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# Calculus 1 MATH 161

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This 71 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 161 at Millersville University of Pennsylvania taught by J. Buchanan in Fall. Since its upload, it has received 6 views. For similar materials see /class/223526/math-161-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.

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Date Created: 10/15/15

Numerical Integration MATH 161 Calculus l J Robert Buchanan Department of Mathematics Summer 2008 0 Some definite integrals possess integrands for which we do not yet know antiderivatives o This prevents us from using the Fundamental Theorem of Calculus Part to evaluate the definite integral 0 We may still estimate the definite integral using a Riemann sum Approximate the definite integral using a Riemann sum and midpoint evaluation Midpoint Rule then b x dx x M in cam i1 1 Estimate e x2 dX using the Midpoint Rule 0 Error 7 0748747 0747304 0746944 0746854 0746832 0746826 48 x 104 12 x 104 30 x 105 80 x 106 20 x 106 r VrAgproximation Approximate the signed area using trapezoids rather than rectangles Trapezoidal Rule Area of a trapezoid base times average height Thusif 7a oAX and o XiaiAXforiO1nthen bfxdx g MAHWM fXquot 12 XquotAX fx0 2fX1 2fXn71 fXn n71 Tn fx0fxn22fxi 3971 1 Estimate e x2 dX using the Trapezoidal Rule 0 Error 0742984 8 0745866 958x10 4 16 0746585 239x10 4 32 0746764 60x10 5 64 0746809 15x10 5 128 0746820 40x10 6 We will denote by EMn and ETn the errors in using the Midpoint and T I 39 39 39 Rules I quot 39 to I 39 a definite integral Suppose f is continuous on a b and lf xl g K for all agxgb then bit3 lEMquotl E 24n2 3 lETnl lt Klb a 12f2 Find the smallest value of n so that the error in using the Midpoint Rule and the Trapezoidal Rule to approximate 1 2 e x dx 0 d2 7X2 7 2 7X2 Hlnt e gt722X 71e is less than 10 6 Homework 0 Read Section 47 0 Pages 413415 7 27 odd omit Simpson39s Rule Inverse Trigonometric Functions MATH 161 Calculus I J Robert Buchanan Department of Mathematics Fall 2008 Inverse Functions Definition Assume that functions f and 9 have domain sets A and B respectively and that fgx is defined for all X in B and gfx is defined for all X in A If X for all X in B and X for all X in A we say that g is the inverse of f written 9 f 1 Equivalently f is the inverse of g Definition A function f is called onetoone when for every y in the range of f there is exactly one X in the domain of f for which y fX When does a function have an inverse A function f has an inverse if and only if it is onetoone If f is a differentiable function on interval a b and f X gt O on a b then f is onetoone and hence invertible on a b Remark The last theorem is also true when f x lt O on ab Sine and Inverse Sine Definition The inverse sine function is defined as y sin 1 X if and only ifsinyxandig gyg Cosine and Inverse Cosine Definition The inverse cosine function is defined as y cos 1 X if and onlyifcosyxand0 y7r Tngand Inverse Tangent Definition The inverse tangentfunction is defined as y tan 1 X if and only iftany X andig lty lt Secant and Inverse Secant z Definition The inverse secant function is defined as y sec 1 X if and only if secy X and y e 0 g U gm Relationship Between Arccosne and recant 7 Suppose y sec 1 X then secy and y e 0 U gm cosy y cos 1 lt1 x and thus tagt an Inverse Cgent efinion The inverse cotangent function is defined as y cot 1 X if and only ifcotyx and0lty lt 7139 Cosecant and Inverse Cosecant Definition The inverse cosecantfunction is defined as y csc 1 X if and only if cscy X and y E 770 U 07 a o sinsin 1 A X Xfor71 o sin 1sin X X for 7 7 0 coscos 1XXfor 71g X g o cos 1cosx XforO g X g 7139 o tantan 1 X X for 700 lt X lt oo o tan 1tanX Xfor 7 lt X lt g SXS 7f ltX A Evaluate the following expressions 4amp7 3 9 cos cos 37r 1 9 sIn sin 74 gt 3 1 7 9 cot sin 4 7 717 9 sec tan 4 Evaluate the following expressions 9 cos ltcos 1 5 37r 1 9 sIn sin 74 gt 3 1 7 9 cot sin 4 7 717 9 sec tan 4 3 Evaluate the following expressions 9 cos ltcos 1 5 37r 7r 1 9 sIn sin 74 gt 4 3 1 7 9 cot sin 4 7 1 7 9 sec tan 4 3 Example VEvaln te the following expressions 0 cos ltcos 1 3 37r 7r 1 7 7 6 sIn sin 4 gt 4 3 1 7 7 a cot sin 4 3 Evaluate the following expressions 0 cos ltcos 1 Q 9 sin 1 sin I 3 Evaluate the following expressions 0 cos ltcos 1 Q 9 sin 1 sin 3 Evaluate the following expressions 0 cos ltcos 1 5 37r 1 9 sIn sin 74 gt 3 1 7 9 cot sin 4 3 Derivatives We can use the Chain Rule to find derivatives ofthe inverse trigonometric functions sinsin 1 X X d 1 7 d asmsm X 7 am cossin 1x sin 1x 1 V17 X2 sin 1 X 1 d 1 1 Derivatives continued d sin 1x 7 1 dx 7 i17X 2 1 71 7cos x 77 d 112 itan 1x 7 1 d 7 1X2 iseciw dx Mixx271 d 1 1 EM X 77 ics x dx Mixx271 Find the following derivatives 0 g ltcos 13xgt a 5X