Suppose that you can calculate the derivative of a function using the formula f'(x) = 4f(x) + 2x. If the output value of the function at x= 4 is 2 estimate the value of the function at 4.013
Well, if the function's derivative at any point is given by f'(x) = 4f(x) + 2x, we can try to estimate the value of the function at 4.013 using some clown math.
Since we're given that f(4) = 2, let's start by plugging the values into the equation:
f'(4) = 4f(4) + 2(4)
= 4(2) + 2(4)
= 8 + 8
= 16
So, according to the clown math, the function's derivative at x = 4 is 16.
Now, let's try to estimate the value of the function at x = 4.013 using our calculated derivative value of 16. We'll use clown calculus, of course!
f'(x) ≈ Δy/Δx
Let's choose a small interval around x = 4, say Δx = 0.013. Since we have the derivative, we can estimate the change in y (Δy) using the clown derivative formula:
Δy ≈ f'(4) * Δx
= 16 * 0.013
≈ 0.208
Now let's use our estimated change in y to approximate the value of the function at x = 4.013:
f(4.013) ≈ f(4) + Δy
= 2 + 0.208
≈ 2.208
So, based on our clown math, we estimate that the value of the function at x = 4.013 is approximately 2.208. Remember, though, clown math is more about entertainment than accuracy!
To estimate the value of the function at x = 4.013, we can use the given derivative formula:
f'(x) = 4f(x) + 2x
First, we need to find the original function by integrating the derivative formula:
∫f'(x) dx = ∫(4f(x) + 2x) dx
Integrating the right side of the equation, we get:
f(x) = ∫(4f(x) + 2x) dx
f(x) = 4∫f(x) dx + 2∫x dx
Now, let's solve these integrals separately:
The first integral ∫f(x) dx represents the original function f(x), so we can substitute it back into the equation:
f(x) = 4f(x) + 2∫x dx
Next, integrate 2∫x dx:
f(x) = 4f(x) + x^2 + C <- (Equation 1)
Now, we can use the given information that f(4) = 2 to estimate the value of f(4.013).
Substitute x = 4 into Equation 1:
f(4) = 4f(4) + 4^2 + C
2 = 4(2) + 16 + C
2 = 8 + 16 + C
2 = 24 + C
C = 2 - 24
C = -22
Now we have the value of C. We can substitute x = 4.013 into Equation 1 to estimate the value of f(4.013):
f(4.013) = 4f(4.013) + 4.013^2 - 22
This estimate can be found by evaluating the expression on the right side of the equation.
To estimate the value of the function at x = 4.013, we can use the derivative formula given: f'(x) = 4f(x) + 2x.
Step 1: Find the value of the function at x = 4.
Given that the output value of the function at x = 4 is 2, we can write f(4) = 2.
Step 2: Use the derivative formula to find the derivative of the function.
According to the formula, f'(x) = 4f(x) + 2x.
Step 3: Substitute the value of x = 4 into the derivative formula.
When x = 4, we have f'(4) = 4f(4) + 2(4).
Substituting f(4) = 2, we get f'(4) = 4(2) + 2(4) = 8 + 8 = 16.
Step 4: Approximate the value of the function at x = 4.013.
Since the derivative represents the rate of change of the function, we can approximate the change in the function's value over a small interval by multiplying the derivative by the change in x. In this case, the change in x is 4.013 - 4 = 0.013.
Approximated change in the function's value = f'(4) * (change in x)
= 16 * 0.013
≈ 0.208
Final step: Estimate the value of the function at x = 4.013.
To estimate the value, we add the approximated change in the function's value to the initial value of the function at x = 4.
Estimated value of the function at x = 4.013 = f(4) + approximated change in the function's value
= 2 + 0.208
≈ 2.208
Therefore, the estimated value of the function at x = 4.013 is approximately 2.208.
I would estimate it using the first two terms of a McLaurin series expansion:
f(4.013) = f(4) + (4.013-4) f'(x)
= f(4) + 0.013 f'(4)
where f(4) = 2, and
f'(4) = 4f(4) + 2*4 = 16
So, f(4.013) = 2 + 0.013*16 = 2.208