# Important information about Various Calculus problems

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

6. dy/dx = y^3, y() = 1

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).

12. Find the center and radius of the circle described in the equation: 2x^2 + 2y^2 - 6x + 2y = 3

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.

#### Solution Summary

It provides answers to a set of various calculus problems.