1 2. Find the Taylor Polynomial of degree n for f (x) =, centered at %| 1-x X = 0.

Problem 16 (#2.3.34).Let f(x) = ax +b, and g(x) = cx +d. Find a condition on the constants a, b, c, d such that f◦g=g◦f. Proof. By definition, f◦g(x) = a(cx +d) + b=acx +ad +b, and g◦f(x) = c(ax +b) + d=acx +bc +d. Setting the two equal, we see acx +ad +b=acx +bc +d ad +b=bc +d (a−1)d=(c−1)b That last step was merely added for aesthetic reasons.

Find a degree 4 polynomial that approximates the solution of y'=y+x², if y=1 when x=0. a.y=1+x+ b. y=1+x+ d. y=1+x+ a O o O - + 2 c. y=1+x+ + انا C + x² 2 4 4 - 3 2 3 + X + + + +

8. Find the (only) equation of a polynomial p(x) th • Root at 0 of degree 1 • Root at 1 of degree 1 • Root at -3 of degree 2 • p(2) = 5

12. Find a quadratic polynomial p(x) such that p(2) = 11, p′(2) = 13, p″(2) = 8.

4. Find the remainder of f (x) = (1+x)3/2 of degree 3.

1. Find the Taylor polynomial p3 centered at a = 7/3 for f(x) = = COST.

13. Find the minimum and maximum value of the function on the giveninterval.y = 2x 2 + 4x + 5 [-2, 2]

Problem 4. The graph of the function f(x) is shown below: f (a) Explain why T3(x) = 1.6 – 0.8(x – 1) +0.4(x – 1)² – 0.1(x – 1)³ cannot be the third Taylor polynomial of f(x) centered at x = 1. (b) Explain why T3(x) = 2.8+0.5(x – 2) +1.5(x – 2)2 – 0.1(x – 2)³ cannot be the third Taylor polynomial of f(x) centered at x = = 2.