AP Calculus BC
Name
Date
Period _______
A50: Euler’s Method & Slope Field Worksheet
1. Match the slope fields with their differential equation.
i) y ′ = x + y ____
ii) y ′ = .5 x − 1 ____
iii) y ′ = .5 y ____
a)
b)
c)
2. (a) Use Euler’s Method to construct a solution for
iv) y ′ = −
x
____
y
d)
dy x
= with the initial condition y (0) = 1 and let
dx 2
∆x=1. Use four steps.
(b) Solve the equation
dy x
= with the initial condition y (0) = 1 . Then use your solution to find y (4) .
dx 2
(c) The error in using Euler’s Method is the difference between the approximate value and the exact
value. What was the error in your answer? How could you produce a smaller error using Euler’s
Method?
3. Suppose a continuous function f and its derivative f’ have values that are given in the following table.
Given that f(2)=5, use Euler’s Method to approximate the value of f(3).
x
2.0
2.5
3.0
f’(x)
0.4
0.6
0.8
f(x)
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A50: Euler’s Method
4.
AP Calculus BC
dy
1
=
and y (0) = 1 . Find an approximation of y (1) using Euler’s
dx x + 2
Method with two steps and step size ∆x=0.5.
Given the differential equation
5. Given the differential equation
dy
= x + y and y (1) = 3 . Find an approximation of y (2) using Euler’s
dx
Method with two equal steps.
6. The curve passing through (2, 0) satisfies the differential equation
dy
= 4 x + y . Find an approximation
dx
to y (3) using Euler’s Method with two equal steps.
dy
= arcsin(xy ) with the initial
dx
condition f (0) = 2 . What is the approximation for f (1) if Euler’s Method is used, starting at x = 0
with a step size of 0.5?
7. (Acorn Book) Let y = f (x) be the solution to the differential equation
(A) 2
(B) 2 +
π
(C) 2 +
6
π
(D) 2 +
4
π
2
(E) 3
8. Assume that f and f ’ have the values given in the table. Use Euler’s Method to approximate the value
of f(4.4).
x
4
4.2
4.4
f’(x)
-0.5
-0.3
-0.1
f(x)
2
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