AP Calculus AB Unit 7 Differential Equations Questions

Explore AP Calculus Unit 7 questions on modeling, verifying, estimating, and solving first-order differential equations in context.

Covered syllabus points

7.1 Modeling Situations with Differential Equations 7.2 Verifying Solutions for Differential Equations 7.3 Sketching Slope Fields 7.4 Reasoning Using Slope Fields 7.6 Finding General Solutions Using Separation of Variables 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables 7.8 Exponential Models with Differential Equations

[Maximum number: 1]

As an ice block melts, the rate at which its mass, M, decreases is directly proportional to the square root of the mass. Which equation describes this relationship?

$\sqrt{M(t)}=k t$

$\frac{d M}{d t}=k \sqrt{t}$

$\frac{d M}{d t}=k \sqrt{M}$

$\frac{d M}{d t}=\frac{k}{\sqrt{M}}$

Given the following differential equation $\frac{dy}{dx}=x-2y+1$ .

(a)

Sketch the slope field for the differential equation at the nine indicated points on the axes provided.

x	-1	-1	-1	0	0	0	1	1	1
y	-1	0	1	-1	0	1	-1	0	1
d y / d x	2	0	-2	3	1	-1	4	2	0

(b)

For which values of m and b is the line y=m x+b a solution to the differential equation?

m is the slope

$\frac{dy}{dx}$ , so replace into the differential equation: m=x-2(mx+b)+1=(1-2m)x+(1-2b). 1-2m=0 and 1-2b=m, so $m=\frac12$ and $b=\frac14$ .

[Maximum number: 6]

The depth of seawater at a location can be modeled by the function H that satisfies the differential equation $\frac{d H}{d t}=\frac{1}{2}(H-1) \cos \left(\frac{t}{2}\right)$ , where H(t) is measured in feet and t is measured in hours after noon ( t=0 ). It is known that H(0)=4.

(a)

A portion of the slope field for the differential equation is provided. Sketch the solution curve, y=H(t), through the point (0, 4).

Figure for Question (a) — AP Calculus AB

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(b)

Use separation of variables to find y=H(t), the particular solution to the differential equation $\frac{d H}{d t}=\frac{1}{2}(H-1) \cos \left(\frac{t}{2}\right)$ with initial condition H(0)=4.

Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.

Figure for Question (b) — AP Calculus AB

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[Maximum number: 1]

The slope field shown is for the differential equation $\frac{d y}{d x}=k y-2 y^{2}$ , where k is a constant. What is the value of k ?

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Covered syllabus points

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Question 2

Question (a)

Question (b)

Question 3

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Question 4