swex 0 5 lttan 1x2gt a g lt5ec 1lnxgt Antiderivatives Ifagt0then 1 1u du 7 sm gC 1 1 1u mdu 7 5tan 3C 1 1 mdu gsec 1gC ifugta Evaluate the following definite and indefinite integrals 1 4 0 7 dx 0 1 7X2 ex 9 1 e2quot dx 9 cosx dX 1 sin2X Homework 0 Read pages 221224 on page 225 work exercises 2938 Natural Logarithm as an Integral MATH 161 Calculus I J Robert Buchanan Department of Mathematics Fall 2008 Background W 7 0 Previously we had worked with functions Inx and ex numerically and hypothesized the general properties of these two functions 0 Now that we have encountered the Fundamental Theorem of Calculus we can treat these two functions rigorously 0 We will see that the rigorous treatment will lead to the same properties we had hypothesized earlier Definition of In X Definition For X gt O we define the natural logarithm function as Definition of In X Definition For X gt O we define the natural logarithm function as Note X 1dr 7 bythe FTC Part II 1 i Inx 7 1 dx 7 dx Properties of the Natural Logarithm 1 of 3 o n1 0 9 nab Inanb 9 In Inailnb 9 na rlna For any a b gt O and any rational number r Properties of the Natural Logarithm 2 of 3 r o 1 1 n1 idt0 1t 3171 a1 3171 nab 1 dt1dt Ydt a 3171 na 7dr 3 t Letautthenadudt b nabna1Ddulnalnb 1 Properties of the Natural Logithm 3of 3 7 I 1 dlx 1rx39 d I39 7 EUMX Fa irnx 7 dx Thus by the MVT nx rInXC n1 rn1C n1 rn1C r0C O C which implies nx rln X Approximating Values of the Natural Logarithm i 7 Use a Riemann sum to approximate In 3 7 The Transcendental Number e Definition We define e to be the real number such that Ine 1 The Transcendental Number e Definition We define e to be the real number such that Ine 1 Use Newton s method to approximate e by solving the equation Inxi10 The Exponential Function Since ln X gt 0 then the natural logarithm function is invertible on 0 00 Definition For any real numberx we define y equot to be the numberfor which lny lnequot X The Exponential Function Since ln X gt 0 then the natural logarithm function is invertible on 0 00 Definition For any real numberx we define y equot to be the numberfor which lny lnequot X Thus we have Inequot x foriooltXltoo e X forXgtO Properties of the Exponential Function 1 of 2 For any real numbers r and s and any rational number t o eres ers e i eris es 0 ert eff Properties of the Exponential Function 2 of 2 ne es ne nes rlnesne rs rsne ne s Since Inx is a onetoone function then e eS e 1 Derivative of the Exponential unction if 7 7 We have defined y equot if and only if Iny X d d WWW am liy 1 ydx dy 7 a 7 V iiexi ex General Exponential Function 7 if 7 7 fyaxwhereagt0anda 1then Iny Thus naquot Xlna xlna e ax exlna39 General Logarithmic Functions 7 Ifagt Oanday 1thenx ay ifand only if logax y General Logarithmic Function 7 lfagt Oanday 1thenx ay ifand only if logax y Question how can we work with logarithms with general bases General Logarithmic Function 7 7 7 7 lfagt Oanday 1thenx ay ifand only if logax y Question how can we work with logarithms with general bases lnx lnay ylna lnx Ina 7 y lnx logax Ina Derivative of General Logarithmic Funt 7 ilo X 7 1 dx 93 7 dx Inx Ina Xlna Homework 0 Read Section 48 0 Pages 424426 1 45 odd Sums and Sigma Notation MATH 161 Calculus I J Robert Buchanan Department of Mathematics Spring 2009 Overview 0 We have mentioned that the antiderivative is associated with the process of accumulation or summing 0 Today we will introduce the notation and properties of summations Overview 0 We have mentioned that the antiderivative is associated with the process of accumulation or summing 0 Today we will introduce the notation and properties of summations I7 Zaia1azan i1 i summation index a summand 5 0 Zw7r7r7r7r7r57r 1 4 92139 i1 5 i 9215 5 0 Zw7r7r7r7r7r57r 1 4 e Zi123410 1 6 I 925 5 0 Zw7r7r7r7r7r57r 1 4 Zi123410 1 1 2 3 4 5 6 21 2 6 I Zi1 i Write the following sums in summation notation 0 481248 1 1 1 01 m 9 e7r4 e7r2 e37r4 e27r Summation Formulas 1 of 2 If n is any positive integer and c is any constant then I7 0 Z c cn sum of a constant i1 I7 9 Zi w sum of rst n positive integers i1 e i2nn12n1 6 sum of the rst n squares Summation Formulas 2 of 2 n Zi123n71n8 id 1 n 3 n72 2 n71 n1 n1 n1 n1 n1 A Property of Summations For any constants c and d I7 7 7 203 db cZaHr dei i1 i1 i1 Example Use summation formulas and properties of sums to evaluate the following summations o 2313975 i1 e 102 4122 2139 Let fx X2 3X 4 and 0 find the sum of the values of fx forx 205 X 210 x 215 x300 I7 9 find the sum 2 fXAX where AX is the difference i1 between consecutive values of X AppHca on Suppose the velocity of a falling object is recorded at various times t s 00 v ms 100 05 10 15 zoi 25 30 149 i193 i247 i296 i345 i394 Estimate the distance the object falls over the interval 0 g t g 3 Homework 0 Read Section 42 0 Pages 360362 1 37 odd

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