Intro to Mathematical Modeling
Intro to Mathematical Modeling MATH 1101
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This 2 page Class Notes was uploaded by Mrs. Kara Jacobs on Monday October 12, 2015. The Class Notes belongs to MATH 1101 at Georgia College & State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/221930/math-1101-georgia-college-state-university in Mathematics (M) at Georgia College & State University.
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Date Created: 10/12/15
MATH 1101 Mathematical Modeling Review for Final Exam Chapters 1 2 3 81 82 11 Models Functions and Graphs Suggested Problems 1 3 5 7 9 11 15 1 7 19 21 I Mathematical Modeling Formulate Solve Interpret Test I Definition of Function Let A and B be nonempty sets A function f from A to B is a rule that assigns to each element a of set A one and only one element called fa from set B o A function must satisfy the following three requirements I I Start with a pair of not necessarily distinct sets designated as a first set and a second set I 2 Each element of the first set is assigned a partner from the second set I 3 No element from the first set is assigned two or more partners from the second set I Functions may be specified in four ways Analytically Geometrically Numerically Verbally I Domain Codomain Range I Does a Graph Represent a Function use the vertical line test I Increasing Decreasing Constant 12 Constructed Functions SuggestedProblems 1 3 5 7 9 11 13 1 7 19 21 23 25 27 33 I Combining Functions Sum Difference Product Quotient Composition 0 adjusting units as necessary so that combinations of functions make sense in context I Piecewise Defined Functions Break Points I Inverse Functions Horizontal Line Test Reverse of a Function Partner Sharing One to One Function 13 Functions Limits and Continuity Suggested Problems 1 3 5 7 9 11 13 15 19 21 23 27 29 I Limits at Infinity lim lim 0 fx L fx M x gt oo x gt oo o Intuitive As x increases without bound fx gets closer and closer to L o Intuitive As x decreases without bound fx gets closer and closer to M 0 Can be defined more formally 0 Horizontal Asymptote I Limits at a Point 1 lim lim 0 m fx L fx M fx N H x gt a x gt a Twosided limits Onesided Limits Twosided Intuitive As x gets closer and closer to a fx gets closer and closer to L Can be defined more formally Num erically approximating the limit at a point Approximating the limit at a point on a graph I Continuity at a Point O O O O o A function f is continuous at a if and only if I fa is defined 2 1 fX exists 3 1 fX fa 0 Determining continuity at a point from a graph 0 Discontinuous at a point means not continuous at a point also called a point of discontinuity 0 Continuous Continuous with discrete interpretation Discrete 0 Continuity for a piecewise function 14 Linear Functions and Linear Models SuggestedProblems 3 5 7 9 13 21 23 25 29 I A linear function is a function having formula description of the form Lx mx b where m and b are real numbers The slope is the signed vertical change corresponding to a one unit increase in the horizontal coordinate Using similar triangles we can show that every pair of points on a given line determine the slope 2ylx2xl m Pointslope form of the equation of a line 0 yl mx xl where m is the slope and XI yl is a point on the line Slopeintercept form of the equation of a line yLx mx b where m is the slope and 0b is the vertical intercept Linear Models for sets of points which are approximately linear 0 Linear Regression also known as least squares regression minimizing the sum of the squared vertical errors 0 InterpolationExtrapolation 0 Four elements of a model a linear function a description including units of both input and output domain 21 Exponential Functions and Models SuggestedProblems 1 3 9 11 13 15 17 21 27 I An exponential function is a function having formula description of the form Ex bax where b is a real number and a is a real number satisfying a gt0 and a l ltNote in most models we39ll find b gt 0gt Exponential functions change very fast the rate of increase r gt 0 or decrease l lt r lt 0 is ultim ately infinite Exponential functions are characterized by a constant percentage rate of change ie for each one unit increase in the horizontal coordinate the vertical coordinate of an exponential function changes by the same percentage If x y and r s are points on an exponential function Ex bax then ys a H When