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- Consider the function k with domain R, k''(t) =A(e-t- 4e-2t)and A positive and real.Show how you find the exact value(s) of the inflection point(s) of the function k.[6] Find the cubic polynomial function p(x) = a + bx + cx2 + dx3 such that p(1) = 5,p'(1) = 8, p(-1) = -3, and p'(-1) = 12. 2.Find the nth Maclaurin polynomial for the function f(x) = sec x, n = 2
- Find the third Taylor's polynomial f(x) = x3 – x² + 3x + 1 which can easily find the derivative at xo = 0, x = 0.5 solution:Determine X(S) expressed as a rational polynomial with simplified numerator and factored denominator a. x(t) = {t2 +2e-2t + sin(3t)} u(t) b. x(t) = 2tcos(2t)u(t) - use theorem on multiplication by t c. x(t) = d/dt {3e-2tcos(t)u(t)} - use theorem on derivativeFind the linear approximation L(x) of the function f(x) = (1-x)1/2 at a=0 and then use L(x) to approximate the number (.9)1/2
- a) Find a second degree polynomial approximating the function f(x)=ln.x, and passing from the points (1, f(1)),(2, F(2)) and (4, f(4)). b) Approximate f(2.5) using the polynomial in part (a) E„(x) = (x-x))(x-x,)(x-x,)...(x-xx) (N+1)! c) use the error formula to estimate the error at x-2.5 d) Obtain a bound for the uniform error E2(x)Let g (x, y) = 7 sin(xy) – 9x? In(y) + 4. Find the degree 2 polynomial, p, which best approximates g near the point (, 1). (Use symbolic notation and fractions where needed.) Р (х, у) -Determine X(S) expressed as a rational polynomial with simplified numerator and factored denominator a. x(t) = d/dt {3e-2tcos(t)u(t)} - use theorem on derivative