x - Warren County Schools

Slope fields and Differential Equations
Student Saturday Session
The topical outline for AB Calculus specifies that students need to understand the geometric interpretation
of differential equations via slope fields and the relationship between slope fields and solution curves for
differential equations. More specifically, students need to be able to:
• Draw a slope field at a specified number of points by hand.
• Sketch a solution that passes through a given point on a slope field.
• Match a slope field to its differential equation.
• Match a slope field to its solution.
• Determine features of the solution to a differential equation based on its slope field and/or its
solution.
Students also need to be able to solve separable differential equations and find a particular solution using
an initial condition. Students should be able to model a real world situation using a differential equation
(including the study of the equation y' = ky and exponential growth).
Additionally, BC students need to know how to find a numerical solution of a differential equation using
Euler's method and be able to solve logistic differential equations and use them in modeling. Those BC
topics will not be covered in this session.
2004 AB6 - no calc
dy
= x 2 ( y − 1) .
dx
a) On the axes provided, sketch a slope field for the given differential equation at the twelve points
indicated.
Consider the differential equation
b) While the slope filed in part (a) is drawn at only twelve points, it is defined at every point in the
xy-plane. Describe all points in the xy-plane for which the slopes are positive.
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
Multiple Choice Questions:
Match the slope fields with their differential equations.
dy
dy
1.
2.
= sin x
= x− y
dx
dx
3.
(A)
(B)
(C)
(D)
dy
= 2− y
dx
4.
dy
=x
dx
5.
The slope field from a certain differential equation is shown above. Which of the following could be a
specific solution to that differential equation?
(A) y = x 2
(B) y = e x
(C) y = e − x
(D) y = cos x
(E) y = ln x
6.
The slope field for a certain differential equation is shown above. Which of the following could be a
specific solution to that differential equation?
1
(A) y = sin x
(B) y = cos x
(C) y = x 2
(D) y = x3
(E) y = ln x
6
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
7. (2003 BC3 - no calc)
Shown above is a slope field for which of the following differential equations?
dy x
(A)
=
dx y
(D)
8. If
dy x 2
=
dx
y
dy x 2
(B)
=
dx y 2
(E)
dy x 3
(C)
=
dx
y
dy x 3
=
dx y 2
dy
= x 2 y , then y could be
dx
⎛ x⎞
(A) 3 ln ⎜ ⎟
⎝ 3⎠
(B) e
(D) 3 e 2 x
(E)
x3
3
+7
(C) 2e
x3
3
x3
+ 1
3
9. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of
bacteria doubles in three hours, in how many hours will the number of bacteria triple?
(A)
3 ln 3
ln 2
⎛ 27 ⎞
(D) ln ⎜ ⎟
⎝ 2 ⎠
(B)
2 ln 3
ln 2
(C)
ln 3
ln 2
⎛9⎞
(E) ln ⎜ ⎟
⎝ 2⎠
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
1
dy
= −2 y and if y = −1 when t = 0 , what is the value of t for which y = ! ?
2
dt
ln 2
ln 2
1
(A) −
(B) −
(C)
2
2
4
10. If
(D)
11. If
2
2
dy
= 2 y 2 and if y = −1 when x = 1, then when x = 2, y =
dx
(A) −
(D)
(E) ln 2
2
3
(B) −
1
3
(E)
1
3
(C) 0
2
3
12. At each point (x, y ) on a certain curve, the slope of the curve is 3x 2 y . If the curve contains the point
(0, 8), then its equation is
(A) y = 8e x
3
(D) y = ln( x + 1) + 8
(B) y = x 3 + 8
3
(C) y = e x + 7
(E) y 2 = x 3 + 8
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
Free Response Questions:
2005 AB6 - no calc
Consider the differential equation
dy
2x
=− .
dx
y
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points
indicated.
(b) Let y = f (x) be the particular solution to the differential equation with the initial condition
f (1) = −1 . Write an equation for the line tangent to the graph of f at (1, − 1) and use it to
approximate f (1.1) .
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
(c) Find the particular solution y = f (x) to the given differential equation with the initial condition
f (1) = −1 .
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
2007 Form B AB5 - no calc
dy 1
= x + y −1
dx 2
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points
indicated.
Consider the differential equation
(b) Find
d2y
in terms of x and y. Describe the region in the xy-plane in which all solution curves
dx 2
to the differential equation are concave up.
(c) Let y = f (x) be a particular solution to the differential equation with the initial condition
f (0) = 1. Does f have a relative minimum, a relative maximum, or neither at x = 0 ? Justify
your answer.
(d) Find the values of the constants m and b, for which y = mx + b is a solution to the differential
equation.
National Math and Science Initiative 2012
Slope fields and Differential Equations
Student Saturday Session
2006 Form B AB5 - no calc
dy
= ( y − 1)2 cos(π x ).
dx
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points
indicated.
Consider the differential equation
(b) There is a horizontal line with equation y = c that satisfies this differential equation. Find the
value of c.
(c) Find the particular solution y = f (x) to the differential equation with the initial condition
f (1) = 0 .
National Math and Science Initiative 2012

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