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# Calculus I MATH 1351

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GPA 3.62

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Date Created: 10/22/15

15 Jan 11 Critical Ideas TermsDefinitions FactsRulesTheorems Supplementary Problems Math 1351013 Spring 2004 Lecture Summaries distance on a number line absolute value distance in a plane trigonometry solving trigonometric equations real numbers real number line absolute value distance interval notation bounded interval open interval halfopen interval closed interval absolute value equation property tolerance absolute error horizontal change vertical change midpoint analytical geometry graph of an equation unit circle completing the square degree radian order properties tricotomy law transitive law of inequality additive law of inequality multiplicative law of inequality absolute value formula distance formula on real number line properties of absolute value intervals inequality notation interval notation graphical representation theorem distance formula in the plane midpoint formula standard form for the equation of a circle 11 every other odd 145 20 Jan 1213 Critical Ideas TermsDefinitions FactsRulesTheorems slope of a line forms for the equation of a line parallel and perpendicular lines definition of a function functional notation domain of a function composition of functions graph of a function classi cation of functions inclination slope angle of inclination parallel perpendicular function image domain range onto function onetoone function bounded function variables dependent variable independent variable evaluate diiTerence quotient piecewisedefmed function domain convention undefined equal functions hole composite function graph vertical line test y intercept xintercept symmetry symmetric with respect to the yaxis even function symmetric with respect to the origin odd function polynomial function degree leading coef cient constant term constant function linear function quadratic function cubic function quartic function rational function power function algebraic function transcendental function trigonometric functions exponential functions logarithmic functions formula for the slope of a line formula for the angle of inclination of a line forms of the equation of a line standard form slopeintercept form pointslope form twointercept form horizontal line vertical line slope criteria for parallel and perpendicular lines rule for equality of two functions rules for finding the yintercepts and xintercepts of a function test for yaXis symmetry of the graph of a function test for origin symmetry of the graph of a function Supplementary Problems 12 every other odd l45 13 every other odd l6l 22 Jan 14 Critical Ideas inverse functions criteria for existence of an inverse f 1 graph of f 1 inverse trigonometric functions inverse trigonometric identities TermsDe nitions inverse of f onetoone function horizontal line test strictly increasing strictly decreasing strictly monotonic reference triangle FactsRulesTheorems theorem a strictly monotonic function has an inverse procedure for nding the graph of the inverse of a function graphs of sin 391 x tan391 x inversion formulas for trigonometric Jnctions Supplementary Problems 14 every other odd 541 27 Jan 21 Critical Ideas informal computation of limits onesided limits limits that do not exist formal de nition of a limit TermsDe nitions limit of a function righthand limit le hand limit diverge tend to infinity divergence by oscillation epsilondelta de nition FactsRulesTheorems lim f x L lim f x L lim f x L theorem oneside limit theorem limfx oo limfx oo Supplementary Problems 21 every other odd l4l 29 Jan 22 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems computations with limits using algebra to nd limits limits of piecewisede ned functions two special trigonometric limits squeeze rule basic properties and rules for limits constant rule limit of x rule multiple rule sum rule diiTerence rule product rule quotient rule power rule theorem limit of a polynomial function theorem limit of a rational Jnction where de ned theorem limits of sin x trigonometric lnctions where de ned theorem special limits lim l xgt0 cos x 1 11m 0 xgt0 x 22 every other odd 157 03 Feb 23 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems intuitive notion of continuity de nition of continuity continuity theorems continuity on an interval the intermediate value theorem continuous at a point xc discontinuity continuous from the right at a continuous from the le at a continuous on the open interval a b continuous on the halfopen interval 61 b continuous on the halfopen interval a b continuous on the closed interval 61 b suspicious point intermediate value property root theorem continuity theorem polynomials rational functions trigonometric functions inverse trigonometric functions are continuous where defmed theorem properties of continuous functions scalar multiples sums and di erences products quotients where de ned compositions where de ned of continuous functions are again continuous functions theorem intermediate value theorem theorem root location theorem 23 every other odd l4l 05 Feb 24 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems exponential functions logarithmic functions natural exponential and logarithmic functions continuous compounding of interest completeness property exponential function with base b logarithm of x to the base b exponent to the base b natural exponential base natural exponential function natural logarithm common logarithm continuous compounding of interest present value principal interest rate future value theorem properties of exponential functions equality rule inequality rules product rule quotient rule power rules theorem properties of logarithmic functions equality rule inequality rules product rule quotient rule power rule inversion rules special values theorem basic properties of natural logarithm ln 1 0 ln 6 l 6quot x exlnb lney y b theorem change ofbase logb x lln Z n 24 every other odd 161 10 Feb 31 Critical Ideas TermsDe nitions FactsRulesTheorems tangent lines the derivative relationship between the graphs of f and f 39 existence of derivatives continuity and di erentiability derivative notation secant line slope of tangent line di erence quotient derivative off differentiate f at x f di erentiable at x formula for the slope of a tangent line to y f x at x x0 formula for the derivative Supplementary Problems of a function fat x W f x theorem formula for the gt0 equation of a tangent line to y f x at x x0 theorem differentiability implies continuity 31 every other odd 561 12 Feb 32 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems derivative of a constant function derivative of a power function procedural rules for nding derivatives higherorder derivatives first derivative of f second derivative of f third derivative of f nth derivative theorem constant rule theorem power rule theorem basic procedural rules constant multiple sum rule di erence rule linearity rule product rule quotient rule 32 every other odd 149 17 Feb 33 Critical Ideas derivatives of the sine and cosine functions differentiation of the other trigonometric functions derivatives of exponential and logarithmic functions TermsDe nitions FactsRulesTheorems theorem trigonometric functions sin x cos x cos x sin x theorem other trigonometric functions theorem natural exponential function e e theorem natural logarithm function ln x i x Supplementary Problems 33 every other odd 153 19 Feb 34 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems