Slopes of Secant and!angent (ines - 2omework
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1 Slopes o Secant and!angent (ines - omework. For te unction ( x) x +, ind te ollowing. Conirm c) on your calculator. between x and x" at x. at x. ( )! ( ) 4! + +!. For te unction ( x) x!, ind te ollowing. Conirm c) on your calculator. y! x! y x + between x 0 and x" at x. at x. + ( )! ( 0)!! + ( + )! + + as gets close to 0, m +. For te unction x x x 4, ind te ollowing. Conirm c) on your calculator. y + x! y x! 4 between x and x" 6 at x. at x. + ( 6)! ( ) 6!! + ( + )! ( + ) ( + ) + as gets close to 0, m 4. For te unction x x 7x 8, ind te ollowing. Conirm c) on your calculator. y + x! y x! between x - and x" 4 at x -. at x -.! + ( 4)! (! ) 4 +!! (! ) (! )! 7(! ) + 8! 7 (! + )! + as gets close to 0, m! y! 7! x + y! x AB Solutions Stu Scwartz
2 +!. For te unction x x x, ind te ollowing. Conirm c) on your calculator. between x - and x" at x. at x. ( )! (! ) !! as gets close to 0, m y! 8 x! y x! 8 6. For te unction ( x), ind te ollowing. Conirm c) on your calculator. x! between x 4 and x" 6 at x. at x. ( 6)! ( 4)! 6! 4 + "! + % '( (! ) + '* - ') (! ), # & 0 +! 0!! y +! x! 4 y! x! 4 4 as gets close to 0, m! x 7. For te unction ( x)!, ind te ollowing. Conirm c) on your calculator. x + between x and x" at x. at x. ( )! ( )! 4 + " # + + 4! % 4 '( 4 ( + 4 ) + '* - ') 4( + 4), & 4 + 4!! as gets close to 0, m 6 y! x! 4 6 y x! AB Solutions Stu Scwartz
3 + 8. I s t 4t is a measure o eet traveled per second, ind a) te average velocity between t and t b) te inseous velocity at t seconds. s + s s( )! s( ) 4( + ) +! 9 4 t sec 4 t sec! 4 4 t sec 9. I s( t) t + 4 is a measure o eet traveled per second, ind a) te average velocity between t 0 and t 4 b) te inseous velocity at t second.! + s 4 s 0 4! 0 4 t sec s + s ( + ) + 4! ( + ) + As gets close to 0, avg. vel. t sec 0. I s t t t is a measure o miles traveled per our, ind a) te average velocity between t 0 and t 4 b) te inseous velocity at t our. +! s 4 s 0 4! 0 mp s + s! + +!! +! As gets close to 0, avg. vel.! mp. I s t t t is a measure o eet traveled per second, ind a) te average velocity between t and t 7 b) te inseous velocity at t seconds. s 7 s 7! 68 t sec s + s ( + ) + ( + )!! 9 6 ( + + ) As gets close to 0, avg. vel. t sec. I s( t) is a measure o eet traveled per second, ind t + 6 a) te average velocity between t and t 7 b) te inseous velocity at t 4 seconds. s 4 + s 4 s 7 s 7!! t sec 9! % " 6 + % 6 6 ' '! ( + )! &# 6 + & As gets close to 0, avg. vel.! t sec 6 AB Solutions - - Stu Scwartz " # 6 6+
4 Slopes o Secant and!angent (ines - Classwork Diagram Diagram Diagram Q Q ( x) ( x ) Q P P P A line is drawn between points P A line is drawn troug P tat We draw te secant line troug and Q. Tat line is called te touces x in one and only PQ. However, point Q starts to te gent line at P. towards P. So te secant line also moves. As Q gets real close to P note tat te secant line starts to look like te gent line at P. secant line troug P and >. one point. Tat line is called move along te curve x We will say tat as Q gets closer and close to P, tat te secant line PQ gets closer to te gent line troug P and tus te slope o te secant line (called m sec ) approaces te slope o te gent line (called m ).. Let us try and ind te slope o te secant line between x and a value Let s try an example. Suppose x x o x closer and closer to. Complete te cart. Set your calculator to ull loat. x A B.C B.B B.DC B.DB B.DDB B.DDDB B (x) Rise Run E-04 0 msec error Wy can t you ind te gent line between x and x? division by zero Secant lines use ow many points? Tangent lines use ow many points? Wat is your best guess or te slope o te gent line at x? Let us use an analogy. Suppose you leave scool at 0 AM or a trip down te sore. You arrive at PM. Te trip is a total o 00 miles. How ast are you going at AM? unknown We cannot ind your actual velocity at AM (called te inseous velocity). But we can ind te average total disce velocity at AM. We know tat average velocity. So your average velocity between 0 AM total time and PM is 0 mp On te next page, you are given te disce you travel between AM and dierent times. Determine te average velocity between tese two times. AB Solutions Stu Scwartz
