Math Project Part 3.doc - Introduction This is the Part III of the math project in this part we deal with no only f(x but also f ’(x and f ’’(x

# Math Project Part 3.doc - Introduction This is the Part III...

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Introduction: This is the Part III of the math project; in this part, we deal with no only f(x), but also f ’(x) and f ’’(x). f(x) helps find the elevation of any point along the line, f ’(x) helps find the slope at any point of the line, and f ’’(x) tells you the concavity of the line in our project. In this project, first we find the first and second deveriative of the A(x), B(x), and C(x) in the piecewise function. Then we do the midpoint analysis and random point analysis using theses deveriatives. In the function breakdown, for every function, we’ll find the critical points using the first deveriative, and the point of inflection using the second deveriative. For the power, exponential, and logarithmic function, however, we’ll prove that these functions have no critical point and point of infection, and then using the first and second deveriative, we’ll plug in a random x-value to find if the line is increasing or decreasing, and the concavity of the function.
Piecewise function for A(x) .0190353867x 2 - .6133526137x + 53.44611511 .1 ≤ x < 47 (1) 60.06101939(1.002216961) x 47 ≤ x < 95 (2) A(x)= 56698.74303x -1.475707031 95 ≤ x < 173 (3) 1.180637442 E 17x -6. 9681296 173 ≤ x < 185 (4) -.08833x + 36.406856 185 ≤ x 237 (5) .03807077x - .6133526137 .1 ≤ x < 47 (1) .133005558(1.002216961) x 47 ≤ x < 95 (2) A’(x)= -83630.64597x -2.475 95 ≤ x < 173 (3) -8.22683471 E 17x -7.9681296 173 ≤ x < 185 (4) -.08833 185 ≤ x < 237 (5) .03807077 .1 ≤ x < 47 (1) 2.945417615 E -4(1.002216961) x 47 ≤ x < 95 (2) A’’(x) 206985.8488x -3.475 95 ≤ x < 173 (3) 6.555248517 E 18x -8.9681296 173 ≤ x < 185 (4) 0 185 ≤ x < 237 (5) A(x) evaluates the elevation, in feet, above sea level, at any point between 65 th Street Park Avenue and intersection of Spring Street Hudson Street. A’(x) evaluates the slope, in ft/mm at any point between 65 th Street Park Avenue and intersection of Spring Street Hudson Street. A’’(x) evaluates the concavity of the land in ft/mm 2 at any point between 65 th Street Park Avenue and intersection of Spring Street Hudson Street. 2
x represents the number of millimeter away from 65 th Street Park Avenue toward intersection of Spring Street Hudson Street, which is included with the project. Midpoint analysis 1) The midpoint at 65 th Street Park Avenue and intersection of Spring Street Hudson Street is located on 29 th Street between 5 th Avenue and Madison Avenue on my map. It is obtained from Part I of the project. 2) The corresponding x-value at the midpoint is 118.5mm, since the whole distance of A(x) is 237mm, 237/2 = 118.5mm. 3) The elevation for the midpoint is 52.6019ft above sea level. The elevation is obtained by Part II of this project. 4) The slope for the midpoint is –40.6870728ft/mile, which means if I continue with the exact slope, I will evaluate –40.6870728ft vertically for every 1 mile. Plus since the slope is negative, I will be walking downhill. Work: A’(x) = -83638.64597x -2.475 A’(118.5) = -83638.64597(118.5) -2.475 A’(118.5) = -.6164708ft -.6164708ft 66mm = -40.6870728ft = -40.6870728ft/mile 1mm 1mile 1mile 66mm/1miles is the standard unit for the map project, it means that 66mm in our map project equal 1 mile in real life length.

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• DanielMcKinney

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