Reference no: EM132345816

CALCULUS PROBLEMS -

1. Evaluate the following limits.

i) lim_x→∞(x+1/x+3)^x+3

ii) lim_x→∞(x - ln(cosh x))

iii) lim_x→0(a^{x^2}+b^{x^2}/a^x+b^x)^1/x, a, b > 0.

2. If lim_h→0 (f(x+h) - f(x))/h = 1/(1+x²), then find the derivative of the composite function f(1/x).

3. Suppose that f(x) = x - √(x²-2x+4) and g(x) = x + √(x²-2x+4). Find the value of f'(x)g(x) + f(x)g'(x).

4. Show that the function y(x) = 1/(x²+bx+c) satisfies the differential equation yy'' - 2y'² + 2y³ = 0.

5. Show that the second derivative of sequence of functions f_n(x) = (x_n(1 x)ⁿ)/n!, n ∈ N satisfies the recurrence relation: f''_n(x) = 2(1 - 2n)f_n-1(x) + f_n-2(x).

6. Suppose

Show that there exists a point c in the interval (0, 1/π) such that 1/c cot 1/c = 2.

7. Calculate the following indefinite integrals.

i) ∫1/(√x - ³√x) dx

ii) ∫1/(√2 + sinx - cos x) dx

iii) ∫1/(sin³x+cos³x) dx.

8. Calculate the following definite integrals.

i) ₀∫^πxsinx/(1+cos²x) dx

ii) ₀∫^π 1/(1+pcosx) dx, |p| < 1.

9. Find the length of the arc of the parabola y² = 2px, p > 0 on the interval [0, 2p].