AP REVIEW - Belle Vernon Area School District
-AP REVIEW
DIFFERENTIAL EQUATIONS AND SLOPE FIELDS
Multiple Choice
1. (Sample Questions - Noncalculator)
The solution to the differential equation [pic] [pic] is
(A) [pic] (B) [pic] (C) [pic]
(D) [pic] (E) [pic]
________________________________________________________________________________
2. (Sample Questions - Noncalculator)
If [pic], then y =
(A) [pic] (B) [pic] (C) [pic] (D) tan x + 5 (E) [pic]
________________________________________________________________________________
3. (2003 – Noncalculator)
Shown above is a slope field for which of the following differential equations?
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E)[pic]
_______________________________________________________________________________
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) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
Free Response
5. (2005 – Noncalculator)
Consider the differential equation [pic]
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve
points indicated.
(b) Let [pic] be the particular solution to the differential equation with the initial
condition [pic] Write an equation for the line tangent to the graph of f at [pic]
and use it to approximate [pic].
(c) Find the particular solution [pic] to the given differential equation with the initial
condition [pic]
6. (2006 – Noncalculator)
Consider the differential equation [pic]
(a) On the axes provided, sketch a slope field for the given differential equation at the eight points
indicated.
(b) Find the particular solution [pic] to the differential equation with the initial
condition [pic] and state its domain.
Answers
1. E 2. C 3. E 4. E
5. (2005 – Noncalculator)
Consider the differential equation [pic]
(a) On the axes provided, sketch a slope field for the given differential
equation at the twelve points indicated.
(b) Let [pic] be the particular solution to the differential equation
with the initial condition [pic] Write an equation for the line
tangent to the graph of f at [pic]and use it to approximate [pic].
(c) Find the particular solution [pic] to the given differential equation
with the initial condition [pic]
_______________________________________________________________________________________
(a) 1: zero slopes
1: nonzero slopes
(b) The line tangent to f at [pic] is [pic]. 1: equation of the tangent line
Thus, [pic] is approximately [pic]. 1: approximation for [pic]
(c) [pic] 1: separates variables
[pic] 1: antiderivatives
[pic] 1: constant of integration
[pic] 1: uses initial condition
[pic] 1: solves for y
Since the particular solution goes through [pic], Note: max 2/5 [1-1-0-0-1] if no
y must be negative. constant of integration
Thus the particular solution is [pic] Notes: 0/5 if no separation of variables
6. Consider the differential equation [pic]
(a) On the axes provided, sketch a slope field for the given differential equation
at the eight points indicated.
(b) Find the particular solution [pic] to the differential equation with the
initial condition [pic] and state its domain.
_______________________________________________________________________________________
(a) 2: sign of slope at each point and
relative steepness of slope lines
in rows and columns
(b) [pic] 1: separates variables
[pic] 2: antiderivatives
[pic] 1: constant of integration
[pic] 1: uses initial condition
2 = C 1: solves for y
[pic] Note: max 3/6 [1-2-0-0-0] if
[pic] no constant of integration
or Note: 0/6 if no separation of
[pic] variables
1: domain
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