average and instantaneous rate of change introduction to mathematical modeling rectilinear motion modeling in physics falling body problem average rate of change of y with respect to x instantaneous rate of change relative rate of change mathematical modeling abstraction velocity acceleration speed advancing retreating accelerating decelerating position falling body problem 34 every other odd 561 24 Feb 35 Critical Ideas introduction to the chain rule extended derivative formulas justi cation of the chain rule TermsDefinitions horizontal tangent line FactsRulesTheorems theorem chain rule f g x39 f 39 gxg 39x extended power rule uquot 39 n u 1 u39 extended trigonometric rules extended exponential and logarithmic rules Supplementary Problems 35 every other odd 561 26 Feb 36 Critical Ideas general procedure for implicit dilTerentiation derivative formulas for the inverse trigonometric functions logarithmic dilTerentiation TermsDefinitions explicitly defined function implicitly defined function implicit dilTerentiation logarithmic dilTerentiation FactsRulesTheorems theorem dilTerentiation rules for inverse trigonometric functions theorem dilTerentiation of exponential and logarithmic functions with base b Supplementary Problems 36 every other odd 157 02 Mar 37 Critical Ideas TermsDefinitions related rate problems general situation speci c situation FactsRulesTheorems Supplementary Problems 37 every other odd 145 04 Mar 38 Critical Ideas TermsDefinitions FactsRulesTheorems Supplementary Problems tangent line approximation the dilTerential error propagation marginal analysis in economics the NewtonRaphson method for approximating roots linear approximation linearization incremental approximation formula dilTerential of x differential of y propagation of error error in measurement propagated error relative error percentage error marginal cost marginal revenue demand function dilTerential rules linearity rule product rule quotient rule power rule trigonometric rules exponential and logarithmic rules inverse trigonometric rules 38 every other odd 149 09 Mar 41 Critical Ideas extreme value theorem relative extrema absolute extrema optimization TermsDe nitions optimization problems absolute maximum absolute minimum absolute extrema extreme values relative maximum relative minimum relative extrema critical number of f critical point on the graph of f FactsRulesTheorems theorem extreme value of a continuous function on 61 b theorem critical number theorem Supplementary Problems 41 every other odd 113 every other odd 2157 11 Mar 42 Critical Ideas Rolle s theorem statement and proof of the mean value theorem the zeroderivative theorem TermsDe nitions FactsRulesTheorems theorem Rolle s theorem theorem mean value theorem theorem zeroderivative theorem theorem constant dilTerence theorem Supplementary Problems 42 every other odd 541 23 Mar 43 Critical Ideas increasing and decreasing functions the rstderivative test concavity and in ection points the second derivative test curve sketching using the rst and second derivatives TermsDe nitions strictly increasing on an interval strictly decreasing on an interval montonic relative maximum relative minimum not an extremum concave up concave down in ection point of a graph secondorder critical number rstorder critical number diminishing retums FactsRulesTheorems theorem monotone function theorem rst derivative test second derivative test Supplementary Problems 43 every other odd 549 25 Mar 44 Critical Ideas TermsDe nitions limits to in nity in nite limits graphs with asymptotes vertical tangents and cusps a general graphing strategy limits to in nity in nite limits vertical asymptote horizontal asymptote vertical tangent cusp extent symmetry FactsRulesTheorems theorem special limits to in nity limii 0 for r gt 0 xgt 0 x Supplementary Problems 44 every other odd 541 30 Mar 45 Critical Ideas a rule to evaluate indeterminate forms indeterminate forms 00 and other indeterminate forms special limits involving 6 and ln x TermsDe nitions indeterminate forms FactsRulesTheorems theorem l Hopital s rule 00 other indeterminate forms 139 0quot 00 0 theorem limits involving exponentials and logarithms lim In x oo xgt0 x k 11m 11m n 00 H x Supplementary Problems 45 every other odd 153 01 Apr 46 Critical Ideas optimization procedure Fermat s principle of optics and Snell s law TermsDe nitions optimization problems optimization evt convention Fermat s principle of optics Snell s law of refraction relative index of refraction FactsRulesTheorems Supplementary Problems 46 7 9 11 12 13 16 17 18 19 20 21 22 24 26 06 Apr 47 Critical Ideas economics maximizing pro t and marginal analysis business management an inventory model and optimal holding time physiology concentration of a drug in the bloodstream and optimal angle for vascular branching TermsDe nitions discrete lnctions marginal analysis demand function total revenue total pro t marginal cost marginal revenue average cost Poiseuille s resistance to ow law FactsRulesTheorems Supplementary Problems 47 every other odd 129 08 Apr 51 Critical Ideas reversing differentiation antiderivative notation antidiiTerentiation formulas applications area as an antiderivative TermsDe nitions antiderivative slope eld direction eld indefinite integral of f inde nite integration constant of integration area function FactsRulesTheorems theorem any two antiderivatives of a function differ by a constant theorem basic integration rules constant multiple rule sum rule difference rule linearity rule constant n1 rule 0 du 0 c power rule Iuquot du u c n at l exponential rule 71 l e du c logarithm rule 1 du ln u c trigonometric rules inverse u trigonometric rules theorem area as an antiderivative Supplementary Problems 51 odd 129 odd 4151 13 Apr 52 Critical Ideas area as the limit of a sum the general approximation scheme summation notation area using summation notation TermsDe nitions summation notation sigma notation index of summation dummy variable FactsRulesTheorems theorem basic rules for summation constant term rule sum rule scalar multiple rule linearity rule dominance rule Supplementary Problems 52 odd 127 odd 3943 15 Apr 53 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems Riemann sums the de nite integral area as an integral properties of the de nite integral distance as an integral partition Riemann sum norm of partition regular partition f is integrable on a b de nite integral of f from a to b integrand interval of integration lower limit of integration upper limit of integration total distance traveled net distance net displacement theorem integrability of a continuous function f on a b theorem properties of de nite integrals linearity rule dominance rule subdivision rule 53 odd 129 20 Apr 5455 Critical Ideas the rst fundamental theorem of calculus the second fundamental theorem of calculus substitution with inde nite integrals substitution with de nite integrals TermsDe nitions dummy variable FactsRulesTheorems theorem the first fundamental theorem of calculus if F 39 f on 61 b then jb f x dx F b F a theorem the second fundamental theorem of calculus let Gx jxfa dtfor x 6 ab then G39x fx Supplementary Problems 54 every other odd 157 55 every other odd 141 22 Apr 56 Critical Ideas introduction and terminology direction elds separable dilTerential equations modeling exponential growth and decay orthogonal trajectories modeling uid ow through an ori ce modeling the motion of a projectile escape velocity TermsDe nitions dilTerential equation solution general solution solved slope eld direction eld separable exponential change growth