(10 marks) Q3: Use the Finite-Difference method with h = 0.5 to approximate the solution to the boundary-value problem y=-(y)-y+ Inx, 1 ≤x≤2, y(1) = 0, y(2) = In 2 Compute the percentage relative error if the exact solution is y(x) = In x.
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- How are the absolute maximum and minimum similar to and different from the local extrema?Substitute the values from step 2 appropriately into the linear approximation expression to estimate the value of (3.01) (4.02)?. (3.01)3(4.02)2 = f(3 + .01, 4 + .02) x f(3, 4) + f;(3, 4)( ]× ) + f,(3, 4)( [216 ]× ) + 432 432 X ) + 216( 216 |× ) = 432 %3Dy=x/(x2+36) Find critical numbers
- The profit function for an operation is given byP(x, y) = 2200 + 24x − x^2 + 80y − y^2where x is the cost of a unit of labor and y the cost of one item. Find the maximum amount of profit that can be generated from this operation. It is necessryto do a prospective analysis of the profitability of this operation. Determine theapproximate increase or decrease in profit if the units of labor increases from 12to 13 while the price of an item remains the same at 40. Also, what if the laborremains steady at 12 but the price increase from 40 to 41?Q2 (9 points) Find and classify the critical points of the function z = x(2− y)(x + y − 1). That is, find all of the critical points, and apply the second derivative test to each of them, determining whether it is a local maximum, a local minimum, a saddle, or the test fails.Find the values of x, y and z that correspond to the critical point of the function: z = f(x, y) = 3x² + 8x - 7y +5y² + 7xy Enter your answer as a decimal number, or a calculation (like 22/7). (Round to 4 decimal places) x= -11.7273 y= 8.9091 Z= Question Help: Video Submit Question (Round to 4 decimal places) (Round to 4 decimal places) Search W F
- Find the critical points of the functions f(x, y) = x² + 2y² - 4y + 2x, g(x, y) = x² − 12xy + y Use the Second Derivative Test to determine the local minimum, local maximum, and saddle points. (Give your answer as a comma-separated list of points in the form (*, *) where needed. Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if the answer does not exist.) local minimum(s): local maximum(s): DNE saddle point(s): Match f (x, y) and g (x, y) with their graphs. g (x, y) (A) (B) f (x, y)Find the absolute maxima of f (x) = 2x³ – 6x in the interval [0,3] The critical numbers in increasing order are and The absolute maxima (y-value) is . This occurs at x = The absolute minima (y-value) is This occurs at x=Let f(x)=2x^2−2x^4. Find the open intervals on which ff is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).