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]

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2. (Sample Questions - Noncalculator)

If [pic], then y =

(A) [pic] (B) [pic] (C) [pic] (D) tan x + 5 (E) [pic]

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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]

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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]

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(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.

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(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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