decay carbon dating orthogonal trajectory isotherms velocity potential curves escape velocity FactsRulesTheorems Supplementary Problems 56 every other odd 133 odd 4351 27 Apr 57 Critical Ideas TermsDe nitions FactsRulesTheorems Supplementary Problems mean value theorem for integrals modeling average value of a function average value trapezoid rule Simpson s rule natural logarithm inversion formulas theorem mean value theorem for integrals jb f x dx f c b a for some 1 1 61 c e a b average value offon ab is 1 fx dx 57 every other odd 133 29 Apr 5859 Critical Ideas TermsDe nitions FactsRulesTheorems approximation by rectangles trapezoid rule Simpson s rule error estimation natural logarithm as an integral geometric interpretation the natural exponential function 3 trapezoid rule Simpson s rule error estimate in trapezoid rule I E IS bl 6M n 5 3K theorem properties of natural error estimate in Simpson s rule l E is b a 180n logarithm function de ned as In x Jul dt properties of exponential function de ned 1 l as inverse of natural logarithm function Supplementary Problems 58 odd 125 59 2 3 10 Jan 11 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems Math 1351008 Spring 2007 Lecture Summaries distance on a number line absolute value distance in a plane trigonometry solving trigonometric equations real numbers real number line absolute value distance interval notation bounded interval open interval halfopen interval closed interval absolute value equation property tolerance absolute error horizontal change vertical change midpoint analytical geometry graph of an equation unit circle completing the square degree radian order properties tricotomy law transitive law of inequality additive law of inequality multiplicative law of inequality absolute value formula distance formula on real number line properties of absolute value intervals inequality notation interval notation graphical representation theorem distance formula in the plane midpoint formula standard form for the equation of a circle 11 every other odd l45 12 Jan 12 Critical Ideas slope of a line forms for the equation of a line parallel and perpendicular lines TermsDefinitions inclination slope angle of inclination parallel perpendicular FactsRulesTheorems formula for the slope of a line formula for the angle of inclination of a line forms of the equation of a line standard form slopeintercept form pointslope form twointercept form horizontal line vertical line slope criteria for parallel and perpendicular lines Supplementary Problems 12 every other odd l45 17 Jan 13 Critical Ideas TermsDefinitions definition of a function functional notation domain of a function composition of functions graph of a function classification of functions function image domain range onto function onetoone function bounded function variables dependent variable independent variable evaluate difference quotient piecewisedefined function domain convention undefined equal functions hole composite function graph vertical line test yintercept x intercept symmetry symmetric with respect to the yaXis even function symmetric with respect to the origin odd function polynomial function degree leading coefficient constant term constant function linear function quadratic function cubic function quartic function rational function power function algebraic function transcendental function trigonometric functions exponential functions logarithmic functions Facts Rules Theorems Supplementary Problems rule for equality of two functions rules for nding the yintercepts and x intercepts of a function test for yaXis symmetry of the graph of a function test for origin symmetry of the graph of a function 13 every other odd 161 19 Jan 14 Critical Ideas inverse functions criteria for existence of an inverse f 391 graph of f 391 inverse trigonometric functions inverse trigonometric identities TermsDefinitions inverse of f onetoone function horizontal line test strictly increasing strictly decreasing strictly monotonic reference triangle FactsRulesTheorems theorem a strictly monotonic function has an inverse procedure for finding the graph of the inverse of a function graphs of sin 391 x tan 391 x inversion formulas for trigonometric functions Supplementary Problems 14 every other odd 541 22 Jan 21 Critical Ideas informal computation of limits onesided limits limits that do not eXist formal definition of a limit TermsDefinitions limit of a function righthand limit lefthand limit diverge tend to infinity divergence by oscillation epsilondelta definition FactsRulesTheorems lim f x L lim f x L lim f x L theorem oneside limit theorem lim fx 00 lim fx 00 Supplementary Problems 21 every other odd 141 24 Jan 22 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems computations with limits using algebra to find limits limits of piecewisedefined functions two special trigonometric limits squeeze rule basic properties and rules for limits constant rule limit of x rule multiple rule sum rule difference rule product rule quotient rule power rule theorem limit of a polynomial function theorem limit of a rational function where defined theorem limits of trigonometric functions where defined theorem special sinx cosx l 111m 0 xgt0 x limits lim Xgt0 22 every other odd 157 26 Jan 23 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems intuitive notion of continuity de nition of continuity continuity theorems continuity on an interval the intermediate value theorem continuous at a point xc discontinuity continuous from the right at a continuous from the left at a continuous on the open interval ab continuous on the halfopen interval 61 b continuous on the halfopen interval a b continuous on the closed interval ab suspicious point intermediate value property root theorem continuity theorem polynomials rational functions trigonometric functions inverse trigonometric functions are continuous where defined theorem properties of continuous functions scalar multiples sums and differences products quotients where defined compositions where defined of continuous functions are again continuous functions theorem intermediate value theorem theorem root location theorem 23 every other odd l4l 29 Jan 24 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems exponential functions logarithmic functions natural exponential and logarithmic functions continuous compounding of interest completeness property exponential function with base b logarithm of x to the base b exponent to the base b natural exponential base natural exponential function natural logarithm common logarithm continuous compounding of interest present value principal interest rate future value theorem properties of exponential functions equality rule inequality rules product rule quotient rule power rules theorem properties of logarithmic functions equality rule inequality rules product rule quotient rule power rule inversion rules special values theorem basic properties of natural logarithm xlnb ln 1 0 ln 6 1 em x ln ey y b e theorem change ofbase lnx lo x gt 1nb 24 every other odd 161 02 Feb 31 Critical Ideas TermsDefinitions Facts Rules Theorems tangent lines the derivative relationship between the graphs of f and f existence of derivatives continuity and differentiability derivative notation secant line slope of tangent line difference quotient derivative off differentiate f at x f differentiable at x formula for the slope of a tangent line to y f x at x x0 formula for the Supplementary Problems derivative of a function fat x W f x theorem formula gt for the equation of a tangent line to y f x at x x0 theorem differentiability implies