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.)
1/(1 + 2x)^5 ≈ 1 − 10x
Sure, let's verify the given linear approximation at a = 0.
When a = 0, the linear approximation becomes:
1/(1 + 2x)^5 ≈ 1 - 10(0)
Simplifying this expression:
1/(1 + 2x)^5 ≈ 1
So, the given linear approximation at a = 0 is accurate.
Now let's determine the values of x for which the linear approximation is accurate to within 0.1.
We can set up the inequality:
|1/(1 + 2x)^5 - (1 - 10x)| ≤ 0.1
Simplifying the inequality, we get:
|1 - (1 - 10x)| ≤ 0.1
This gives us:
|10x| ≤ 0.1
Dividing both sides by 10, we obtain:
|x| ≤ 0.01
Therefore, the values of x for which the linear approximation is accurate to within 0.1 can be represented as the interval [-0.01, 0.01].
To verify the given linear approximation at a = 0, we need to substitute x = 0 into both sides of the equation and check if they are equal.
1/(1 + 2*0)^5 = 1/(1)^5 = 1
1 - 10*0 = 1
As both sides of the equation are equal to 1, we have verified the linear approximation at a = 0.
To determine the values of x for which the linear approximation is accurate to within 0.1, we need to find the range of values of x that satisfy the inequality:
|1/(1 + 2x)^5 - (1 - 10x)| ≤ 0.1
Let's solve this inequality step by step:
1. Rewrite the inequality by removing the absolute value:
-0.1 ≤ 1/(1 + 2x)^5 - (1 - 10x) ≤ 0.1
2. Add (1 - 10x) to both sides of the inequality:
-0.1 + (1 - 10x) ≤ 1/(1 + 2x)^5 ≤ 0.1 + (1 - 10x)
Simplify:
0.9 - 10x ≤ 1/(1 + 2x)^5 ≤ 1.1 - 10x
3. Multiply all sides of the inequality by (1 + 2x)^5:
(0.9 - 10x)*(1 + 2x)^5 ≤ 1 ≤ (1.1 - 10x)*(1 + 2x)^5
4. Expand the powers:
(0.9 - 10x)*(1 + 10x + 20x^2 + 40x^3 + 80x^4 + 160x^5) ≤ 1 ≤ (1.1 - 10x)*(1 + 10x + 20x^2 + 40x^3 + 80x^4 + 160x^5)
5. Simplify and group the terms:
0.9 + 10x + 20x^2 - 100x^3 - 200x^4 - 400x^5 ≤ 1 ≤ 1.1 + 10x + 20x^2 + 110x^3 + 220x^4 + 440x^5
6. Rearrange the inequality:
0 ≤ 0.1 + 10x + 210x^3 + 420x^4 + 840x^5 ≤ 0.2
7. Solve the middle part of the inequality to find the values of x:
0.1 + 10x + 210x^3 + 420x^4 + 840x^5 ≤ 0.2
Rearrange the inequality:
10x + 210x^3 + 420x^4 + 840x^5 ≤ 0.1
To solve this equation, you can use numerical methods or software such as graphing calculators or mathematical software like Mathematica.
Unfortunately, it is not practical to solve for the exact values of x manually because the equation involves higher-order polynomials. Using numerical methods or software will provide accurate solutions.