Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.) √x11 x11 dx
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- Let f be the function defined as, Ր f(x) = [* sin (t³) dt. Determine an expression for f(x) as a power series.Please show all work ง 5. Evaluate the vector surface integral JJ (+2², x + 2y + 32², 5x − 2y + 2³) . Ndo. S Here, S is the sphere Sy²+2 1. = Since S is a closed surface, the outside of S is considered as the positive side on S. Hint: use the divergence theorem.Let f(x) = √√x, g(x) = x + 8 and h(x) = 5. (f + g)(16) X Compute the following function value.
- Find the optimal solution for the following problem. Note: Round your answers to 3 decimal places. Maximize C = 13x + 9y subject to and 6x + 11y ≤ 18 16x + 21y ≤ 41 x ≥ 0, y ≥ 0. a. What is the optimal value of x? X b. What is the optimal value of y? c. What is the maximum value of the objective function?Find the derivative of the function f by using the rules of differentiation. 7 f(x) = -X417 4 f'(x) =A system is described by the action S[r, r'] = [ dt (x² + 4rª + mr³4), r(t) where m is a constant. Find the Euler-Lagrange equation for r = which extremises S. Interpret each of the terms that appear in the Euler- Lagrange equations in terms of Newton's second law. Find a conserved quantity for this system.
- Consider the following utility function: u = 400 (1 + 1)−1. xy a. Is this utility function homogenous? Briefly explain. b. Use implicit differentiation to find the MRCS. c. Write down an expression for the indifference curve if u = 20.Why is this formula very important F = P( I + i)?In order to find a local minimum or maximum of a function, how do we manipulate the fırst derivative? Make it equal to zero O Make it positive O Make it negative Make it equal to f(x)