5 between BB am G :0 : :0 :0 :0 :0 :00:0 :00:0 :00 disce traveled time duration 4 miles miles 0 miles. miles miles.8 miles. miles 80 t average velocity 48 mp mp 60 mp 66 mp 60 mp 48 mp 6.6 mp 4. mp??? velocity at BB AM??????????????????????????? Again, you cannot actually ind te instaenous velocity at AM? Wy? division by zero But, as te time duration becomes smaller we ind tat te inseous velocity at AM is muc more likely to be very close to wic o tese values? 4. mp Wy? less time or variation Average velocity uses ow many times? Inseous velocity uses ow many times? 0 P x, y 0 0 ( x) Q x, y x 0 x y! y 0 x - axis However, since anoter way o writing y is x Let us now take te two points and give tem general coordinates P x, y and Q x, y We draw te rigt triangle below te curve. Note tat te lengt o te orizontal line is x! x 0 and te lengt o te vertical line is y! y 0 We now ind te slope o te secant line PQ and denote it as. m sec rise run m sec y! y0 x! x ( ( x )! ( x0) ), we can say tat m sec x! x 0 Now, let s concentrate on te gent line. In order to ind te slope o te gent line, te above ormula does not work. Wy? division by zero So, let us deine te variable as te orizontal disce between te two points P and Q. Tus: x! x 0 and it ollows tat x x0 + x x0, it ollows tat m sec x! x Since m sec 0 ( x0 + )! ( x0)!k LKM: Remember tat te gent line on te previous page was deined as te line created wen Q gets closer and closer to P. As Q gets closer and closer to P, x gets closer and closer to x 0. So, as Q gets closer to P, (te orizontal disce between P and Q) gets close to zero. Tereore, we can now state te starting point or dierential calculus: ( x0 + )! ( x0) m as gets ininitely close to zero. Note tat cannot equal zero. Wy not? div. by 0 AB Solutions - - Stu Scwartz
6 Tree ormulas you will need to know are: te slope o te secant line: x x m 0 sec x! x0 te slope o te gent line: x x m + te point-slope equation o a line: y! y m x! x 0 0 as gets ininitely close to zero. Example ) For te unction ( x) x +, ind te ollowing. Conirm c) on your calculator. between x and x" at x. at x. ( )! ( ) 4! ! ( 4 + ) 4 + as gets close to 0, m Example ) For te unction ( x) x +, ind te ollowing. Conirm c) on your calculator. 4 y! 4 x! y 4x! between x - and x" 4 at x. at x. ( 4)! (! ) 4 + +! + +! as gets close to 0, m y! x! y x + Example ) For te unction x x 4x, ind te ollowing. Conirm c) on your calculator. between x - and x" - at x -. at x -.! + ( )! (! ) + 0! (! ) (! ) + 4(! )! + as gets close to 0, m 0 y + 0 x + y! AB Solutions Stu Scwartz
7 !! Example 4) For te unction x x x, ind te ollowing. Conirm c) on your calculator. between x 0 and x" at x. at x. ( )! ( 0)!! 0! + Example ) For te unction x x x! +! + + ( + ) + as gets close to 0, m AB Solutions Stu Scwartz, ind te ollowing. Conirm c) on your calculator. y + x! y x! between x - and x" at x. at x. + ( )! (! ) +! ( + ) +! as gets close to 0, m Example 6) For te unction ( x), ind te ollowing. Conirm c) on your calculator. x + y! x! y x between x and x" 4 at x. at x. ( 4)! ( )! 4! " # +! %( ( + ) + '* - &) ( + ),! + (!+ ) y!! x! 9 x y! as gets close to 0, m! 9 Let us suppose tat an object is traveling along a straigt line according to te ormula s( t) t + were t is measured in seconds and s t is measured in eet. Complete te cart. t 0 4 s t t O C P Q BB BO + I we want to calculate te average velocity between t 0 and t 4, we know average velocity So, average velocity equals measured in t sec total disce total time
8 But, i we wis to calculate te inseous velocity at t seconds, we are interested in exactly ow ast we are traveling at t. Tis is not as easy to do. Using te analogy above, we can now state ormulas wic allow us to ind bot average and inseous velocity. Two ormulas you will need to know are: Given s t s( t)! s( t) Average velocity t! t Inseous velocity: s t + s t as te disce traveled in time t ( )! ( ) as gets ininitely close to zero. So in te problem above, let s ind te inseous velocity at t. s( + )! s( ) ( + ) +! +! t sec Note tat te average velocity is te same as te inseous velocity. I you are in a car were your average speed is te same as your inseous speed, wat is tat called? cruise control + Example 6) I s t t is a measure o eet traveled per second, ind, a) te average velocity between t 0 and t b) te inseous velocity at t seconds. s + s s( )! s( 0) ( + ) +! 7 t sec! 0 t sec Example 7) I s( t) t! t is a measure o eet traveled per second, ind a) te average velocity between t 0 and t b) te inseous velocity at t seconds. s + s s s 0! 0 +!! 0 t sec AB Solutions Stu Scwartz! +! + 0 ( + ) + As gets close to 0, avg. vel. t sec Example 8) I s t t t t is a measure o eet traveled per second, ind a) te average velocity between t and t b) te inseous velocity at t second. s s! 9 t sec + + ( + )! ( + )!! s + s !!!! As gets close to 0, avg. vel. 4 t sec
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