horizontal coordinates are equally spaced the date is exactly exponential if and only if first ratios are constant Ex bax blrx if r gt 0 a gt1 then growth factor lr percentage growth rate is lOOr percent Ex bax blrx if 1 lt r lt 0 a gt1 then decay factor lr percentage decay rate is lOOlrl percent For Ex bax 0 b is the vertical intercept the horizontal axis is a horizontal asymptote no vertical asymptotes If data looks exponential but neither end approaches zero a vertical shift may result in a better fit 2 2 Logarithmic Functions and Models Suggested Problems 1 2 3 4 5 7 9 11 13 I A logarithmic function is a function having formula description of the form lx a b lnx where b is a real number satisfying b 0 and a is a real num er I e39ab 0 is the horizontal intercept no horizontal asymptotes the vertical axis is a vertical asymptote I Increases without bound bgt0 or decreases without bound blt0 but does so veg slowly I No inflection points Always concave down bgt0 OR Always concave up blt0 I Logarithmic functions are only defined for positive real numbers the vertical axis is a vertical asymptote I Alignment may be necessary if not all horizontal coordinates are positive 23 Logistic Functions and Models Suggested Problems 7 9 11 15 19 23 25 I A logistic function is a function having formula description of the form Zx Ll Ae39Bquot where A is a positive real number and b is a real number satisfying B 0 I y L and y 0 are horizontal asymptotes 0 L1A is the vertical intercept 0 As A approaches 0 the vertical intercept approaches 0 L 0 As A approaches infinity the vertical intercept approaches 0 0 I The function either increases Bgt0 OR decreases Blt0 as lBl approaches infinity the graph is steeper in the middle I One in ection point a point around which concavity changes I If data looks logistic but neither end approaches zero a vertical shift may result in a better fit I Alignment may be necessary if not all horizontal coordinates are close to zero 24 Quadratic and Cubic Functions and Models SuggestedProblems 1 3 5 7 11 17 19 21 23 25 I A quadratic function is a function having formula description of the form qx ax2 bx c where a b and c are real numbers and a 0 lt Also can be written in form qx a xh 2 k where hk is the vertex Note h b2a gt Parabolic shape ushape concave up ushape opens up for a gt 0 concave down ushape opens down for a lt 0 If horizontal coordinates equally spaced the data is exactly quadratic when second differences are nonzero constant A cubic function is a function having formula description of the form Cx ax3 bx2 cx d where a b c and d are real numbers and a 0 lt Note that cubic functions have one point of inflectiongt I If a gt 0 end behavior ofC is the same as y x3 while ifa lt 0 the end behavior ofC is the same as y 9 If horizontal coordinates equally spaced the data is exactly cubic when third differences are nonzero constant I 25 Choosing a Function to Fit Data SuggestedProblems 1 2 3 4 5 6 7 9 11 13 15 I Shape of scatter plot may focus our attention on particular models anticipated end behavior should also be a factor I Linear and Exponential are characterized for Quadratic and Cubic there are characterizations for equally spaced data W SuggestealPVOblems 7 9 11 13 15 17 I A sinusoidal function is a function having formula description of the form Sx a sin bx h k where a b h and k are real numbers and b gt 0 The basic sinusoidal function is y sinx I The amplitude of a function is Max 7 Min 2 provided this number exists 0 The amplitude of a sinusoidal is lal ie the absolute value of a I A function f is periodic if there exists a number p for which fx p fx for all x in the domain of f o The period of a periodic function f is the smallest positive number p for which fx p fx for all real numbers x in the domain of f provided such a number p exists The period of a sinusoidal function is 2pib I The horizontal shift of a sinusoidal function Sx a sin bx h k is ihb I The equilibrium of a sinusoidal function is Max Min2 This corresponds to the vertical shift which is k 82 Sinusoidal Functions as Models SuggestedProblems 1 3 5 7 9 11 I For a given set of quotperiodicquot data 7 ie data we expect to repeat a pattern be able to o producing a scatter plot 0 produce an approximate sinusoidal model by hand approximating the values of a b h and k 0 produce an approximate sinusoidal model using regression
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