ' ( exists (and is finite) 3) lim ' ( '(+) Heggen % lifting your pencil Ex: Draw examples of ways in which a function could be discontinuous Continuous Cts Removable discontinuity : ~ Jump discontinuity : * s from the left : fig fc )f( a) Cts from the right tsnatfkkfca ) 7 in

27 I Ex: Where are each of the following functions discontinuous? a) f(x) 0C 7 07( G I k 2k 07( xz :L K ^ 7 b) f(x) Y c) f(x) B 0 C 2' ( 0 1 2' ( 0 0 C 707( 07( 2' ( 2 1 2' ( 2 discontinuous O # because at ljngfcx ) isn't fnile kiyotfco ) diafa*fa, 2 discontinuous at 2 d) f(x) [[x]]

28 Theorem 7 The following are continuous at every value in their domains: 1) Polynomials 2) Rational Functions 3) Root Functions 4) Trig Functions 5) Inverse Trig Functions 6) Exponential Functions 7) Logarithmic Functions Ex: Evaluate lim 0 r sut 0 (8uvs 0 } threatens of discontinuity Steps: A) Is the numerator continuous? Yes ( at * F) B) Is the denominator continuous? Yes C) Are there any points where the rational expression isn t defined? No > If the function is continuous everywhere, then lim 0 r (0) f(2) tins' *zstis 0 Theorem 8 If f is continuous at b and lim 4 0 6, then lim (4 0 ) (6), which means lim (4 0 ) (lim 4 0 ) (See Example 8, Page 124)

29 Ex: Show that there is a root of the equation 4x 3 6x 2 + 3x 2 0 between 1 and ( 8) ±Not tae n " are cts the graph must pass axis, * Since polynomials Intermediate Value Theorem * through the x between Xlmdx2

30 26 Limits at Infinity Horizontal Asymptotes Look at the graph of f(x) B 0 aft( As x > +, f(x) > from As x >, f(x) > As x > 0 +, f(x) > As x > 0 f(x) > The line y 0 is called a horizontal asymptote of f(x), since lim +, 0 0 x to Note: lim 0 x +, / can be read as: above ) 0 ( from below ) 1) The limit of f(x), as x approaches infinity, is L 2) The limit of f(x), as x becomes infinite, is L 3) The limit of f(x), as x increases without bound, is L Theorem 5 If r > 0 is a rational number, then 1 lim 0 x, y 0 If r > 0 is a rational number such that, y is defined for all x, then lim 0 7x 1, y 0

31 Ex: Evaluate lim \0 C 707( 0 C 8K08B sinaee?e (Using Theorem 5 above) (Note: Limit Laws only hold for finite limits, not infinite limits, so use the above technique to rewrite the following function) Ex: Evaluate lim 0 x (0 C 8B \07@ (Note:, (, *t sna x jff him Ft or

32 Ex: Evaluate lim 0 x (, ( + 1,) Hint: Use the conjugate time Etten x ' timed 'm ka [* D Ex: Evaluate lim (, (,) Hint: Look at this as a product 0 x it D txiszxcx Ex: Evaluate lim >?@A B Hint: Let t 1/x, and then look at the graph of arctan 0 A 5 0 ing grant E i#i H tiao[#±n#

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34 or Formal Limits Let f be a function defined on some interval (a, ) Then lim ' ( 0 x means that for every positive number M, there is a corresponding positive number N such that if x > N then f(x) > M Ex: Prove that lim 0 x,0 Given M We need an N st ( et N M en > M en > M #M > ( need Kelvinthis N > KM en > M e M Better : We need to find Given N st M en>mfkgtnhm+ f ( N ) > M f(n)en ekmt ' >ehmm *,

35 27 Derivatives and Rates of Change Let s look at secant lines again: For a given curve y f(x), we re interested in what the tangent line at P (a, f(a) ) looks like So we look at another point, Q (x, f(x) ), to the right of P, and investigate what the secant line through P and Q looks like: Kroft rise a at Tangent : : ' Since we now understand a little about limits, we can look at what happens as we move Q closer and closer to P by letting x approach a The slope of PQ z{ } * R 0 7R(J) z~y 0 07J so, as we nudge Q closer to P, we get

36 R 0 7R(J) m lim 07J This is the slope of the tangent line at P (Provided the limit exists) Ex: Find an equation for the tangent line to the parabola y x 2 at the point P(1, 1) m ti f;, 'I ;afhx±eitiyf # Y ' ' ling, c,*# lying, 'htd 1+12 If we look at the original idea of points P(a, f(a) ) and Q(x, f(x) ), but, instead, consider the x values as x and x + h, the new graph looks like this: y y M2 l ' 14 1 'hnl* ' ' ), ) ) 12 2 y

37 372A I Ex: (#14 in the book) If a rock is thrown upward on the planet Mars with a velocity of 10m/s, its height (in meters) after t seconds is given by H 10t 186 t 2 a) Find the velocity of the rock after one second ' ft4 si4%ayyfh H C 814 m b) Find the velocity of the rock when t a tin fk+hyf( [ loath ) foal 86D ] lim h 0 L ;m Ha+'0ht#iy37IaT h 0 lim h o 10h 372 h ah j n[0 372A 86D A % c) When will the rock hit the surface? HCt)O HCE ) lot 101 t ( or t #Fa 5376 Seconds d) With what velocity will the rock hit the surface? Fonb, velocity 10 Ys a % ) velocity or 6

38 HH ) lot 186T ' n aμ timothy, Instantaneous of f ( lth ) At h o h Rate change at FCD hix@ckh) 1864M 186 km ' h [ ] 372 thing ahi% Is6n] 6,28% [10186] '

39 28 The Derivative as a Function!'($) lim b A! $ + h!($) h Ex: If f(x) 2x 2 4x + 3, find f'(x) faith )2(x+h)t4( xth ) xhtzh h ftpyat#t4xh+2h2hx4hhxt4/xb tin 4x_hth4h h O hoi n o # tin 1 h K(4xht#4 in (4 +44) Notation: f'(x) y' Å2 Å0 ÅR Å0 Å Å0 f(x) D f(x) D x f(x) first derivative f''(x) ÅC 2 Å f'(x) second derivative Å0 C Å0 derivative change or speed or rate of change

40 A function f is differentiable at a if f'(a) exists It is differentiable on an open interval (a, b) or (a, ) or (, a) or (, ) if it is differentiable at every number in the interval Ex: Where is f(x) x differentiable? Some ways a function can fail to be differentiable: the Corner/kink: ; Discontinuous: * Vertical Tangent Line (with or without a corner): tax

41 Ex: If f(x) 4x 3 2x 2 + 3x 12, find f'(x), f''(x), f'''(x) and f (4) (x) fino fatalism fat#*' li ;okx+hf2c +h x+d 2x2txID h # hte2h#4xhykkhd#3xxed lim thing h [ f' G)

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