# Solving Differential Equations, Volumes of Solids and Taylor Series

See attached file for full problem description.

1. First find a general solution of the differential equation dy 3y

dx

= . Then

find a particular solution that satisfies the initial condition that

y(1) = 4.

2. Solve the initial value problem dy y3

dx

= , y(0) = 1.

3. 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?

4. Find the center and radius of the circle described in the equation

2x2 + 2y2 - 6x + 2y = 3.

5. Find an equation of the ellipse with center (-2, 1), horizontal major axis

10, and eccentricity 2

5

.

6. Determine whether or not the sequence n 2n 3

na = 2 + converges and find

its limit if it does converge.

7. Write the Taylor series with center zero for the function

f (x)=ln(1+x2).

8. Given a = 2i + 3j, b = 3i + 5j, and c = 8i + 11j express c in the form ra +

sb where r and s are scalars.

9. Given a = <4, -3, -1> and b = <1, 4, 6>, find a × b.

10. Find the arc length of the curve given by x = cos 3t, y = sin 3t,

z = 4t, from t = 0 to t =

2

π .

11. Compute the first-order partial derivatives of f (x,y) 2x

x y

=

−

.

12. 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.

13. Evaluate

3 1

1 0

(2x 3y)∫ ∫ − dxdy .

14. Find the volume of the solid bounded by x = 0, y = 0, z = 0, and

x + 2y + 3z = 6 by triple integration.

15. Calculate the divergence and curl of the vector field

F(x, y, z) = 2xi + 3yj + 4zk.

16. Use Green's theorem to evaluate

C

��∫ Pdx +Qdy

P(x, y) = xy, Q(x, y) = ex; 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.

See attached file for full problem description.

#### Solution Summary

Solving Differential Equations, Volumes of Solids and Taylor Series are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.