Math 1351 011 Fall 2006 Lecture Summaries 28 Aug 1 1 Critical Ideas distance on a number line absolute value distance in a plane trigonometry solving trigonometric equations Terms Definitions real numbers real number line absolute value distance interval notation bounded interval open interval half open 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 Facts Rules Theorems 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 Supplementary Problems 1 1 every other odd 1 45 30 Aug 1 2 Critical Ideas slope of a line forms for the equation of a line parallel and perpendicular lines Terms Definitions inclination slope angle of inclination parallel perpendicular Facts Rules Theorems 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 slope intercept form point slope form two intercept form horizontal line vertical line slope criteria for parallel and perpendicular lines Supplementary Problems 1 2 every other odd 1 45 01 Sep 1 3 Critical Ideas Terms Definitions 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 one to one function bounded function variables dependent variable independent variable evaluate difference quotient piecewise defined function domain convention undefined equal functions hole composite function graph vertical line test y intercept xintercept symmetry symmetric with respect to the y axis 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 rule for equality of two functions rules for finding the y intercepts and xintercept s of a function test for y axis symmetry of the graph of a function test for origin symmetry of the graph of a function Supplementary Problems 1 3 every other odd 1 61 06 Sep 1 4 Critical Ideas inverse functions criteria for existence of an inverse f 1 graph of f 1 inverse trigonometric functions inverse trigonometric identities Terms Definitions inverse of f one to one function horizontal line test strictly increasing strictly decreasing strictly monotonic reference triangle Facts Rules Theorems theorem a strictly monotonic function has an inverse procedure for finding the graph of the inverse of a function graphs of sin 1 x tan 1 x inversion formulas for trigonometric functions Supplementary Problems 1 4 every other odd 5 41 08 Sep 2 1 Critical Ideas informal computation of limits one sided limits limits that do not exist formal definition of a limit Terms Definitions limit of a function right hand limit left hand limit diverge tend to infinity divergence by oscillation epsilon delta definition Facts Rules Theorems lim f x L lim f x L lim f x L theorem one side limit theorem x c x c x c lim f x lim f x x c Supplementary Problems 11 Sep 2 2 Critical Ideas x c 2 1 every other odd 1 41 computations with limits using algebra to find limits limits of piecewise defined functions two special trigonometric limits Terms Definitions squeeze rule Facts Rules Theorems 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 sin x cos x 1 limits lim 1 lim 0 x 0 x 0 x x Supplementary Problems 2 2 every other odd 1 57 13 Sep 2 3 Critical Ideas intuitive notion of continuity definition of continuity continuity theorems continuity on an interval the intermediate value theorem Terms Definitions continuous at a point x c discontinuity continuous from the right at a continuous from the left at a continuous on the open interval a b continuous on the half open interval a b continuous on the half open interval a b continuous on the closed interval a b suspicious point intermediate value property root Facts Rules Theorems 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 Supplementary Problems 2 3 every other odd 1 41 15 Sep 2 4 Critical Ideas exponential functions logarithmic functions natural exponential and logarithmic functions continuous compounding of interest Terms Definitions 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 Facts Rules Theorems 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 e 1 e ln x x ln e y y b x e x ln b theorem change of base ln x log b x ln b Supplementary Problems 2 4 every other odd 1 61 20 Sep 3 1 Critical Ideas tangent lines the derivative relationship between the graphs of f and f existence of derivatives continuity and differentiability derivative notation Terms Definitions secant line slope of tangent line difference quotient derivative of f differentiate f at x f differentiable at x Facts Rules Theorems formula for the slope of a tangent line to y