-AP REVIEWDIFFERENTIAL EQUATIONS AND SLOPE FIELDS

Multiple Choice1. (Sample Questions - Noncalculator)

The solution to the differential equation is

(A) (B) (C)

(D) (E)

________________________________________________________________________________2. (Sample Questions - Noncalculator)

If , then y =

(A) (B) (C) (D) tan x + 5 (E)

________________________________________________________________________________3. (2003 – Noncalculator)

Shown above is a slope field for which of the following differential equations?

(A) (B) (C) (D) (E)

_______________________________________________________________________________

4. (Sample Questions – Noncalculator)

The slope field for a certain differential equation is shown above. Which of the following could be a specific solution to that differential equation?(A) (B) (C) (D) (E)

Free Response5. (2005 – Noncalculator)

Consider the differential equation

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(b) Let be the particular solution to the differential equation with the initial

condition Write an equation for the line tangent to the graph of f at

and use it to approximate .

(c) Find the particular solution to the given differential equation with the initial

condition

6. (2006 – Noncalculator)

Consider the differential equation

(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.

(b) Find the particular solution to the differential equation with the initial

condition and state its domain.

Answers1. E 2. C 3. E 4. E

5. (2005 – Noncalculator)

Consider the differential equation

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (b) Let be the particular solution to the differential equation

with the initial condition Write an equation for the line

tangent to the graph of f at and use it to approximate .

(c) Find the particular solution to the given differential equation

with the initial condition _______________________________________________________________________________________(a) 1: zero slopes

1: nonzero slopes

(b) The line tangent to f at is . 1: equation of the tangent line

Thus, is approximately . 1: approximation for

(c) 1: separates variables

1: antiderivatives

1: constant of integration

1: uses initial condition

1: solves for y

Since the particular solution goes through , Note: max 2/5 [1-1-0-0-1] if no y must be negative. constant of integration Thus the particular solution is Notes: 0/5 if no separation of variables

6. Consider the differential equation

(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.(b) Find the particular solution to the differential equation with the

initial condition and state its domain._______________________________________________________________________________________(a) 2: sign of slope at each point and relative steepness of slope lines

in rows and columns

(b) 1: separates variables

2: antiderivatives

1: constant of integration

1: uses initial condition 2 = C 1: solves for y Note: max 3/6 [1-2-0-0-0] if no constant of integration or Note: 0/6 if no separation of variables

1: domain