1. Applications of Taylor Series Solve the definite integral: 2 1.2 x fo} =1- e_{7} dx 0 3. Expand the value in the integral as a Taylor Series centered at o = 1, through the fourth derivative. b. Integrate the Taylor's series using the limits of the definite integral that you are approximating. After you have done this, and to scale your answer so that the total area under the entire curve [from -0 to on] would be equal to l, multiply your answer by % \I 11' c. This integral represents the r'Normal Distribution" when the population mean of some measurable attribute is centered at x = U. The upper integration limit indicates the number of standard deviations (abbreviation: o} the attribute is away from the mean. Add your answer for "b" above to 05 and multiply the result by 100% to estimate the percentage of the population with a certain attribute less than 1.2 standard deviations above the mean: Now subtract your answer for "b" above from 0.5 and, again, multiply the result by 100% to estimate the percentage of the population with a certain attribute less than 1.2 standard deviations below the mean: Polar and Cartesian Coordinates Plots a. Give the domain of the ellipse {below} in Cartesian coordinates: x2 2 _+y_=1 25 36 b. Convert the equation for the ellipse into one where y is an explicit function of 31'. Then draw or plot the ellipse in Cartesian coordinates. c. Convert the ellipse in "a" above to polar coordinates, where ris an explicit function of 9. Then draw or plot the ellipse in polar coordinates. Given the overlapping cardiods at right: Cardiods : r = 3(1 + cos0), r = 2(1 - cose) a. Find the areas of each of the two 90 cardiods. 120 60 150 -180 -150 -30 -120 -60 -90 vert horiz 30 60 -30 -60 r1 - 12 b. Find the angles of the intersections between the two cardiods. c. Find the area of the overlap between the two cardiods