Find the dimension of a cylindrical can open at the top and

Question

1. Find the dimension of a cylindrical can, open at the top and of fixed surface area K, which maximize the volume.

2. Find the non-negative number x and y such that 2x + y = 30 and xy² is maximized.

3. Find the most general antiderivative of the following functiuons:

a) x² + 2x^1/3 + 3x^-4 + 5x^-1 + e^π

b) 2^3x

c) 2/3x+4

d) (x² + 1)²

e) x(1+x²)^1/3

f) sec²(3x)

g) 3/1+4x²

4. Evaluate the following limit lim_n→∞i=1∑ⁿ 3i/n + 4(e^2i/n/n).

5. Evaluate the following integrals:

a) ₀∫¹x²e^{-x^3} dx

b) ₀∫^πx³cos(x⁴) dx

c) _-1∫¹(x/1 + x² + sin²x)dx

6. Evaluate the following integrals by interpreting it as an area:

0∫6 (√(9-(x-3)²) + 2) dx.

7. Find the area between the line y = x and parabola y = x² from x = 0 to x = 2.

8. Find the derivative of the following functions:

f(x) = _sinx∫^{e^x} ln(1+t²)dt, g(x) = sec x ₀∫^x (t/2+tan³t) dt.

9. The velocity at time t of a particle moving along the x axis is given by v(t) = 2t³ + 1. Find s(t), the position of the particle at time t given that it is at x = 10 at the initial time t = 0.