Exercise 4.1 Solutions

2.

(a)

Table[Series[Cos[x], {x, 0, k}], {k, 4, 8, 2}]

(b)

\(\(E\_8\) \((x)\) = \(\(\(f\^\((9)\)\) \((c)\) x\^9\)\/\(9!\)\ \ so\  | \(E\_8\) \((x)\) |  = \(\(|\)\(\(f\^\((9)\)\) \((c)\) || x\)\( | \^9\)\)\/\(9!\) ≤ 1\/\(9!\)\ for\ \  ∣ 
            x ∣  ≤ 
            1\ since\  | \(f\^\((9)\)\) \((c)\) | \(\(≤\)\(1\)\)\)\)

(c)

Normal[Series[Cos[x], {x, π/4, 4}]]

3.

Series[(x^(1/2)), {x, 0, 8}]

Series[(x^(1/2)), {x, 1, 8}]

5.

Series[(E^(-(x^2)/2)), {x, 0, 3}]

8.

(a)

Series[(((2 + x))^(1/2)), {x, 2, 3}]

(b)

N[Normal[Series[(((2 + x))^(1/2)), {x, 2, 3}]] /. 
        x -> 1]

(c)

Plot[
        D[(((2 + t))^(1/2)), {t, 4}] /. t -> x, {x, 1, 
          3}];

11.

Series[(∫_(-1)^x) Cos[(t^2)] \[DifferentialD]t, {x, 0, 
        8}]

Normal[
      Series[(∫_(-1)^x) Cos[(t^2)] \[DifferentialD]t, {x, 0, 
          8}]]

% /. x -> 0.1

13.

(a)

Table[D[Log[1 + x], {x, k}], {k, 1, 4}]

(b)

Table[Normal[Series[Log[1 + x], {x, 0, k}]], {k, 1, 4}]

Exercise 4.2 Solutions

2.

P[x_] := -0.04 (x^3) + 0.14  (x^2) - 0.16  x + 2.08

P[0]

P[1]

P[2]

P[4]