continuity 31 every other odd 561 05 Feb 32 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems derivative of a constant function derivative of a power function procedural rules for nding derivatives higherorder derivatives rst derivative of f second derivative of f third derivative of f nth derivative theorem constant rule theorem power rule theorem basic procedural rules constant multiple sum rule difference rule linearity rule product rule quotient rule 32 every other odd 149 09 Feb 33 Critical Ideas derivatives of the sine and cosine functions differentiation of the other trigonometric functions derivatives of exponential and logarithmic functions TermsDefinitions FactsRulesTheorems theorem trigonometric functions sin x cos x cos x sin x theorem other trigonometric functions theorem natural exponential function e 39 e theorem natural logarithm function ln x i x Supplementary Problems 33 every other odd 153 12 Feb 34 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems average and instantaneous rate of change introduction to mathematical modeling rectilinear motion modeling in physics falling body problem average rate of change of y with respect to x instantaneous rate of change relative rate of change mathematical modeling abstraction velocity acceleration speed advancing retreating accelerating decelerating position falling body problem 34 every other odd 561 14 Feb 35 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems introduction to the chain rule extended derivative formulas justification of the chain rule horizontal tangent line theorem chain rule fgx39 f gxg39x extended power rule uquot 39 nuHu39 extended trigonometric rules extended exponential and logarithmic rules 35 every other odd 561 1619 Feb 36 Critical Ideas general procedure for implicit differentiation derivative formulas for the inverse trigonometric functions logarithmic differentiation TermsDefinitions explicitly defined function implicitly defined function implicit differentiation logarithmic differentiation FactsRulesTheorems theorem differentiation rules for inverse trigonometric functions theorem differentiation of exponential and logarithmic functions with base b Supplementary Problems 36 every other odd 157 21 Feb 37 Critical Ideas TermsDefinitions related rate problems general situation specific situation FactsRulesTheorems Supplementary Problems 37 every other odd 145 23 Feb 38 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems tangent line approximation the differential error propagation marginal analysis in economics the NewtonRaphson method for approximating roots linear approximation linearization incremental approximation formula differential of x differential of y propagation of error error in measurement propagated error relative error percentage error marginal cost marginal revenue demand function differential rules linearity rule product rule quotient rule power rule trigonometric rules exponential and logarithmic rules inverse trigonometric rules 38 every other odd 149 28 Feb 41 Critical Ideas extreme value theorem relative extrema absolute extrema optimization TermsDefinitions optimization problems absolute maximum absolute minimum absolute extrema extreme values relative maximum relative minimum relative extrema critical number of f critical point on the graph of f FactsRulesTheorems theorem extreme value of a continuous function on ab theorem critical number theorem Supplementary Problems 41 every other odd 113 every other odd 2157 05 Mar 42 Critical Ideas Rolle s theorem statement and proof of the mean value theorem the zero derivative theorem TermsDefinitions FactsRulesTheorems theorem Rolle s theorem theorem mean value theorem theorem zero derivative theorem theorem constant difference theorem Supplementary Problems 42 every other odd 541 0709 Mar 43 Critical Ideas increasing and decreasing functions the firstderivative test concavity and in ection points the second derivative test curve sketching using the first and second derivatives TermsDefinitions strictly increasing on an interval strictly decreasing on an interval montonic relative maximum relative minimum not an extremum concave up concave down in ection point of a graph secondorder critical number f1rstorder critical number diminishing returns FactsRulesTheorems theorem monotone function theorem first derivative test second derivative test Supplementary Problems 43 every other odd 549 1921 Mar 44 Critical Ideas TermsDefinitions Facts Rules Theorems limits to in nity in nite limits graphs with asymptotes vertical tangents and cusps a general graphing strategy limits to infinity infinite limits vertical asymptote horizontal asymptote vertical tangent cusp extent symmetry theorem special limits to infinity lim 4 0 for r gt 0 Hoe x Supplementary Problems 44 every other odd 541 2326 Mar 45 Critical Ideas a rule to evaluate indeterminate forms indeterminate forms 00 and 00 00 other indeterminate forms special limits involving ex and In x TermsDefinitions indeterminate forms FactsRulesTheorems theorem l Hopital s rule 00 00 00 other indeterminate forms 1 O 0 O 00 0 00 0 oo 00 theorem limits involving eXponentials and logarithms l l 1 11m nx cgto 11m nx011me oo xgt0 xquot xgtoo xquot xgtoo x Supplementary Problems 45 every other odd 153 28 Mar 46 Critical Ideas optimization procedure Fermat s principle of optics and Snell s law TermsDefinitions optimization problems optimization evt convention Fermat s principle of optics Snell s law of refraction relative index of refraction FactsRulesTheorems Supplementary Problems 46 7 9 11 12 13 16 17 18 19 20 21 22 24 26 47 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems economics maximizing pro t and marginal analysis business management an inventory model and optimal holding time physiology concentration of a drug in the bloodstream and optimal angle for vascular branching discrete functions marginal analysis demand function total revenue total pro t marginal cost marginal revenue average cost Poiseuille s resistance to ow law 47 every other odd 129 02 Apr 51 Critical Ideas TermsDefinitions Facts Rules Theorems reversing differentiation antiderivative notation antidifferentiation formulas applications area as an antiderivative antiderivative slope field direction field indefinite integral of f indefinite integration constant of integration area function theorem any two antiderivatives of a function differ by a constant theorem basic integration rules constant multiple rule sum rule difference rule linearity Supplementary Problems n1 u c nab l n1 rule constant rule IO du 0 c power rule Iuquot du exponential rule I du e c logarithm rule Ii du ln l u l 0 u trigonometric rules inverse trigonometric rules theorem area as an antiderivative 51 odd 129 odd 4151 06 Apr 52 Critical Ideas area as the limit of a sum the general approximation scheme summation notation area using summation notation TermsDe nitions summation notation sigma notation index of summation dummy variable FactsRulesTheorems theorem basic rules for summation constant term rule sum rule scalar multiple rule linearity rule dominance rule Supplementary Problems 52 odd 127 odd 3943 11 Apr 53 Critical Ideas Riemann sums the de nite integral area as an integral properties of the definite integral distance as an integral TermsDe nitions partition Riemann sum norm of partition regular partition f is integrable on ab de nite integral of f from a to b integrand interval of integration lower limit of integration upper limit of integration total distance traveled net distance net displacement FactsRulesTheorems theorem