Use the Euler Method with a step size of 0.2 to estimate f(3) where f′(x)=2x−7 and f(2)=4.
Your answer should be correct to two decimal places.
f(2.2) = f(2) + f'(2) (.2)
f(2.2) = 4 + (2*2-7) (.2)
f(2.2) = 4 + (4-7)(.2)
f(2.2) = 4 + (-3)(.2)
f(2.2) = 3.4
now do f(2.4) the same way
then f(2.6) etc up to f(3)
<img src="http://www.mathscoop.com/image-gallery/slope-fields/2*x-75-55-5ffffff3300661-1120400ff0000240_23190.png " alt="Slope Field by MathScoop.com" />
from
http://www.mathscoop.com/calculus/differential-equations/euler-method-calculator.php
now exact
dy/dx = 2 x - 7
integrate
y = x^2 - 7 x + c
when x = 2, y = 4
4 = 4 -14 + c
c = 14
so
y = x^2 - 7 x + 14
so if x = 3
y = 9 -21 + 14
y = 2 would be exact
If that is the case, Damon, wouldn't the answer just be 1?
To use the Euler Method to estimate the value of f(3), we will approximate the derivative of the function f(x) using the given derivative equation f′(x) = 2x - 7. We need to find the value of f(x) at each step by using the previous value and the estimated derivative.
Let's start by calculating the step size using the given value: h = 0.2. We will then calculate the number of steps required to reach x = 3. In this case, we are going from x = 2 to x = 3, and the step size is 0.2, so the number of steps is (3 - 2) / 0.2 = 5.
Now let's set up the Euler Method iteration:
Step 1: Initialize the values
x0 = 2 (initial value of x)
f0 = 4 (initial value of f)
Step 2: Iterate over the steps
for i = 1 to 5:
xi = x0 + (i * h) // Calculate xi (next x)
fi = f0 + (h * (2 * x0 - 7)) // Calculate fi (estimate f at xi)
x0 = xi // Update x0 for the next iteration
f0 = fi // Update f0 for the next iteration
Step 3: Calculate the final estimated value
f(3) ≈ f0
Let's calculate the estimated value using the above steps:
Step 1:
x0 = 2
f0 = 4
Step 2:
i = 1:
xi = 2 + (1 * 0.2) = 2.2
fi = 4 + (0.2 * (2 * 2 - 7)) = 4 + (0.2 * -3) = 4 - 0.6 = 3.4
i = 2:
xi = 2.2 + (2 * 0.2) = 2.6
fi = 3.4 + (0.2 * (2 * 2.2 - 7)) = 3.4 + (0.2 * -2.6) = 3.4 - 0.52 = 2.88
i = 3:
xi = 2.6 + (3 * 0.2) = 3
fi = 2.88 + (0.2 * (2 * 2.6 - 7)) = 2.88 + (0.2 * -1.8) = 2.88 - 0.36 = 2.52
i = 4:
xi = 3 + (4 * 0.2) = 3.8
fi = 2.52 + (0.2 * (2 * 3 - 7)) = 2.52 + (0.2 * -1) = 2.52 - 0.2 = 2.32
i = 5:
xi = 3.8 + (5 * 0.2) = 4
fi = 2.32 + (0.2 * (2 * 3.8 - 7)) = 2.32 + (0.2 * -0.4) = 2.32 - 0.08 = 2.24
Step 3:
f(3) ≈ 2.24
Therefore, using the Euler Method with a step size of 0.2, the estimated value of f(3) is 2.24, correct to two decimal places.