2.la Definition of the Derivative =D Derivative measures the instantaneous rate of change of a function at a particular input. S Sty Soup = D Velocity = rate. Velocitey = Change in clistance Change in time L> how can we measure such a thing at an instant, when the change in time is 0? The answer involDon't forget about the age old question of What are the two kinds of social structures?
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ves limits + derivatives. Average velocity = Average Velocitu - h(t)- hlal a 19999999999911111111 19999999999911111 The Derivative of a function fat Point a f'(a) = lim f(x) – fraj - X-> x-a (prime) indicates we are talking about the clerivative of f. L> instantaneous rate of change of fat a Is The slope of the tangent line to the graph at (a,fca)) Ex. What is the derivative of f(x) = x?+ at x = 2?. f'(a): lim f(x) = f(2) X-2 X-2 lim X2+1-5 X->2 X-2 lim x2-4 - X->2 X -2 lim (x+2)(x-2) = x-> (x-2) - lim (x+2) lim(x+2) Y-)2 = 4 Alternate form of derivative formula: f'ca) - lim fcx) - fra) Studs = xa x-a -> fias= lim fcath) - fcas 20 h Ex. What is the derivative of f(x) = ŕ at x=1? f'(o)= him flith) – f(1). im len talimś lith-1) ho ho lim Ima h-) h-> o -G) - PPPPPPPPPPPPPPPPPPPPPPDDDDDD0000 - J 12 -3 - limy Find the equation of the line tangent to the graph - f(x)=√x-3 at 12,3) in lim f(12+h)-f(12) lim 512th 3 - 512-3 Slope = f(12) = no ho ho h ima v 9th - J12-3. lim sath-3Jath +3 - h- o h I n vath +3 lim 9th-q lim - = = ho hath+3) h-so 9th+ 3 59 + 0 +3 6 y = 3+ = (x-12) y = 3 + 7-2 = n o y = x + 1 Equation of a Tangent line Point - slope form of a line y=yo+m(x-xo) . Tangent line equation @x=a y = Yo + f '(a)(x-xo). - fla) + f(al(x-a) 2.2a The Derivative as a function & Graphing the Derivative * what is the derivative of fat -1 is the same question as what is the slope of the tangent line when X=-1? * For every x value, there is a corresponding value of the derivative. I The derivative itself is a function of X. The Derivative as a function For any function f, we define the derivative function, f' by f'(x), lim f(x+h)-f(x) F CAN h- o h ใใใใใใใใใใใใใใใใใใใใใใใใใใใใใใใใใ * Difference btwn f'(x) + fica) I The result of f'(x) will be an expression in x, + defines a function of x. i f'ca) is a number but f'(x) is a function of x dy df difference in y-values * f'(x) = y . dr - dx difference in x-values * If we want to indicate the value of a derivative at a specific number a w. the Leibniz Notation -> For y = f(x) CIX SO ficado Ix=a Ex. Let f(x)= x2-3x. Find f '(x) f'(x)= lim f(x-1) - f(x) lim (x+h)3-3(x+h)-(x3-3x) cho nho - lim (x3+ 3x? h+3xha+h3)-3x-3h - X2+3x Sh. - lim 3x? h+3xha+h3-3h a 3x2+3xh+ h2-3 ho = 3x2 + 3x.0 +02-3 = 342-3 ho ay If f'>0 on an interval, then f is increasing over that interval. If f'<o on an interval, then f is clecreasing over that interval on f Position (f')'(x) = f "cx) a (f")'(x) = F"(x) - co by f' velocity | f"')'(x) = f(u(x) in cel ales Crateof change of position) f" Acceleration Crate of Change of velocity) 2.26 Properties of the Derivative LLLLLLL LLLLLLLLLPPPPPPPPPPPPPT -> A fonction is differentiable at a point if it has a derivative there. Lyf is differentiable at a if then the following limit exists when does the Derivative of f Exist? Afunction fis differentiable at a if € Calexists. It is clifferentiable on an open interval (a, b) if it is differentiable at every number in the interval. f is differentóc ble at a if f'(a): 8- ho ha exists When can