Reference no: EM132176741

Part -1:

Question 1: . Suppose
(a) f is continuous for x ≥ 0,
(b) f'(x) exists for x > 0,
(c) f(0) = 0,
(d) f' is monotonically increasing.

Put

g(x) = f(x)/x     (x > 0)

and prove that g is monotonically increasing.

Question 2. Suppose f' is continuous on [a, b] and ε> 0. Prove that there exists δ > 0 such that

|(f(t)- f(x))/(t-x) - f'(x)| < ε

Question 3. Suppose f is a real function on (-∞, ∞). Call x a fixed point of f if f(x) = x.

(a) If f is differentiable and f '(t) ≠ 1 for every real t, prove that f has at most one fixed point.

(b) Show that the function f defined by

f(t)= t + (1 + e^t)^-1

has no fixed point, although 0 <f'(t) < 1 for all real t.

(c) However, if there is a constant A < 1 such that |f'(t)| ≤ A for all real t, prove that a fixed point x off exists, and that x = lim x_n where x_i is an arbitrary real number and

x_n+1 = f(x_n)

for n = 1, 2, 3, ... .

(d) Show that the process described in (c) can be visualized by the zig-zag path

(X₁, x₂) → (x₂, x₂) (x₂ , x₃) → (x3, x3) →(x3, x4) →

Question 4. Suppose f is twice differentiable on [a, b], f(a) < 0, f(b) > 0, f'(x) ≥ δ ≥ 0, and 0 ≤ f"(x) ≤ M for all x ∈ [a, b]. Let ξ be the unique point in (a, b) at which f(ξ) = 0.

Complete the details in the following outline of Newton's method for computing ξ.

(a) Choose x₁ ∈ (ξ, b), and define (x_n) by

X_{n +1} = X_n - f(x_n)/f'(x_n)

Interpret this geometrically, in terms of a tangent to the graph off.

(b) Prove that x_n+1 < x_n and that

lim n→∞ x_n = ξ.

(c) Use Taylor's theorem to show that

x_n+1 - ξ = f''(t_n)/2f'(x_n)(x_n - ξ)²ⁿ.

for some t_n ∈ (ξ, x_n).

(d) If A = M/2δ, deduce that

0 ≤ x_n+1 - ξ ≤ –6 A -[,4(xt - O]2".
(Compare with Exercises 16 and 18, Chap. 3.)

(e) Show that Newton's method amounts to finding a fixed point of the function g defined by

g(x) = x - f(x)/f'(x).

How does g'(x) behave for x near ξ?

(f) Put f(x) = x^1/3 on (-∞, ∞) and try Newton's method. What happens?

Part -2:

Question 1: Suppose f ≥ 0, f is continuous on [a, b], and _a∫^b f(x) dx = 0. Prove that f(x) = 0 for all x ∈ [a, b]. (Compare this with Exercise 1.)

Question 2: Define

                          f(x) = _x∫^x+1 sin(t²)dt.

(a) Prove that |f (x)| < 1/x if x > 0.

Hint: Put t² = u and integrate by parts, to show that f(x) is equal to

cos (x²)/2x - cos [(x + 1)²]/2(x+1) - _x²∫^(x+1)2 cos u/4u^3/2 du.

Replace cos u by -1.

(b) Prove that

2xf(x) = cos (x²) - cos [(x + 1)²] + r(x)

where |r(x) < c/x and c is a constant.

(c) Find the upper and lower limits of xf(x), as x → ∞.

(d) Does ₀∫^∞ sin (t²) dt converge?