Suppose that you can calculate the derivative of a function using the formula f′(x)=2f(x)+4x. If the output value of the function at x=4 is 4 estimate the value of the function at 4.006.
f(4+0.006) = f(4) + f'(4) * 0.006
f' is the slope of the tangent line. Thus, since ∆y/∆x ≈ dy/dx,
the tangent line at x=4 is
y-f(4) ≈ f'(4) (x-4)
That is, the equation shown above
To estimate the value of the function at x = 4.006, we need to use the derivative formula provided: f′(x) = 2f(x) + 4x.
Given that the output value of the function at x = 4 is 4, we can set up an initial condition for the function:
f(4) = 4
Now, let's use this information to estimate the value of the function at x = 4.006. We will need to use an approximation technique, such as Euler's method, which uses the derivative formula to iteratively update the function value.
1. Calculate the derivative at x = 4:
f′(4) = 2f(4) + 4(4)
= 2(4) + 16
= 8 + 16
= 24
2. Use the derivative to estimate the function value at x = 4.006:
To do this, we will take a small step size (h) and perform the following calculations iteratively until we reach x = 4.006:
a. Calculate the change in x: Δx = 4.006 - 4
b. Set the step size h, for example, h = 0.001
c. Determine the number of steps needed: n = Δx / h
d. Update the function value using Euler's method:
for i = 1 to n:
f(4.006) = f(4) + f′(4) * h
f(4) = f(4.006)
3. Plug in the values and calculate f(4.006):
a. Calculate the change in x: Δx = 4.006 - 4
Δx = 0.006
b. Set the step size: h = 0.001
c. Determine the number of steps: n = Δx / h
n = 0.006 / 0.001
= 6
d. Update the function value using Euler's method:
for i = 1 to 6:
f(4.006) = f(4) + f′(4) * h
= 4 + 24 * 0.001
Perform the calculation iteratively:
i = 1: f(4.006) = 4 + 24 * 0.001
i = 2: f(4.006) = f(4.006) + 24 * 0.001
= f(4) + 2 * 24 * 0.001
...
i = 6: f(4.006) = f(4.006) + 24 * 0.001
= f(4) + 6 * 24 * 0.001
e. Calculate the final value of f(4.006):
f(4.006) = 4 + 6 * 24 * 0.001
Therefore, the estimated value of the function at x = 4.006 is f(4.006) = 4 + 6 * 24 * 0.001.
To estimate the value of the function at x=4.006, we can use the derivative formula to approximate the function's change in value in a small interval around x=4.
Step 1: Calculate the derivative of the function at x=4.
Given the derivative formula f′(x)=2f(x)+4x, we can substitute x=4:
f′(4) = 2f(4) + 4(4)
Step 2: Calculate the value of the function at x=4.
Given that the output value of the function at x=4 is 4, we can substitute f(4) with 4 in the derivative formula:
f′(4) = 2(4) + 4(4)
Step 3: Calculate the approximate change in the function's value.
To estimate the change in the function's value from x=4 to x=4.006, we need to approximate the derivative at x=4 and then multiply it by the difference between x=4.006 and x=4:
Change in value ≈ f′(4) * (4.006 - 4)
Step 4: Estimate the value of the function at x=4.006.
To estimate the value of the function at x=4.006, we add the approximated change in value to the initial value of the function at x=4:
Estimated value = 4 + Change in value
By following these steps, we can estimate the value of the function at x=4.006 based on the given derivative formula and the output value at x=4.