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- consider a continuous function y = f(x) whose first derivative is y = x(x – 2). The graph of f is concave up on the interval(s) Choices: 0 < x < 2 x > 1 x > 2 x < 0 or x > 2Compute dy/dx using the limit definition of the first derivative for f(x) = 3x2Compute the second derivative of f(x)=ex+x at x = 1 using a step size h = 0.2
- 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.true or false with reason if f is continuous in[a,b] then integrationba x f(x)dx= x integrationbaf(x)dxLet z= x+iy and determine where the function f(z) = Log(z2 + 1) is holomorphic. What is its derivative in the region where it is holomorphic?
- Where is the function f(z) = Log(z2 + 1) holomorphic? What is its derivative in the region where it is holomorphic?Use Richardson extrapolation to estimate the first derivative of y = cos x at x = π/ 4 using step sizesof h1 = π/ 3 and h2 = π/ 6. Employ centered differences of O(h2) for the initial estimates.Let f(x, y) = x2y4 sin(1/ (sqrt(x2+y2)) where (x, y) cannot be (0, 0) and f(0, 0) = 0 Is f continuous in origo?
- Find the linear approximation of f(x)=lnx at x=1and use it to estimate ln(1.3) L(x)=? ln 1.3≈?Suppose that a nonnegativefunction y = ƒ(x) has a continuous first derivative on [a, b] . LetC be the boundary of the region in the xy-plane that is bounded below by the x-axis, above by the graph of ƒ, and on the sides by the lines x = a and x = b. Show that ∫ƒ(x) dx = - ∮C y dx.Evaluate in terms of Gamma function: y^5√(b^2 – y^2) dy LIMIT: 0 to b