AP Calculus that pertains to differential equations covered in the sixth to eighth module of the Florida Virtual Learning calculus curriculum.

Let y = f(x) be the particular solution to the differential equation dy over dx equals negative 7 times x times y cubed comma with the initial condition f(1) = −1. Which of the following gives an expression for f(x) and the domain for which the solution is valid?

(NOT) f of x equals square root of quantity 1 over 7 times x squared minus 6 end quantity for open parenthesis negative infinity comma negative square root of six sevenths close parenthesis and open parenthesis square root of six sevenths comma infinity close parenthesis

The rate at which a particular medication leaves an individual's bloodstream is proportional to the amount of this medication that is in the bloodstream. An individual takes 250 mg of the drug initially. The amount of medication is halved every 13 hours. Approximately how many mg of the medication remain in the individual's bloodstream after 6 hours?

182 mg

The rate at which a particular medication leaves an individual's bloodstream is proportional to the amount of this medication that is in the bloodstream. An individual takes 275 mg of the drug initially. The amount of medication is halved every 14 hours. Approximately how many mg of the medication remain in the individual's bloodstream after 8 hours?

185 mg

The pressure valve to a garden hose is opened, causing the hose to experience an increase in pressure. The rate of increase in pressure with respect to time, t, is proportional to the sum of the initial pressure, po, and the instantaneous pressure, p.

dp over dt equals c times quantity p plus p sub o

Let y = f(x) be the particular solution to the differential equation dy over dx equals negative 3 times x times y cubed comma with the initial condition f(1) = 1. Which of the following gives an expression for f(x) and the domain for which the solution is valid?

f of x equals square root of quantity 1 over 3 times x squared minus 2 end quantity for open parenthesis negative infinity comma negative square root of two thirds close parenthesis and open parenthesis square root of two thirds comma infinity close parenthesis

Determine the value of k that will make y = 2cos(2x) a solution to y ″ − ky = y ″.

k = 0

Determine the value of k that will make y = 2cos(2x) a solution to ky + y ″ = y.

k = 5

What is the general solution to the differential equation dy over dx equals x times y plus 3 times x?

y equals e to the x squared over 2 plus C power minus 3

What is the general solution to the differential equation dy over dx equals x times y plus 4 times x?

y equals e to the x squared over 2 plus C power minus 4

Which of the following is the solution to the differential equation dy over dx equals 2 times x times y all over quantity x squared plus 2 end quantity comma with the initial condition y(2) = 3?

y equals one half times x squared plus 1

Which of the following is the solution to the differential equation dy over dx equals 2 times x times y all over quantity x squared plus 1 end quantity comma with the initial condition y(3) = 6?

y equals one half times x squared plus three halves

The general solution of the differential equation dy over dx equals 2 times y over x, where x ≠ 0, is a family of which of the following?

Parabolas

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x plus y comma which of the following statements is true?

The slopes are all positive in Quadrant 1.

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x minus 2 times y comma which of the following statements is true?

The slopes are all positive in Quadrant 4.

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x minus y comma which of the following statements is true?

The slopes are all positive in Quadrant 4.

The general solution of the differential equation d y over d x equals x over y where y ≠ 0, is a family of which of the following?

Hyperbolas

Select the general solution to x dy over dx equals 2 plus 3y. I. one third ln absolute value of quantity 2 plus 3y close absolute value equals ln absolute value of x value plus C II. ln absolute value of quantity 2 plus 3y close absolute minus C equals ln absolute value of x

I

Select the general solution to x dy over dx equals y over quantity 2 plus 3y. I. 2ln|y| + 3y = ln|x| + C II. one third ln absolute value of quantity 2 plus 3y all over y close absolute value equals ln x plus C

I

Select the general solution to x squared dy over dx equals 3 plus 2y. I. one half ln absolute value of quantity 3 plus 2y close absolute value equals ln absolute value of x squared close absolute value plus C II. one half ln absolute value of quantity 3 plus 2y close absolute value equals negative 1 over x plus C

II

Determine the value of k that will make y = 2cos(2x) a solution to ky − y ″ = y.

NOT k = 5

A heat source is applied to a metal surface, causing the temperature of the metal to increase with respect to time at a rate that is proportional to the sum of the metal's instantaneous temperature, T, and the metal's original temperature, To. Select the differential equation that represents the relationship.

dT over dt equals c times quantity T plus T sub o