Notes: DERIVATIVES. Velocity and Other Rates of Change
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1 Notes: DERIVATIVES Velocity and Oter Rates of Cange
2 I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b, f ( b )
3 B.) Ex.- Find te average rate of cange of f(x) on [, 5] for te following functions: 1.) ( ) f x x x 1.) f( x) e f(5) f() m e 5 m 5 4 m 7 3 as x canges from to 5, te y-values increase at an average rate of 7 to 1. e 3 4 1
4 C.) Def.- Te average rate of cange of f(x) on te interval [a, a+] is f(a) f( a+ ) f( a) f(a+) a a+
5 II. Instantaneous Rate of Cange A.) Def.- Te instantaneous rate of cange of f(x) at x a (if it exists) is f '( a) 0 tangent f( a+ ) f( a) m to f ( x ) at x a slope of tge curve f( x) at x a te derivative of f( x) at x a
6 III. Examples A.) Find te instantaneous rate of cange of te following functions at te given x values. Ten, give te equation of te tangent line at tat point. 1.) f( x) x, at x 3 f(3 + ) f(3) 0 ( ) ( )
7 Tangent line f (3)3 9 y 9 6( x 3) 5.) f( x), at x 1 x f(1 + ) f(1) (1 + ) (1 + ) (1 ) (1+ 0) f (1)5 y 5 5( x 1)
8 IV. Normal Line A.) Def. Te normal line to a curve at a point is te line perpendicular to te tangent line at tat point. B.) Ex. Find te equation of te normal line to te following grap: m m ( ) (9 ) f( x) 9 x, x y 5 ( x ) 1 4
9 V. Free Fall Motion A.) st ( ) 16 t + vt+ s or st ( ) 4.9t + vt+ s B.) A rock is dropped from a cliff 1,04 ig. Answer te following: 1.) Find te average velocity for te first 3 seconds. st () s(3) s(0) (3 ) 16(0 ) st ( ) 48 ft/sec 3
10 ) Find te instantaneous velocity at t 3 seconds. s(3 + ) s(3) (3 ) 16(3 ) 16(9 6 ) 16(9) ( ) ft/sec
11 t 3.) Find te velocity at te instant te rock its te ground. 16t (8 ) 16(8 ) 0 56 ft/sec
12 VI. Rectilinear Motion A.) Def: Te motion of a particle back and fort (or up and down), along an axis s over a time t. B.) Te displacement of te object over te time interval from is t to t s s( t+ t) st () i.e., te cange in position.
13 C.) Te average velocity of te object over te same time interval is s t s t t st ( + ) () t displacement t D.) Te instantaneous velocity of te object at any time t is v t ds st ( + t) st () s'( t) dt t ( ) t 0 E.) Te speed of te object at any time t is speed v t ( ) ds dt
14 F.) Te acceleration of te object at any time t is dv d s a( t) v () t s () t dt dt G.) A particle in rectilinear motion is speeding up if te signs of te velocity and te acceleration are te same. H.) A particle in rectilinear motion is slowing down if te signs of te velocity and te acceleration are te opposite.
15 VII. Definitions Te derivative of a function f(x) at any point x is A.) f '( x) 0 f( x+ ) f( x) provided te it exists! B.) Alternate Def. f f( x) f( a) '( x) x a x a provided te it exists!
16 f VIII. Calculating Derivatives A.) Using te formal definition of derivative, calculate te derivative of f( x) x+ f( x+ ) f( x) '( x) 0 0 x+ + + x+ x+ + x x+ + + x+ ( ) ( ) x+ + x+ x+ + + x+ 0 x+ + + x+ 0 ( x+ + ) ( x+ ) ( ) x x 1 f '( x), x> x +
17 B.) Using te alternate definition of derivative, calculate te derivative of f( x) x+ x a f( x) f( a) f '( x) x a x a x a x+ a+ 1 x a x a x a x a x+ a+ x+ + a+ x a x a x+ + a+ ( x a)( x+ + a+ ) ( ) x a ( x+ ) ( a+ ) ( x a)( x+ + a+ ) 1 f '( x), a > a +
18 C.) Using bot definitions, find te derivative of te following: gx x x ( ) You sould get g'( x) 6x+ 1
19 IX. Derivative Notation A.) Te following all represent te derivative y ' f '( x) d dx At a point - dy dx df dx d dx [ f( x) ] ' y ' ( ) ( ) f df dx x
20 X. Differentiability A.) Def. If f (x 0 ) exists for all points on [a, b], ten f (x) is differentiable at x x 0. B.) In order for te derivative to exist, f( x+ ) f( x) f( x+ ) f( x) Bot must exist and be equal to te same real number!
21 C.) One-sided derivatives- For any function on a closed interval [a, b] f( a+ ) f( a) te rigt-and derivative at a. + 0 f( b+ ) f( b) te left-and derivative at b. 0 D.) Visual Representation: RT-and Deriv. LT-and Deriv.
22 XI. Examples A.) Does f(x) x ave a derivative at x 0? f( x) Left-and: 0 xx, < 0 xx, 0 (0 + ) ( 0) Rigt-and: (0 + ) (0)
23 f '( x) 1, x < 0 1, x > 0 Since te left-and and rigt-and derivatives equal different values, tis function does not ave a derivative at x 0.
24 B.) Does te following function ave a derivative at x 1? 3 x, x< 1 f( x) 6x 3, x 1 Left-and: (1 + ) 3(1) 3(1 ) Rigt-and: 6(1 + ) 3 (6(1) 3)
25 Since te left-and and rigt-and derivatives bot equal 6 as x approaces 1, tis function does ave a derivative at x 1.
26 XII. Grapical Relationsips Since te derivative of f at a point a is te slope of te tangent line to f at a, we can get a good idea of wat te grap of te function f looks like by estimating te slopes at various points along te grap of f.
27 Ex. 1 Given te grap of f below, estimate te grap of f. y f (x) y f (x)
28 Ex. Given te grap of f below, estimate te grap of f. y f (x) y f (x)
29 Ex. 3 Given te grap of f below, estimate te grap of f. y f (x) y f (x)
f a h f a h h lim lim
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