Please help with the following problems. See attached.
1. First find a general solution of the differential equation: dy/dx = 3y
Then find a particular solution that satisfies the initial condition that y(1) = 4.
2. Given a = <4, -3, -1> and b = <1, 4, 6>, find a x b
3. Use the method of Lagrange multipliers to find the extreme values of 3x - 4y + 12z on the spherical surface with equation x2 + y2 + z2 = 1.
4. Nyobia had a population of 3 million in 1985. Assume that this country's population is growing continuously at a 5% annual rate and that Nyobia absorbs 40,000 newcomers per year. What will its population be in the year 2015?
5. Evaluate the double integral
7. Determine whether or not the sequence converges and find its limit if it does converge.
8. Find the volume of the solid bounded by: x = 0, y = 0, z = 0, and x + 2y + 3z = 6 by triple integration.
9. Use Green's theorem to evaluate: P(x,y) = xy, Q(x,y) = e^x; C is the curve that goes from (0, 0) to (2, 0) along the x-axis and then returns to (0, 0) along the parabola y = 2x - x2.
10. Compute the first-order partial derivatives of: f(x, y) = 2x/(x - y)
11. Write the Taylor series with center zero for the function f(x) = ln(1 + x2).
13. Find the arc length of the curve given by x = cos 3t, y = sin 3t, z = 4t,
14. Given: a = 2i + 3j, b = 3i + 5j, and c = 8i + 11j, Express c in the form ra + sb where r and s are scalars.
15. Find an equation of the ellipse with center (-2, 1), horizontal major axis 10, and eccentricity 2/5
16. Calculate the divergence and curl of the vector field F(x,y,z) = 2xi + 3yj + 4zk.
It provides answers to a set of various calculus problems.