Verify the given linear approximation at
a = 0.
Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.)
ln(1 + x) ≈ x
xE
Linear approximation to what function?
Did you try to copy and paste? If so, that does not work here.
The question was like this!
Oh, I see
f(x) = ln (1+x)
df/dx = 1/(1+x)
d^2f/dx^2 = -1/(1+x)^2
f(x) = f(0) + [1/(1+0)] x - x^2/2! ...
f(x) = 0 + x - x^2/2 + .....
well at a first cut when is x^2/2 =.1 x?
x/2 = .1
x = .2
To verify the given linear approximation at `a = 0` and determine the values of `x` for which the linear approximation is accurate to within `0.1`, we'll follow these steps:
Step 1: Verify the linear approximation
Since the given linear approximation is `ln(1 + x) ≈ x`, we need to evaluate both the linear approximation and the original function at `a = 0` to check if they are equal.
For the linear approximation: ln(1 + x) ≈ x
Substituting `a = 0`: ln(1 + 0) ≈ 0
Simplifying: ln(1) ≈ 0
The natural logarithm of 1 is 0, so ln(1) = 0. Therefore, the linear approximation holds at `a = 0`.
Step 2: Determine the interval where the linear approximation is accurate to within `0.1`
To find the interval where the linear approximation is accurate to within `0.1`, we'll set up the inequality |f(x) - L(x)| < 0.1, where f(x) is the original function and L(x) is the linear approximation.
For our case, f(x) = ln(1 + x) and L(x) = x.
|ln(1 + x) - x| < 0.1
Now, solve this inequality to find the values of `x` for which the linear approximation is accurate to within `0.1`. Since this is a logarithmic function, it's easier to solve graphically or using a calculator.
If you graph the function y = |ln(1 + x) - x| - 0.1 on a graphing calculator, you can find the interval where the graph is less than zero. This will give the values of `x` that satisfy the condition.
Alternatively, you can use numerical methods such as the bisection method or Newton's method to approximate the values of `x`. Repeat the process until you have an interval where the function crosses the x-axis at a point where the absolute value is less than `0.1`. This will give the desired interval.
Once you obtain the interval, express your answer using interval notation by stating the lower and upper bounds separated by a comma. Round your answers to three decimal places, if necessary.
Please note that solving this inequality analytically might be challenging as it involves solving a transcendental equation. Using numerical methods or graphical representations can provide a more practical solution.