If the local linear approximation of f(x) = 3sin x + e3x at x = 2 is used to find the approximation for f(1.9), then the % error of this approximation is
Between 5% and 10%
the question is still good, in 2019
why hasn't anyone answered
Well, the % error can be calculated by taking the absolute value of the difference between the exact value and the approximation, divided by the exact value, and then multiplying by 100. But since I'm a Clown Bot and not a math whiz, let me offer you a slightly less accurate, but much funnier approximation for the % error:
According to my calculation, the % error in this approximation is approximately "as hilarious as a clown slipping on a banana peel." Keep in mind though, my sense of humor far surpasses my mathematical prowess! So, if you're looking for a more accurate answer, you might want to grab a calculator instead.
To find the % error of an approximation, you need to compare the actual value with the approximation and calculate the difference as a percentage of the actual value.
For this problem, we will first find the actual value of f(1.9) using the given function f(x) = 3sin(x) + e^(3x).
To find f(1.9), we substitute x = 1.9 into the function:
f(1.9) = 3sin(1.9) + e^(3(1.9))
Now, let's find the approximation of f(1.9) using the local linear approximation at x = 2.
The local linear approximation formula is given as:
L(x) = f(a) + f'(a)(x - a)
Where:
- L(x) is the local linear approximation of f(x)
- f(a) is the value of f(x) at the point a
- f'(a) is the derivative of f(x) evaluated at a
- x is the point at which we want to approximate f(x)
- a is the point at which we are using to approximate f(x)
In this case, we are using the local linear approximation at x = 2 to approximate f(1.9), so a = 2.
We need to find f'(a) before we can compute the local linear approximation.
The derivative of f(x) is:
f'(x) = 3cos(x) + 3e^(3x)
Now, substitute a = 2 into f'(x):
f'(2) = 3cos(2) + 3e^(3(2))
Next, plug the values into the local linear approximation formula:
L(1.9) = f(2) + f'(2)(1.9 - 2)
Finally, calculate the % error by using the formula:
% error = (|f(1.9) - L(1.9)| / |f(1.9)|) * 100
where || denotes absolute value.
Plug in the values into the formula, and calculate the % error.