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- Suppose that the quadrature formula has degree of accuracy atleast 2 integral f(x)dx with upper limit 2 and lower limit -2 = a1f(−2) + a2f(0) + a3f(2) Use the definition of degree of accuracy to determine the constantsa1, a2, a3, and then determine the exact degree of accuracy of the quadratureformula. Is there any quadrature formula that has higher degree of accuracyand that also uses three functional evaluations?How do I use L'hopitales Rule for (x)2 / (x-2)2 as x approaches 2 ?2. Give an example showing that, even if f0(c) = 0, it could happen that f(x) does not have a localmaximum or minimum at x = c. How is this consistent with Fermat’s Theorem in the book?
- construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be determined. Then show that the critical point at the origin is of the indicated type. 1.dxdt=−x3+xy2,dydt=−2x2y−y3; asymptotically stableArgue graphically why a function of the form ax2 + bry + cy2 with an infinite number of critical points must be either positive semidefinite or negative semidefinite.Where are the two local maximums located?
- A rectangular fence is to be constructed with one of its face adjacent to an existing wall. Given that the available materials for the fence are good for 24m, what dimensions will produce a maximum area for the fence? What is/are the function's critical point/sWhat values of a and b make ƒ(x) = x3 + ax2 + bx have a. a local maximum at x = -1 and a local minimum at x = 3? b. a local minimum at x = 4 and a point of inflection at x = 1?Prove that that the y=0 is the horiz.asymp. and x=-1 and x=1 are vert.asymp, of y=coth-1x through computinglimits.