integrability of a continuous function f on 61 b theorem properties of de nite integrals linearity rule dominance rule subdivision rule Supplementary Problems 53 odd 129 13 Apr 54 Critical Ideas Terms De nitions Facts Rules Theorems Supplementary Problems the rst fundamental theorem of calculus the second fundamental theorem of calculus dummy variable theorem the first fundamental theorem of calculus if F 39 f on ab then f fx dx Fb Fa theorem the second fundamental theorem of calculus 1a Gx j ft dt for x e abthen Gm fx 54 every other odd 157 1618 Apr 55 Critical Ideas Terms De nitions Facts Rules Theorems Supplementary Problems substitution with inde nite integrals substitution with de nite integrals dummy variable 55 every other odd l4l 20 Apr 56 Critical Ideas introduction and terminology direction elds separable differential equations modeling exponential growth and decay orthogonal trajectories modeling uid ow through an ori ce modeling the motion of a projectile escape velocity TermsDe nitions differential equation solution general solution solved slope eld direction eld separable exponential change growth decay carbon dating orthogonal trajectory isotherms velocity potential curves escape velocity FactsRulesTheorems Supplementary Problems 56 every other odd 133 odd 4351 23 Apr 57 Critical Ideas mean value theorem for integrals modeling average value of a function TermsDe nitions average value trapezoid rule Simpson s rule natural logarithm inversion formulas FactsRulesTheorems theorem mean value theorem for integrals f f x dx f c b a for some a e a b average value offon 61 b is b1 jb fx dx a a Supplementary Problems 57 every other odd 133 25 Apr 58 Critical Ideas approximation by rectangles trapezoid rule Simpson s rule error estimation TermsDe nitions FactsRulesTheorems trapezoid rule Simpson s rule error estimate in trapezoid rule Supplementary Problems 3 5 l E I b 6 M error estimate in Simpson s rule I E I b azK 12n 180n 58 odd 125 27 Apr 59 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems natural logarithm as an integral geometric interpretation the natural exponential function theorem properties of natural logarithm function de ned as In x dt 1 properties of exponential function defined as inverse of natural logarithm function 59 2 3 28 Aug 11 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems Math 1351011 Fall 2006 Lecture Summaries distance on a number line absolute value distance in a plane trigonometry solving trigonometric equations real numbers real number line absolute value distance interval notation bounded interval open interval halfopen interval closed interval absolute value equation property tolerance absolute error horizontal change vertical change midpoint analytical geometry graph of an equation unit circle completing the square degree radian order properties tricotomy law transitive law of inequality additive law of inequality multiplicative law of inequality absolute value formula distance formula on real number line properties of absolute value intervals inequality notation interval notation graphical representation theorem distance formula in the plane midpoint formula standard form for the equation of a circle 11 every other odd l45 30 Aug 12 Critical Ideas slope of a line forms for the equation of a line parallel and perpendicular lines TermsDefinitions inclination slope angle of inclination parallel perpendicular FactsRulesTheorems formula for the slope of a line formula for the angle of inclination of a line forms of the equation of a line standard form slopeintercept form pointslope form twointercept form horizontal line vertical line slope criteria for parallel and perpendicular lines Supplementary Problems 12 every other odd l45 01 Sep 13 Critical Ideas TermsDefinitions definition of a function functional notation domain of a function composition of functions graph of a function classification of functions function image domain range onto function onetoone function bounded function variables dependent variable independent variable evaluate difference quotient piecewisedefined function domain convention undefined equal functions hole composite function graph vertical line test yintercept x intercept symmetry symmetric with respect to the yaXis even function symmetric with respect to the origin odd function polynomial function degree leading coefficient constant term constant function linear function quadratic function cubic function quartic function rational function power function algebraic function transcendental function trigonometric functions exponential functions logarithmic functions Facts Rules Theorems Supplementary Problems rule for equality of two functions rules for nding the yintercepts and x intercepts of a function test for yaXis symmetry of the graph of a function test for origin symmetry of the graph of a function 13 every other odd 161 06 Sep 14 Critical Ideas inverse functions criteria for existence of an inverse f 391 graph of f 391 inverse trigonometric functions inverse trigonometric identities TermsDefinitions inverse of f onetoone function horizontal line test strictly increasing strictly decreasing strictly monotonic reference triangle FactsRulesTheorems theorem a strictly monotonic function has an inverse procedure for finding the graph of the inverse of a function graphs of sin 391 x tan 391 x inversion formulas for trigonometric functions Supplementary Problems 14 every other odd 541 08 Sep 21 Critical Ideas informal computation of limits onesided limits limits that do not eXist formal definition of a limit TermsDefinitions limit of a function righthand limit lefthand limit diverge tend to infinity divergence by oscillation epsilondelta definition FactsRulesTheorems lim f x L lim f x L lim f x L theorem oneside limit theorem lim fx 00 lim fx 00 Supplementary Problems 21 every other odd 141 11 Sep 22 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems computations with limits using algebra to find limits limits of piecewisedefined functions two special trigonometric limits squeeze rule basic properties and rules for limits constant rule limit of x rule multiple rule sum rule difference rule product rule quotient rule power rule theorem limit of a polynomial function theorem limit of a rational function where defined theorem limits of trigonometric functions where defined theorem special sinx cosx l 111m 0 xgt0 x limits lim Xgt0 22 every other odd 157 13 Sep 23 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems intuitive notion of continuity de nition of continuity continuity theorems continuity on an interval the intermediate value theorem continuous at a point xc discontinuity continuous from the right at a continuous from the left at a continuous on the open interval ab continuous on the halfopen interval 61 b continuous on the halfopen interval a b continuous on the closed interval ab suspicious point intermediate value property root theorem continuity theorem polynomials rational functions trigonometric functions inverse trigonometric functions are continuous where defined theorem properties of continuous functions scalar multiples sums and differences products quotients where defined compositions where defined of continuous functions are again continuous functions theorem intermediate value theorem theorem root location theorem 23 every other odd l4l 15 Sep 24 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems exponential functions logarithmic functions natural exponential and logarithmic functions continuous compounding of interest completeness property exponential function with base b logarithm