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- Use Richardson extrapolation to find an improved estimate for the first derivative of y = cos x at x =π/4 using step sizes of h1 = π/3 and h2 = π/6. Employ centered differences of O(h2) for the initial estimates. Discuss from your results the effect of using Richardson extrapolation on the error of using centered differences.At the x value which is closest to 44, calculate the derivative of f(x) using a suitable second order accurate finite difference method with a) h = 25, b) a different step size. (You should use the same finite difference method in (a) and (b)). Using the values you calculated at parts (a) and (b) apply Richardsonextrapolation to improve your calculation.1. Determine the -values where relative extrema occur for y = (x ^ 2 + x - 12) ^ 2 . Use the second derivative test, if possible.
- The tangent line to the graph of y=f(x) at the point (7,55) has the equation y=8x−1 (a) Find f′(7)f′(7) = (b) Find the instantaneous rate of change of y with respect to x at x=7Instantaneous Rate of Change =If (x, y) moves from (x0, y0) to a point (x0 + dx, y0 + dy) nearby, how can you estimate the resulting change in the value of a dif-ferentiable function ƒ(x, y)? Give an example.The sales of a book publicationare expected to grow according to the function S = 300000(1 - e-0.06t), where t is the time, given in days. 1. Show using differentiation that the sales never attains an exact maximum value. 2. What is the limiting value approached by the sales function?
- The linearization of ex at x = 0 Derive the linear approximation ex = 1 + x at x = 0.Consider the function on the interval (0, 2?). f(x)= x/2+sin(x) b) Apply the First Derivative Test to identify the relative extrema. relative maximum (x, y) = relative minimum (x, y) =Beginning at rest, an automobile drives around a circular trackof radius R = 300 m, accelerating at a rate of 0.3 m/s2. After howmany seconds will the car begin to skid if the coefficient of friction isμ = 0.6?
- From the partial differential eqiuation by eliminating the arbitrary function z=(x2+y2+z2)Is there a smooth (continuously differentiable) curve y = ƒ(x) whose length over the interval 0 ≤x ≤ a is always √2a? Give reasons for your answer.1. Given the function f(x) =(x-2)^2 +(y-3)^2 (i) Find the critical points. (ii) Using the second derivative test, Classify the critical points .