Tangent Lines-1. Tangent Lines

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1 Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property tat tey intersect te circle in exactly one point. Any curve tat can be obtained from a circle by projection (casting sadows) as tangent lines tat are te projections (sadows) of te tangent lines to te circle. Tese curves are ellipses, parabolas, and yperbolas, and teir tangents ave te same intersection property.

2 Tangent Lines-2 If we wis to define tangent lines to oter curves, we cannot expect te single intersection property to still old true. If we are given te grap of a function wit equation y f(x), and wis to define te tangent to te grap at a point P (a, f (a)) on te grap, ten all we need is to define its slope m, and we can use te point-slope form y f (a) m(x a)

3 Tangent Lines-3 Slope of a Tangent Line If P (a, f (a)) is a point on te grap of a continuous function f, te secant line troug P and anoter point Q (b, f(b)) as slope 8 y m PQ f(b) f (a). If m PQ Q P b a exists and equals m, f(b) f (a) P Q we define te tangent line to y f(x) at P to be te line troug P wit slope m x

4 Tangent Lines-4 Te Point-Slope Form equation of tis tangent line is y f (a) m(x a) We can rewrite te it calculation, using and b a +, in te form Q P m PQ b a f(a+ ) f (a) f(b) f (a) f(a+ ) f (a) (a + ) a

5 Tangent Lines-5 Example : f(x) x 2, P (2, 4). m PQ f(b) f(2) b 2 b b 2 f(b) f (a) (b 2)(b + 2) b 2 b + 2, so m m PQ Q P (b + 2) b b 2 b + 2 b 2 Alternative it formulation: f(2 + ) f(2) m (2 + ) (2) (4 + )

6 Tangent Lines-6 Te equation of te tangent line is y 4 4(x 2) y x (2,4)

7 Tangent Lines-7 Example 2: f(x) x 2, P (a, a 2 ). m PQ f(b) f (a) b2 a 2 ()(b + a) b + a, so m m PQ b + a exists (and equals Q P b a b + a a + a 2a for any number a.) b a b a

8 Tangent Lines-8 If we take a to be, ten m m PQ (b + ) + 2, Q P b a so te Point-Slope Form equation of te tangent line to te curve y x 2 at te point (, ) is y 2(x ) Since te slope of te tangent line at (a, a 2 ) is 2a, its equation in Point-Slope Form is y a 2 2a(x a)

9 Tangent Lines-9 Using te alternative it formulation, we ave, for arbitrary a: m f(a+ ) f (a) a 2 + 2a + 2 a 2 2a + 2a + 0 2a (a + ) 2 a 2 2a + 2 (2a + )

10 Tangent Lines-0 Example 3: f(x) ( x, P 2, ). 2 m PQ ( b f(b) f (a) )( ) ( 2 b 2 b b 2 b 2 2b )( ) 2 b 2 2b, f(b) f(2) b 2 ( ) ( 2 b 2b 2b b 2 ) b 2 b 2 2 b 2b b 2 so m m PQ Q P b 2 2b 2(2) 4 exists, and tus te equation of te tangent line is y 2 4 (x 2)

11 Tangent Lines- Using te alternative it formulation, we ave m f(2 + ) f(2) (2 + )2 2 + (2 + )2

12 Tangent Lines-2 2 (2 + ) (2 + )2 (2 + )2 (2 + )2 (2 + )2 (2 + 0)2 4 y x

13 Tangent Lines-3 Example 4: f(x) x, P (4, 2). m PQ f(b) f (a) f(b) f(4) b 4 b 4 b 4 b 2 b 4 ( ) b 2 b + 2 b 4 b + 2,so b + 2 ( b) (b 4)( b + 2) b 4 (b 4)( b + 2) m PQ Q P b 4 b exists, so te equation of te tangent line is y 2 4 (x 4)

