Notes: DERIVATIVES Velocity and Oter Rates of Cange
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 )
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
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+
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
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 ( ) 3+ 3 0 9+ 6+ 9 0 + 6 + 0 0 + 6 6 0 6 ( )
Tangent line f (3)3 9 y 9 6( x 3) 5.) f( x), at x 1 x f(1 + ) f(1) 0 5 5 1+ 1 0 0 5 5(1 + ) (1 + ) 5 + 0 (1 ) 5 5 0 (1+ 0) f (1)5 y 5 5( x 1)
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 0 + 9 ( ) (9 ) 4+ 4 0 f( x) 9 x, x y 5 ( x ) 1 4
V. Free Fall Motion A.) st ( ) 16 t + vt+ s or st ( ) 4.9t + vt+ s 0 0 0 0 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 0 16(3 ) 16(0 ) st ( ) 48 ft/sec 3
0 0 0 0.) Find te instantaneous velocity at t 3 seconds. s(3 + ) s(3) + + 16(3 ) 16(3 ) 16(9 6 ) 16(9) + + + + 144 96 16 144 0 96 16 ( ) 96 16 0 96 ft/sec
t 3.) Find te velocity at te instant te rock its te ground. 16t 104 8 0 + + 16(8 ) 16(8 ) 0 56 ft/sec
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.
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
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.
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!
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+ 1 0 0 x+ + + x+ ( ) ( ) x+ + x+ x+ + + x+ 0 x+ + + x+ 0 ( x+ + ) ( x+ ) ( + + + + ) x x 1 f '( x), x> x +
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 +
C.) Using bot definitions, find te derivative of te following: gx x x ( ) 3 + 5 You sould get g'( x) 6x+ 1
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
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) + 0 0 Bot must exist and be equal to te same real number!
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.
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 + (0 + ) (0) 1 + 1 0
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.
B.) Does te following function ave a derivative at x 1? 3 x, x< 1 f( x) 6x 3, x 1 Left-and: 0 0 0 3(1 + ) 3(1) 3(1 ) 3 6+ 3 6 + + 0 + 0 Rigt-and: 6(1 + ) 3 (6(1) 3) + 6+ 6 3 6+ 3 6 6 + 0
Since te left-and and rigt-and derivatives bot equal 6 as x approaces 1, tis function does ave a derivative at x 1.
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.
Ex. 1 Given te grap of f below, estimate te grap of f. y f (x) y f (x)
Ex. Given te grap of f below, estimate te grap of f. y f (x) y f (x)
Ex. 3 Given te grap of f below, estimate te grap of f. y f (x) y f (x)