Math 1121 Calculus (II)

Math 1121 Calculus (II)

Homework 5-2

(Hand in Problem 1, 3, 4, 6 )

1. Let f(x) be a function with sufficiently many times derivative at 0 and P_n,0,f(x) be the Taylor polynomial of degree n for f at 0.

(a) Letg(x) = f(x²). Prove that P_2n,0,g(x) =P_n,0,f(x²).

(Hint: Use the result that P_2n,0,g(x) is the unique polynomial with degree ≤2n such that lim

x→0

g(x)−P(x) x²ⁿ = 0.)

(b) For all i∈N, find g⁽ⁱ⁾(0) in terms of the derivative of f at 0.

(c) In general, leth(x) = f(x^k), find h⁽ⁱ⁾(0) in terms of the derivatives of f at 0.

(d) Letk(x) = cos(x²). Find the Taylor polynomial P_2n,0,k(x).

(e) Compute k⁽²⁰¹⁶⁾(0).

2. Let F(x) = Z x

0

t

1 +t² dt. Find the Taylor polynomial P_2n,0,F(x).

3. For the following given functionf(x) and given point a , express the remainderRn,a,f(x) in the integral form.

(a) f(x) = e^2x, a= 0.

(b) f(x) = sin(x²), a= 0.

(c) f(x) = lnx, a= 1.

4. Let f(x) = tan⁻¹x. Determine a number n∈N such that

|tan⁻¹(0.8)−P_n,0,f(0.8)|< 1 10000

and explain your reason. (Not necessary to find the smallest number n.) 5. In class, we prove that ifR(x) is a polynomial with degree ≤n and satisfies

x→0lim R(x)

xⁿ = 0

then R(x)≡0. But, this statement could be false if R(x) is not a polynomial.

Construct a nonzero function f(x) which satisfies (i) lim

x→0

f(x) xⁿ = 0 (ii) lim

x→0

f(x)

x^n+δ 6= 0 for allδ >0.

6. Let f and g be two functions with sufficiently many times derivative ata, andP_n,a,f(x) and P_m,a,g(x) be the corresponding Taylor polynomials of degree n and m for f and g respectively at a.

Determine that each of the following statements is true or false and explain it.

(a) The Taylor polynomial of degreen−1 forf⁰(x) at a is

P_n,a,f(x)⁰

. That is, P_n−1,a,f⁰(x) = d

dx

P_n,a,f(x)

(b) The Taylor polynomial of degreen+k for “(x−a)^kf(x)” at ais “(x−a)^kP_n,a,f(x)”.

(c) If k ≤ n ≤ m, the Taylor polynomial of degree k for “f(x)g(x)” at a is “ the first k+ 1 terms of the polynomial P_n,a,f(x)P_m,a,g(x)”.

That the first k+ 1 terms of a polynomial

P(x) =c0+c1(x−a) +· · ·+ck(x−a)^k+· · ·+c`(x−a)<sup>`</sup> is

c₀+c₁(x−a) +· · ·+c_k(x−a)^k. (d) Is the statement(c) still true if k > n?