Reference no: EM132608029

Question 1. Compute the following

(i) Let f (x) = x, 0 ≤ x ≤ 1 and P = {0, 1/3, 2/3, 1}. Then compute U (P, f ) and L(P, f ).

(ii) Let P_n = {0, 1/n, 2/n, ..., n/n}. Find lim_n→∞ U (P_n, f ) and lim_n→∞ L(P_n, f ) for

(a) f (x) = x, 0 ≤ x ≤ 1
(b) f (x) = x², 0 ≤ x ≤ 1

Question 2. Evaluate the following limits

(i) lim_n→∞ 1/n [e^3/n + e^6/n ....e^3n/n]

(ii) lim_n→∞ 1/n [sin Π/n + sin 2Π/n ....sin nΠ/n ]

(iii) lim_n→∞ [1/n+1 + 1/n+2 + ....1/2n ]

(iv) lim_n→∞ 1/n[(1/n)² + (1/n)² + ... (n/n)²]

Question 3. Using the definition of Riemann integrability, check if the following functions are Rie- mann integrable?

(i) f (x) = 1 x < 1 , x ∈ [0, 1]
              0 x = 0, x ∈ [0, 1]

(ii) f (x) = sin(1/sin x) x ≠ 0, Π, 2Π, x ∈ [0, 2Π]

              0              x = 0 Π, 2Π, x ∈ [0, 2Π]

(iii) f (x) = x[1/x],  x ∈ [0, 1],  x ∈ [0, 1]

                  0       x=0, x ∈ [0, 1]

(iv) f(x) = cos Π/x          x ∈ Q, x ∈ R

                 0                x /∈ Q, x ∈ R
(v)) f (x) = 1 + x x ∈ Q, x ∈ R
                1 - x  x /∈ Q, x ∈ R

(vi) sgn( x) = 1          x > 0, x ∈ R
                   0         x = 0, x ∈ R

                   -1        x < 0, x ∈ R

Question 4. Test the convergence of the following improper integrals

(i) ₁∫^∞ x cos xdx (ii) ₀∫^∞ e^-x cos x (iii) ₁∫^∞ dx/(x²+√x)

(iv) ₁∫^∞ (dx/(1+x³))^1/3 dx (v) ₁∫^∞ √x/1+x²dx.

Question 5. Show that if _a∫^b f (x) dx = 0, for a non-negative continuous function f on [a, b] then f(x) ≡ 0.

Question 6. Suppose that f : [a, b] → R is Riemann integrable. Then prove that

m(b-a)≤ _a∫^b f ≤ M(b-a),

where M and m are supremum and infumum of f over [a, b] respectively.

Question 7. Test the convergence of the following improper integrals

(i) ₀∫¹ sin x/x^3/2.dx (ii)₀∫¹(log 1/x)/√ x (iii) ₀∫³ log x/√|2-x| (iv) _a-1∫^a+11/(x-a)^1/3 (v) ₁∫² √x/logx dx.

Question 8. True or False? Give justifications.

(i) If f is integrable over [a, b] then |f| is also integrable on [a, b].

(ii) If |f| is integrable over [a, b] then f is also integrable on [a, b].

(iii) If f is integrable over [a, b] and f (x) = g(x) except countable number of points x ∈ [a, b], then g is integrable over [a, b]

(iv) If f is integrable over [a, b] and f (x) = g(x) except finite number of points x ∈ [a, b], then g is integrable over [a, b]