a function failto be differentiable? lim f(a+h)-f(a) v sono 1 x + x Sharp Corner discontinuity Vertical Tangent L Differentiable or continuous at O ? f(x) = 1x) +2 - g(x) = sin hcx) = linx X continuous neither continuous t differentiable (x) = XY2 ) mcx) = In (X42) continuous ใใใใใใใใใใใใใใใๆ ใใใใใใ continuous + clifferentiable * If a function is differentiable at a point, it must be continuous there StudySou: f (B)-f(2) 2 -1 c Get the slope Concep In! The graph of a function y = f(x). For each, use the figureto decide which is large. a) f (3) or f(4) b) f(3) f (2) or f (2)-f(1) c) f(2)-f(1) ar f13) - f12) f(a)-fci) 2-1 3-2 c) f'l orf'4 @f(4) > f(3) ! 2 3 4 bf (2) -- FU) > f(3)-f(2) © f(2)-f(1), f(3) - f(2) @ fll> 614 Free response Propiem on midterm 1 Ex. Find the equation of the line tangent to the graph of f(x) = 2x at(1, 1) bali) m = f'(1) Ly To write the equation of a lin: (i) (1, 1) (i) slope - M flas- lim f(a+h)-f(a) (ii) a point (X, Yo) - hoh Point - slope form f'() im Flith) - fl!) " hoh y = m ( x - Xo) + yo 57 1 [ f(x) = 2x I lima h) 0 - 4= *(x-1)+! <7 flitW211th) atha (1th) 12th) | fli) = 2.2.1 no +ń -' -> 2 h nh X+! uselimit definition of derivative 211+h)-1 ttn +. X+1 h ņ muttiplig by lithiti -im 201+h) - Warm Hohet y = x - +1 y = 1 / 2 x + 1 / 2 th lim A. i ho ath Te slope of 'f'li) - him in a tangent Tline Study Soup Study 2/13/17 The Derivative as a function hy we find the derivative of a function, f, ata point a by: fW) = f(a+h)-f(a) Now we let a vary to get the derivative function f '(x): f'xilim f(x +h)-f(x) Sud I g-f(x) 1 y = f '(x) Take derivativesofa y = f(x) to form -> apply Rule of Thumb: f(x) = x2 -> f'(x) = ? - using the limit def. Es apply foolim f(x+h)-f(x), f(x) = x2 -> f(1) = ? Ls apply f'(x) = him f(1+h)-f(1) + compute Ex. Find f'(x) if f(x) = 2 2 2 (К-1200 ompute & compute ho his mother on m (50) (18) Tx Jxth x S xth x x x x = 250-250th 12 +24x+h) hux xth 12. -2 JXth) X3/2 4x-4(Xth) - 4K - direct substitutia Thurs xth ( 258 +2 58th) Insx Sxth (2 586 +25x75 l of h00 x x (25 x + 2x) Study ou SEO Sketch the derivative of the function on graph in clecreasing i (-) Slope (-) derivative www. * Draw tangent unes, work out slope increasing *Identify it (+) Slope it is becoming . more or less SE (+) derivative negative move (+) Î Slope is becoming uss neg., thin Tuss (-), thin i positive, Slope thin i positive o 45 Connection between f t f our Ls tangent line is horizontal = f'-o L> f is decreasing -> f'(x) <0 Lfis increasing -> f'(x) > 0 Ex where is f(x)= x3-6x3 increasing? where is f'(x) >0 ? f(x) -> f'(X) = 372-12x > 0 i 4 3x(x-4)>0 increasing when X>0 and X24 SEO Studio 2/13/17 Given f '(x) Ex. On what intervals is the function of increasin? A) (0,x) b) (0x2) C) (0,X2) + (x4.xs) d) (X2, X3) | Sour Lecture - Week 4 Feb. 13, 2017 Question on midterm: Find derivative using limit definition -> which of the following class represent f'(a)? @ lim fla+h)-fch) - ha a flath) Lo lim f(a+h)-f(a) ho h @ lim f(x) -fra) to f(a) Ix-a X-a These two W im Fla)-f(x) are the same. a lim flath)-f(a)_ f'(a) The Derivative of a function at a point ho - la numberl L> The derivative of a function tata point a, denoted by fila), is F(X) L fia) - lim f(x)-fla) - xa x-a 1. flab fca) = lim flath)-f(a) x-) a a-x ath > lavsou -->0 on a 9999999999999911111111111 - f'(a) is the slope of the tangent Ine to the graph of of eat a - f'(a) represents the instantaneous rate of change of fata lim f(x) = f(a) = f'(a) X-) A X -a