Problem 23 - Thomas Calculus

Question

Estimate the error if \(\cos t^{2}\) is approximated by \(1-\frac{t^{4}}{2}+\frac{t^{8}}{4 !}\) in the integral \(\int_{0}^{1} \cos t^{2} d t\).

Step-by-Step Solution

Answer

The error is approximately \(-\frac{1}{9360}\).

Step 1: Understand the Problem

We are given the function \( \cos t^2 \) and its approximation with a polynomial series, \( 1 - \frac{t^4}{2} + \frac{t^8}{4!} \). We need to estimate the error between the true integral \( \int_{0}^{1} \cos t^{2}\, dt \) and the integral of the approximation.

Step 2: Set Up the Error Term

The error in approximating the cosine function using its Taylor series is given by the next term in the series (in Lagrange form of the remainder for Taylor series). The next term after \( \frac{t^8}{4!} \) for \( \cos t^2 \) is \(-\frac{t^{12}}{6!}\). Thus the error term can be set up as \( -\int_0^1 \frac{t^{12}}{6!} dt \).

Step 3: Compute the Error Integral

Calculate the integral of the error term \( -\int_0^1 \frac{t^{12}}{6!} dt \). This becomes \(-\frac{1}{6!} \left[ \frac{t^{13}}{13} \right]_0^1 = -\frac{1}{13 \times 6!} \).

Step 4: Simplify the Results

Compute \( 6! = 720 \). Plug this into the formula to find the error: \(-\frac{1}{13 \times 720} = -\frac{1}{9360} \).

Step 5: Interpret the Error

The error from using the polynomial approximation of \( \cos t^2 \) over the interval \([0, 1]\) is \(-\frac{1}{9360}\). This is a small negative value, indicating that the approximation slightly overestimates the integral.

Key Concepts

Taylor Series

The Taylor series is a powerful mathematical tool used to approximate complex functions with polynomials. Calculating higher degree terms allows for a more precise approximation of the original function. A Taylor series expands a function f(x) around a point (usually zero) in terms of its derivatives. The general formula for a Taylor series centered at zero is:\[ f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \ldots \]This approach helps to estimate functions like the cosine function, especially where direct integration is tricky.

• The polynomial approximation includes terms of ascending powers of x.
• Each term becomes smaller as higher powers are used, giving a more refined approximation.

We use Taylor series when it's difficult to compute the function directly, as they offer a simple way to approximate complex functions.

Integral Approximation

Integral approximation involves estimating the value of an integral when it's challenging to solve it analytically. When we approximate integrals, especially with polynomials, we inevitably introduce some error. The error estimation in integral approximation via Taylor series can be computed by considering the next term in the series that wasn't included.

• For the integral \( \int_{0}^{1} \cos t^2 \, dt \), we use a polynomial to approximate \( \cos t^2 \).
• The error is the integral of the next term in the series that wasn't used in the approximation.

The integral of this unused term tells us how much our approximation differs from the exact value. This helps us determine how precise our approximation is.

Cosine Function

The cosine function, denoted as \( \cos x \), is periodic and even. This means it repeats its shape every 2π units, and it is symmetric about the y-axis. The cosine function is common in trigonometry, describing the x-coordinate of a point on a unit circle as the point moves around the circle.

• Often used in physics, engineering, and mathematics for simplifying oscillatory behavior.
• The power series for the cosine function is an infinite sum of terms.

Approximations like the one provided in Taylor series help when you need to work with cosine without using calculus directly on the function. These approximations are especially useful when integrals involving \( \cos \) are hard to solve analytically.