14 Tangent Lines-4 Using te alternative it formulation, we ave f(4 + ) f(4) m ( ) ( 4 + ) ( ) ( ) ( )

15 Tangent Lines-5 ( ) ( ) ( ) ( ) (2 + 2) 4 y x

16 Tangent Lines-6 Example 5: f(x) x 3, P (, ). m PQ f(b) f (a) (b )(b 2 + b + ) b Q P m PQ b f(b) f() b b 2 + b +, so b 2 + b + 3 exists, so te equation of te tangent line is b3 3 b y y 3(x ) x

17 Tangent Lines-7 Using te alternative it formulation, we ave m f( + ) f() ( + ) () 2 + 3() ( ) ( ) (3 + 3(0) ) 3

18 Tangent Lines-8 Velocity Te Average Velocity of a moving object is defined to be te Distance Travelled divided by te Time Elapsed : Average Velocity Distance Travelled Time Elapsed Te Average Speed is te absolute value of te average velocity. So, if we drive 20 km in 90 minutes, our average speed and velocity are bot equal to: 20km 90min 4 3 km min 4 3 km min 60min our 80 km our

19 Tangent Lines-9 If te moving object is assumed to be moving on a straigt line, and its distance from a fixed reference point at time t is given by a function s(t), ten te average velocity from time t to time t 2 will be s(t 2 ) s(t ) t 2 t

20 Tangent Lines-20 Example: s(t) 2t 2 6t + 3 Time Interval [t,t 2 ] Average Velocity s(t 2) s(t ) t 2 t [0, ] s() s(0) [0, 2] [0, 3] [0, 4] [3, 4] s(2) s(0) 2 0 s(3) s(0) 3 0 s(4) s(0) 4 0 s(4) s(3)

21 Tangent Lines-2 On te general interval [a, b] te Average Velocity is s(b) s(a) 2b2 6b + 3 (2a 2 6a + 3) 2(b 2 a 2 ) 6() 2()(b + a) 6() 2(b + a) 6

22 Tangent Lines-22 Let us look at te Average Velocity near t 3: Time Interval [3,t 2 ] Average Velocity 2(3 + t 2 ) 6 [3, 4] 2(3 + 4) 6 8 [3, 3.] 2(3 + 3.) [3, 3.0] 2( ) [3, 3.00] 2( ) [3, 3.000] 2( ) [3, ] 2( )

23 Tangent Lines-23 and Time Interval [t, 3] Average Velocity 2(t + 3) 6 [2, 3] 2(2 + 3) 6 4 [2.9, 3] 2( ) [2.99, 3] 2( ) [2.999, 3] 2( ) [2.9999, 3] 2( ) [ , 3] 2( )

24 Tangent Lines-24 If we take te it as t approaces 3, we get: s(t) s(3) t 3 t 3 t 3 2(t + 3) 6 2(3 + 3) Tis it is called te Instantaneous Velocity at time t 3.

25 Tangent Lines-25 In general, te Instantaneous Velocity at time t a is defined to be s(t) s(a) t a t a Wit our example function s(t) 2t 2 6t + 3 we ten ave s(t) s(a) t a t a t a 2(a + t) 6 2(a + a) 6 4a 6

26 Tangent Lines-26 Anoter equivalent and very useful formula for te instantaneous velocity at time t is: s(t + ) s(t) Wit our example function s(t) 2t 2 6t + 3 we ten ave s(t + ) s(t) 2(t + ) 2 6(t + ) + 3 (2t 2 6t + 3) 2(t 2 + 2t + 2 ) 6t t 2 + 6t 3 2t 2 + 4t t t 2 + 6t 3 4t t t + 2(0) 6 4t 6

27 Tangent Lines-27 Oter Rates of Cange If a quantity Q(t) varies wit time, its average rate of cange over te time interval [t 0,t ] is defined to be Q(t ) Q(t 0 ) t t 0 Te instantaneous rate of cange is defined to be if tis it exists. Tis it is also equal to Q(t ) Q(t 0 ) t t 0 t t 0 Q(t 0 + ) Q(t 0 )