Precalc review: modeling with functions 1 PDF

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Precalc Review: Modeling with Functions 1...

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PRECALC REVIEW: MODELING with FUNCTIONS I Modeling with Linear Functions • Basic Linear Function is y = mx + b, with slope m and y−intercept b. Alternate (point-slope) form y = y1 + m(x − x1), given one data point (x1, y1). • Important Application is Data Fitting (linear regression): given x,y data, find the “best-fit” line for the data. Example: Columbia River Depth vs Flow Velocity Data (Problem 1.2.45)

Data value lists for x and y x = [ .7 2 2.6 3.3 4.6 5.9 7.3 8.6 9.9 10.6 11.2]; y = [1.55 1.11 1.42 1.39 1.39 1.14 .91 .59 .59 .41 .22];

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PRECALC REVIEW: MODELING with LINEAR FUNCTIONS Modeling problem: how to find good m and b values for y = mx + b. Some possibilities: a) use first two data values: slope m = (y2 − y1)/(x2 − x1) ≈ −.3385. b) use first and last data values: slope m = (y11 − y1)/(x11 − x1) ≈ −.1267. c) use all data values to find “best-fit” line: using calculus and Matlab m = −.1165, b = 1.681.

Estimated river velocity with 12 ft depth is ?

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Math formulas for best-fit line slope and intercept: P P P P P n P xiyi − xi yi m = n x2 yi −nm xi ; P 2 , b= − ( x) i Matlab instructions for previous graphi x = [.7 2 2.6 3.3 4.6 5.9 7.3 8.6 9.9 10.6 11.2]; y = [1.55 1.11 1.42 1.39 1.39 1.14 .91 .59 .59 .41 .22]; p = polyfit( x, y, 1 ); m = p(1), b = p(2) m1 = (y(2)-y(1))/(x(2)-x(1)); m2 = (y(11)-y(1))/(x(11)-x(1)); plot(x,y,’*’,x,y(1)+m1*(x-x(1)),x,y(1)+m2*(x-x(1)),x,b+m*x)

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PRECALC REVIEW: MODELING with FUNCTIONS CONT. Modeling with Periodic Functions • Basic Periodic Function is y = a cos(bx) or y = a cos(bx − c) with amplitude |a|, period 2π/b, and phase shift c/b; could also use y = a cos(bx − c) + d or sin instead of cos. • Model Application: need to find good values for a, b, and maybe c, d. Example: some plotted data values

a?, b?, c?, d?.

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PRECALC REVIEW: MODELING with FUNCTIONS CONT. Modeling with Exponential Functions • Basic Exponential Function is y = abx or y = abx + c, with base b > 0, x ≥ 0 . Equivalent forms y = aekx or y = aekx + c, where e ≈ 2.71828182846, the base for natural logarithms. • Common Exponential Function Applications: a) Population Growth (b > 1, k > 0) or Decay (b < 1, k < 0) b) Financial Models • Typical Sketch Graphs?

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PRECALC REVIEW: MODELING with FUNCTIONS CONT. • Some Population Growth Examples a) If bacteria population P (t) at time t is modeled with exponential model, and P (0) = 25000, P (10) = 40000, determine when P (t) = 50000.

b) (Text 1.4.39) If Carbon C-14 has a half-life of 5,730 years, how much is left of 500g after t years?

c) (Text 1.4.44) Mice tumors grow exponentially with doubling time of 2.9 days. If initial tumor size is .1 cm3, what is tumor size after t days? Determine when tumor size is .5 cm3....