Find the Maclaurin’s polynomial of degree 3 for  ( a ) f ( x ) = (1 + x ) e 2 x ( b ) f (x) = ln(3 + ex )

consider the polynomial f (x) = 7x4- (2 + 2x + 6x2 + 5x3) -8x5. In x = 0.7, use the approximation in divided differences centered for the second derivative of F (x). Use a rack of Δx = 0.2 step.   F '' (x) ≈

find the  Derivative

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)

Find f′(0), where f(x)= (x|x|)/(2+⌊x⌋)

Find Newton’s polynomial that interpolates f(x) = x1 at x = 1, x=2,x=4,andx=5. Usethetableweusedinclasstofindthe coefficients.

F(x) = 2ln(x). What is F′(2)?

Find a formula for f-¹(x) and (ƒ˜¹)'(x) if ƒ(x) = √√ f-¹(x) = 0 (ƒ-¹)'(x) =

p"(x;) = 0. p'(xi) Let p(x) be a polynomial function of degree n with simple roots x1,..., xn. Show that

Consider the function f(x)=x3+x+1. Let's suppose that it has an inverse g so that g(f(x))=x.What is g(3)?What is f′(1)?What is g′(3)?