of x to the base b exponent to the base b natural exponential base natural exponential function natural logarithm common logarithm continuous compounding of interest present value principal interest rate future value theorem properties of exponential functions equality rule inequality rules product rule quotient rule power rules theorem properties of logarithmic functions equality rule inequality rules product rule quotient rule power rule inversion rules special values theorem basic properties of natural logarithm xlnb ln 1 0 ln 6 1 em x ln ey y b e theorem change ofbase lnx lo x gt 1nb 24 every other odd 161 20 Sep 31 Critical Ideas TermsDefinitions Facts Rules Theorems tangent lines the derivative relationship between the graphs of f and f existence of derivatives continuity and differentiability derivative notation secant line slope of tangent line difference quotient derivative off differentiate f at x f differentiable at x formula for the slope of a tangent line to y f x at x x0 formula for the Supplementary Problems derivative of a function fat x W f x theorem formula gt for the equation of a tangent line to y f x at x x0 theorem differentiability implies continuity 31 every other odd 561 25 Sep 32 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems derivative of a constant function derivative of a power function procedural rules for nding derivatives higherorder derivatives rst derivative of f second derivative of f third derivative of f nth derivative theorem constant rule theorem power rule theorem basic procedural rules constant multiple sum rule difference rule linearity rule product rule quotient rule 32 every other odd 149 27 Sep 33 Critical Ideas derivatives of the sine and cosine functions differentiation of the other trigonometric functions derivatives of exponential and logarithmic functions TermsDefinitions FactsRulesTheorems theorem trigonometric functions sin x cos x cos x sin x theorem other trigonometric functions theorem natural exponential function e 39 e theorem natural logarithm function ln x i x Supplementary Problems 33 every other odd 153 29 Sep 34 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems average and instantaneous rate of change introduction to mathematical modeling rectilinear motion modeling in physics falling body problem average rate of change of y with respect to x instantaneous rate of change relative rate of change mathematical modeling abstraction velocity acceleration speed advancing retreating accelerating decelerating position falling body problem 34 every other odd 561 02 Oct 35 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems introduction to the chain rule extended derivative formulas justification of the chain rule horizontal tangent line theorem chain rule fgx39 f gxg39x extended power rule uquot 39 nuHu39 extended trigonometric rules extended exponential and logarithmic rules 35 every other odd 561 0406 Oct 36 Critical Ideas general procedure for implicit differentiation derivative formulas for the inverse trigonometric functions logarithmic differentiation TermsDefinitions explicitly defined function implicitly defined function implicit differentiation logarithmic differentiation FactsRulesTheorems theorem differentiation rules for inverse trigonometric functions theorem differentiation of exponential and logarithmic functions with base b Supplementary Problems 36 every other odd 157 09 Oct 37 Critical Ideas TermsDefinitions related rate problems general situation specific situation FactsRulesTheorems Supplementary Problems 37 every other odd 145 1 1 Oct 38 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems tangent line approximation the differential error propagation marginal analysis in economics the NewtonRaphson method for approximating roots linear approximation linearization incremental approximation formula differential of x differential of y propagation of error error in measurement propagated error relative error percentage error marginal cost marginal revenue demand function differential rules linearity rule product rule quotient rule power rule trigonometric rules exponential and logarithmic rules inverse trigonometric rules 38 every other odd 149 16 Oct 41 Critical Ideas extreme value theorem relative extrema absolute extrema optimization TermsDefinitions optimization problems absolute maximum absolute minimum absolute extrema extreme values relative maximum relative minimum relative extrema critical number of f critical point on the graph of f FactsRulesTheorems theorem extreme value of a continuous function on ab theorem critical number theorem Supplementary Problems 41 every other odd 113 every other odd 2157 20 Oct 42 Critical Ideas Rolle s theorem statement and proof of the mean value theorem the zero derivative theorem TermsDefinitions FactsRulesTheorems theorem Rolle s theorem theorem mean value theorem theorem zero derivative theorem theorem constant difference theorem Supplementary Problems 42 every other odd 541 2325 Oct 43 Critical Ideas increasing and decreasing functions the firstderivative test concavity and in ection points the second derivative test curve sketching using the first and second derivatives TermsDefinitions strictly increasing on an interval strictly decreasing on an interval montonic relative maximum relative minimum not an extremum concave up concave down in ection point of a graph secondorder critical number frrstorder critical number diminishing returns FactsRulesTheorems theorem monotone function theorem first derivative test second derivative test Supplementary Problems 43 every other odd 549 2730 Oct 44 Critical Ideas TermsDefinitions Facts Rules Theorems limits to in nity in nite limits graphs with asymptotes vertical tangents and cusps a general graphing strategy limits to infinity infinite limits vertical asymptote horizontal asymptote vertical tangent cusp extent symmetry theorem special limits to infinity lim 4 0 for r gt 0 Hoe x Supplementary Problems 44 every other odd 541 01 Nov 45 Critical Ideas a rule to evaluate indeterminate forms indeterminate forms 00 and 00 00 other indeterminate forms special limits involving ex and In x TermsDefinitions indeterminate forms FactsRulesTheorems theorem l Hopital s rule 00 00 00 other indeterminate forms 1 O 0 O 00 0 00 0 oo 00 theorem limits involving eXponentials and logarithms l l 1 11m nx cgto 11m nx011me oo xgt0 xquot xgtoo xquot xgtoo x Supplementary Problems 45 every other odd 153 03 Nov 46 Critical Ideas optimization procedure Fermat s principle of optics and Snell s law TermsDefinitions optimization problems optimization evt convention Fermat s principle of optics Snell s law of refraction relative index of refraction FactsRulesTheorems Supplementary Problems 46 7 9 11 12 13 16 17 18 19 20 21 22 24 26 47 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems economics maximizing pro t and marginal analysis business management an inventory model and optimal holding time physiology concentration of a drug in the bloodstream and optimal angle for vascular branching discrete functions marginal analysis demand function total revenue total pro t marginal cost marginal revenue average cost Poiseuille s resistance to ow law 47 every other odd 129 08 Nov 51 Critical Ideas TermsDefinitions Facts Rules Theorems reversing differentiation antiderivative notation antidifferentiation formulas applications area as an antiderivative antiderivative slope field direction field indefinite integral of f indefinite integration constant of integration area function theorem any two antiderivatives of a function differ by a constant