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mathua

mathua

Description

School: NYU School of Medicine
Department: Engineering
Course: MATHUA 121: Calculus I
Professor: Selin kalaycioglu
Term: Spring 2017
Tags: Math, Calculus, and Derivatives
Cost: 25
Name: Readings and Lecture Notes_ Week 4
Description: These notes cover the readings and lecture from week 4: - Definition of the Derivative - The Derivative as a Function and Graphing the Derivative - Properties of the Derivative
Uploaded: 02/16/2017
10 Pages 117 Views 0 Unlocks
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What is the derivative of f(x) = ŕ at x=1?




What is the derivative of f(x) = x?




L> how can we measure such a thing at an instant, when the change in time is 0?



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?
We also discuss several other topics like What are the energy sources for convection heat?
We also discuss several other topics like  How can we apply Piaget’s theory to education?
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Don't forget about the age old question of chm 372 purdue
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

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