Math 213
Name:
Midterm Exam
1. [12 points] Use guess & check to find a particular solution to the following differential equation:
y00 + 7y = 9 sin 2x.
2. [6 points] Solve the following initial-value problem:
y00 = −4y,
y(0) = 6,
y0 (0) = 8.
3. [8 points] Solve the following initial-value problem:
3
y0 = − ,
y
y(0) = 4.
4. [10 points] A chemist prepares a 15 g sample of a radioactive isotope. Initially, the mass of the isotope
is found to be decreasing at a rate of 0.0054 g/day. What is the half-life of the isotope?
5. [14 points] During a biology experiment, a culture of cells grows according to the logistic equation,
with a maximum population of 150,000 cells. At the beginning of the experiment, the culture has
a population of 25,000 cells, and the population is increasing at a rate of 1,000/min. What will the
population be after 1 hour? (Round your answer to the nearest thousand.)
6. [10 points] In the following figure, the point p is the center of the sphere, and has coordinates (3, 1, 4).
Find the coordinates of the point q.
7. [10 points] The following figure shows a right triangle in R2 . Find the coordinates of the point p.
8. [8 points] Find parametric equations for the line of intersection of the following planes:
2x + y + 3z = 2
and
4x + 3y + 8z = 5.
9. [6 points] The following figure shows an equilateral triangle in the plane. Given that p = (6, 8), compute the dot product p · q.
10. [6 points] Find the value of a for which the planes
2x + 5y + 3z = 8
are perpendicular to one another.
and
ax − 3y + 7z = 10
11. [10 points] The following figure shows a square in R2 . Find the coordinates of the point p.