theorem basic integration rules constant multiple rule sum rule difference rule linearity Supplementary Problems n1 u c nab l n1 rule constant rule IO du 0 c power rule Iuquot du exponential rule I du e c logarithm rule Ii du ln l u l 0 u trigonometric rules inverse trigonometric rules theorem area as an antiderivative 51 odd 129 odd 4151 13 Nov 52 Critical Ideas area as the limit of a sum the general approximation scheme summation notation area using summation notation TermsDe nitions summation notation sigma notation index of summation dummy variable FactsRulesTheorems theorem basic rules for summation constant term rule sum rule scalar multiple rule linearity rule dominance rule Supplementary Problems 52 odd 127 odd 3943 15 Nov 53 Critical Ideas Riemann sums the de nite integral area as an integral properties of the definite integral distance as an integral TermsDe nitions partition Riemann sum norm of partition regular partition f is integrable on ab de nite integral of f from a to b integrand interval of integration lower limit of integration upper limit of integration total distance traveled net distance net displacement FactsRulesTheorems theorem integrability of a continuous function f on 61 b theorem properties of de nite integrals linearity rule dominance rule subdivision rule Supplementary Problems 53 odd 129 17 Nov 54 Critical Ideas Terms De nitions Facts Rules Theorems Supplementary Problems the rst fundamental theorem of calculus the second fundamental theorem of calculus dummy variable theorem the first fundamental theorem of calculus if F 39 f on ab then f fx dx Fb Fa theorem the second fundamental theorem of calculus 1a Gx j ft dt for x e abthen Gm fx 54 every other odd 157 20 Nov 55 Critical Ideas substitution with inde nite integrals substitution with de nite integrals TermsDe nitions dummy variable FactsRulesTheorems Supplementary Problems 55 every other odd 141 27 Nov 56 Critical Ideas introduction and terminology direction elds separable differential equations modeling exponential growth and decay orthogonal trajectories modeling uid ow through an ori ce modeling the motion of a projectile escape velocity TermsDe nitions differential equation solution general solution solved slope eld direction eld separable exponential change growth decay carbon dating orthogonal trajectory isotherms velocity potential curves escape velocity FactsRulesTheorems Supplementary Problems 56 every other odd 133 odd 4351 29 Nov 57 Critical Ideas mean value theorem for integrals modeling average value of a function TermsDe nitions average value trapezoid rule Simpson s rule natural logarithm inversion formulas FactsRulesTheorems theorem mean value theorem for integrals f f x dx f c b a for some a e a b average value offon 61 b is b1 jb fx dx a a Supplementary Problems 57 every other odd 133 01 Dec 58 Critical Ideas Terms De nitions Facts Rules Theorems Supplementary Problems approximation by rectangles trapezoid rule Simpson s rule error estimation trapezoid rule Simpson s rule error estimate in trapezoid rule 3 5 l E I b 6 M error estimate in Simpson s rule I E I b azK 12n 180n 58 odd 125 04 Dec 59 Critical Ideas TermsDefinitions Facts Rules Theorems Supplementary Problems natural logarithm as an integral geometric interpretation the natural exponential function theorem properties of natural logarithm function de ned as In x dt 1 properties of exponential function defined as inverse of natural logarithm function 59 2 3 MATH 1 351 Calculus I Course Author Dr Harold Bennett Yourgmder my be di ferentfmm ne nu ior MATH 1351 features 0 3 hours credit 0 14 lessons each containing Objectives How to Proceed and Lesson Assignment 0 1 comprehensive final examination 1 textbook Prerequisite MATH 1350 MATH 1550 score of 7 or higher on the Mathematics Placement Examination or score of 5 on the Mathematics Placement Examination and MATH 1321 o All lesson assignments must be submitted via surface mail only MATH 1 35 1 v1 Published by Division of Outreach and Distance Education Texas Tech University BOX 42 191 Lubbock TX 794092191 Outreach 81 Distance Education Course Development Instructional Designer Dr Harold Bennett Copyright 2002 by the Board of Regents for the Department of Mathematics and Statistics acting for and on behalf of Texas Tech University Lubbock Texas 79409 All rights reserved TABLE OF CONTENTS Introduction to MATH 1351 Calculus I V Course Lessons Lesson One Functions and Graphs PartI 1 Lesson Two Functions and Graphs Part II 3 Lesson Three Limits and Continuity PartI 5 Lesson Four Limits and Continuity Part II 7 Lesson Five Dy quot Part I 9 Lesson Six Dy quot Part II 1 1 Lesson Seven Differentiation Part III 13 Lesson Eight Applications of the Derivative PartI 15 Lesson Nine Applications of the Derivative Part II 17 Lesson Ten Applications of the Derivative Part III 19 Lesson Eleven Integration PartI 21 Lesson Twelve Integration Part II 23 Lesson Thirteen Integration Part III 25 Lesson Fourteen Integration PartIV 27 Final inn dirertinn 7Q introduction Calculus I elcome to Calculus I the introductory calculus course for Outreach overview 8 Distance Education at Texas Tech University In this course you will study limits and continuity the differentiation of algebraic and transcendental functions applications of the derivative differentials and definite and indefinite integrals To enroll in this course you must have taken at least one of the Prerequisites following i A ng After completing this course you will be able to comprehend and work college analytical geometry at Texas Tech MATH 1350 college precalculus at Texas Tech MATH 1550 high school precalculus with a score of B or better or at least one semester of high school trigonometry and at least one semester of high school analytical geometry both with a score of B or better Course Objectives problems successfully in the following subjects and their subtopics functions and graphs limits and continuity differentiation additional applications of the derivative integration MATH 1351 V10 Introduction v Required Textbook About the Textbook Texas Tech University The required textbook for this course is Strauss Monty J Gerald L Bradley and Karl Smith Calculus 3rd ed Upper Saddle River NJ Prentice Hall 2002 If you are planning to take Calculus 11 andor Calculus III with Outreach 8 Distance Education at Texas Tech this is the textbook for those courses too so be sure to keep it Prentice Hall says you can order a student solutions manua quot that goes with the textbook too This manual contains solutions to the odd numbered problems with all work shown not just the answers as in the textbook 1 am not requiring you to buy this manual because I think it adds expense without a lot of value There are plenty of examples in the textbook But if you think you would benefit from such a manual you can always give it a look Unlike some ODE courses this basic level calculus course does not include discussions or introductions in the course guide rather the introductions discussions examples and problem solving strategies you need are to be found in the carefully selected textbook I chose this approach deliberately because I believe studying math in a distance learning course is hard enough without making students feel that they must master two sets of material in the textbook and in the course guide If a textbook is available that is comprehensive and fully explanatory which certainly is not always the case but is so with this course the best that discussions in a course guide can do is to repeat the textbook material unnecessarily At worst they can restate