QuestionFind , the degree 5 Taylor polynomial of the function at . Find all values of for which this approximation is within 0.004794 of the right answer. Assume for simplicity that we limit ourselves to . ote: You can earn partial credit on this problem.
Studdy Solution
STEP 1
1. We are working with the function f(x) = cos(x).
2. We need to find the Taylor polynomial of degree 5 centered at a = 0.
3. We will use the Taylor series expansion for cosine.
4. For the error bound, we'll use the Lagrange remainder theorem.
5. We are limiting our analysis to |x| ≤ 1.
STEP 2
1. Calculate the derivatives of f(x) up to the 5th order.
2. Evaluate these derivatives at a = 0.
3. Construct the Taylor polynomial T₅(x).
4. Determine the error bound using the Lagrange remainder theorem.
5. Solve the inequality for x to find the range where the approximation is within 0.004794.
STEP 3
Let's calculate the derivatives of f(x) = cos(x) up to the 5th order:
f(x) = cos(x)
f'(x) = -sin(x)
f''(x) = -cos(x)
f'''(x) = sin(x)
f⁽⁴⁾(x) = cos(x)
f⁽⁵⁾(x) = -sin(x)
STEP 4
Now, let's evaluate these derivatives at a = 0:
f(0) = cos(0) = 1
f'(0) = -sin(0) = 0
f''(0) = -cos(0) = -1
f'''(0) = sin(0) = 0
f⁽⁴⁾(0) = cos(0) = 1
f⁽⁵⁾(0) = -sin(0) = 0
STEP 5
We can now construct the Taylor polynomial T₅(x) using the formula:
T₅(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)² + (f'''(a)/3!)(x-a)³ + (f⁽⁴⁾(a)/4!)(x-a)⁴ + (f⁽⁵⁾(a)/5!)(x-a)⁵
Substituting the values we found:
T₅(x) = 1 + 0(x-0) + (-1/2!)(x-0)² + 0(x-0)³ + (1/4!)(x-0)⁴ + 0(x-0)⁵
Simplifying:
T₅(x) = 1 - (1/2)x² + (1/24)x⁴
This is our final Taylor polynomial of degree 5 for cos(x) at a = 0.
STEP 6
To determine the error bound, we'll use the Lagrange remainder theorem. The error R₅(x) is bounded by:
|R₅(x)| ≤ (M₆/6!)|x|⁶
where M₆ is the maximum value of |f⁽⁶⁾(x)| on the interval [0, x].
The 6th derivative of cos(x) is cos(x), which has a maximum absolute value of 1 on any interval.
So, M₆ = 1, and our error bound is:
|R₅(x)| ≤ (1/6!)|x|⁶ = |x|⁶/720
STEP 7
To find where the approximation is within 0.004794 of the right answer, we need to solve:
|x|⁶/720 ≤ 0.004794
720 * 0.004794 = |x|⁶
3.45168 = |x|⁶
|x| ≤ ∛∛3.45168
|x| ≤ 0.9999...
Since we're already limited to |x| ≤ 1, our final answer is:
|x| ≤ 1
This means the approximation is within 0.004794 of the true value for all x in the given interval [-1, 1].
The Taylor polynomial T₅(x) for cos(x) at a = 0 is:
T₅(x) = 1 - (1/2)x² + (1/24)x⁴
And the range of x for which this approximation is within 0.004794 of the right answer, given |x| ≤ 1, is:
|x| ≤ 1
Was this helpful?