material in a slightly different way so that the similarity is clear to someone who already understands the subject matter but to a neophyte who doesn t understand that the two sources are saying the same thing in slightly different terms it can seem that there is twice as much material to master In this situation more energy is spent in resolving the differences between sources than in learning and understanding the material So since this textbook does its job so well I have centered the course around using the various features of the textbook to fullest advantage At the beginning of each chapter in the book are two sections titled Preview and Perspective The Preview lists what specific subjects you will cover in that chapter while the Perspective section will give you an vi Introduction MATH 1351 V10 Outreach amp Distance Education idea of why it matters where the theory came from who developed it what sorts of practical applications it has and so on To avoid confusion the lesson objectives in each lesson in this course guide are taken directly from the Contents listed at the beginning of each textbook chapter Periodically the text includes short passages called Historical Questsquot Reading these for your own interest will give you a good outline of the history of math Chapters are divided into sections designated by decimal points 11 12 13 etc When you read each chapter section you should start by reading the list at the beginning called In This Section which announces the subjects and tasks that you will be studying and doing Then read the text in that section paying particular attention to the copious examples that are worked out in detail be sure you look closely at those and really understand what s going on and how a solution was reached Then you should work some of the odd numbered questions in the problem set at the end of the section Answers for those odd numbered questions are at the back of the book This would be a good place for me to mention a phrase that I repeat to all my students frequently both in the traditional classroom and at a distance quotMath is not a spectator sport What I m getting at is the need for participation in a math course especially one that you are attempting at a distance You must build your skills from the ground up You simply cannot understand this material to the level that will be necessary for you to do well on the final exam without working a lot of problems And the more problems of every kind that you work the more successful you will be After you finish this process with a section do the problems assigned for that section in the lesson assignment at the end of each lesson in this course guide These problems are selected from the even numbered questions in the problem set with an occasional odd numbered one thrown in rarely If you have trouble with an even numbered question on a lesson assignment try working the problems before and after it and checking your answers with those in the back of the book Be sure to show all your work also be sure that your calculations are neat enough to be read easily and that you indicate your answer clearly You will repeat this process with each section in a chapter until you complete the chapter it will take more than one lesson to do so At the MATH 1351 v10 Introduction vii Texas Tech University end of each chapter is a review with a proficiency examination and supplementary problems It will be worthwhile for you to do these problems as well whenever you reach the end of a chapter every few lessons or so In fact you can make the proficiency exam very helpful by simulating test conditions Set a timer turn off music and TV and do the problems without looking back at the chapter or your notes If you train for the test this way not only will you get a realistic idea of how you would have done on an actual exam over that material you will also avoid the test shockquot that can happen if you haven t done anything other than the graded lesson assignments which allow you to take all the time and get all the help you need A couple of potentially confusing points in the textbook Theorems and figures in the textbook are numbered separately without regard for what section of the chapter they re in so that may be a bit confusing for instance Figure 19 is in Section 12 an unfortunate and potentially difficult system of numbering Just be aware in case you re looking for a particular figure N The numbering system for examples in the textbook restarts at 1 for each section of each chapter so if you remember that Example 5quot helped you and you re looking back at an Example 5quot that looks nothing like the one you remember be sure you re in the right section Grading The 14 lesson assignments count 40 of the final grade the final exam counts 60 Grading is on the standard percentage scale A F A 90 100 B 80 89 C 70 79 D 60 69 F 59 or below You need to know two critical things about the final exam 1 N0 computers or calculators will be allowed on the final You will be expected to do your own calculations and show your work N As with all Outreach 8 Distance Education courses you must pass the final to pass the course no matter what your grades on the lesson assignments have been or what your final weighted viii Introduction MATH 1351 v10 Outreach amp Distance Education average would be because of those assignment grades However if you fail on a first try at the final exam you may apply for one retake Dr Harold Bennett PhD Arizona State is a full professor in the AbOUt the Allthor Department of Mathematics and Statistics at Texas Tech University His interests include sports and reading but he mostly enjoys spending time with his three children He even claims to enjoy doing math in his spare time MATH 1351 v10 Introduction ix one Functions and Graphs Partl fter completing this lesson you will be able to understand and work Lesson ObieCtiveS problems successfully concerning distance in a number line absolute value distance in the plane trigonometry and trigonometric equations slope of a line forms for the equation of a line parallel and perpendicular lines 1 Be sure you have read the Lesson Objectives above and the In This HOW to Proceed Section list for Section 11 in Chapter 1 of your textbook Also read the Preview and Perspectives sections at the beginning of the chapter 2 Survey read and take notes on Section 11 paying close attention to how the examples are worked out in the book Make sure you understand not only the theory but also how that theory is illustrated in the examples 3 Work some unassigned odd numbered problems at the end of the section and check your answers with those listed in the back of the text MATH 1351 v10 Lesson One 1 Lesson One Assignment 6 Texas Tech University Complete the questions listed for Section 11 in the Lesson One Assignment Show all of your work clearly and indicate your answers legibly Repeat steps 1 4 above for Section 12 When you have completed the Lesson One Assignment to your satisfaction send it to Outreach 8 Distance Education according to the instructions given in the Policies 8 Forms Guide Please be sure to staple the cover sheet provided to your lesson assignment Review your notes and keep working practice problems until you complete the course After you have finished this lesson you may proceed to Lesson Two Complete the following problems Show all of your work clearly and legibly and be sure to indicate your answers Section Problems 11 468 14 18 20 22 26 30 36 40 42 52 12 2 4 6 8 12 14 16 20 24 28 30 46 50 2 Lesson One MATH 1351 V10

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