Problem 30 - Calculus

Question

Use the fourth-degree Taylor approximation for \(x\) near 0 $$ \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} $$ to explain why \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}\)

Step-by-Step Solution

Answer

The limit is \(\frac{1}{2}\).

Step 1: Understand the Given Taylor Approximation

We are given the fourth-degree Taylor polynomial for \(\cos x\) near \(x = 0\): \[ \cos x \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} \]This polynomial approximation simplifies the expression of \(\cos x\) around 0 for small values of \(x\).

Step 2: Write the Expression for \(1 - \cos x\)

Substitute the Taylor approximation into \(1 - \cos x\):\[ 1 - \cos x \approx 1 - \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} \right) \]Simplifying this, we get:\[1 - \cos x \approx \frac{x^2}{2!} - \frac{x^4}{4!}\]

Step 3: Divide by \(x^2\)

We need to find the limit of \(\frac{1 - \cos x}{x^2}\) as \(x\) approaches 0. Start by dividing the simplified expression by \(x^2\):\[ \frac{1 - \cos x}{x^2} \approx \frac{\frac{x^2}{2!} - \frac{x^4}{4!}}{x^2} = \frac{x^2}{2!x^2} - \frac{x^4}{4!x^2}\]Simplify this to:\[ \frac{1}{2} - \frac{x^2}{24}\]

Step 4: Evaluate the Limit as \(x\) Approaches 0

As \(x\) approaches 0, the term \(\frac{x^2}{24}\) approaches 0 because it is multiplied by \(x^2\), which goes to zero faster than any other polynomial term that is not of degree 0.Thus, we have:\[ \lim_{x \to 0} \left( \frac{1}{2} - \frac{x^2}{24} \right) = \frac{1}{2}\]Hence, the limit is \(\frac{1}{2}\).

Key Concepts

Limit

Limits are a fundamental concept in calculus that allows us to determine the behavior of a function as its input approaches a certain value. In the specified exercise, we are asked to find the limit of \( \frac{1-\cos x}{x^2} \) as \(x\) approaches 0.
This means we want to understand how the expression \( \frac{1-\cos x}{x^2} \) behaves or what value it approaches as \(x\) gets closer and closer to zero. By using Taylor series approximation, we simplify the \(1-\cos x\) term using its Taylor polynomial, making the expression easier to handle.

• The key is to simplify complex trigonometric expressions into polynomials, which are easier to work with.
• Calculating limits of simpler polynomial expressions becomes straightforward as we analyze the behavior as \(x\) nears a point.

In this case, the limit helps in showing how the polynomial approximated derivative relates back to the original trigonometric function.

Cosine Function

The cosine function \(\cos x\) is one of the basic trigonometric functions and describes the shape of the 'cosine wave'. It is an even function, meaning \(\cos(-x) = \cos(x)\), which implies that its graph is symmetrical about the y-axis. The function is periodic with a period of \(2\pi\), repeating its values over that interval.
Understanding the cosine function's behavior near 0 is crucial for approximating it using Taylor series, as it transitions from its maximum value of 1 at \(x=0\).

• Considering only small values of \(x\), where \(\cos x\) is close to 1, is essential for approximation.
• Near \(x=0\), the small differences between \(1\) and \(\cos x\) make polynomial approximations practical.

Utilizing cosine’s properties in Taylor series approximations allows us to explore its behavior algebraically rather than relying solely on its trigonometric representation.

Polynomial Approximation

Polynomial approximation, particularly the Taylor Series, is a method of representing complex functions using polynomials of a finite degree.
For small values of \(x\), a polynomial can provide a very good approximation to a function like \(\cos x\). In the Taylor series approximation for \(\cos x\) specified in the exercise:\[\cos x \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!}\]we see only terms up to the fourth degree of \(x\), which simplifies calculations near zero.

• The first term, 1, is the value of \(\cos x\) at \(x=0\).
• Successive terms, like \(-\frac{x^2}{2!}\), reflect the effects of \(\cos x\)'s curvature at small \(x\).
• The approximation becomes more accurate as we include higher degree terms, but near zero, few terms suffice.

These approximations allow tackling otherwise intricate functions using basic algebra.

Derivatives

Derivatives are a key tool in calculus for understanding how a function changes with respect to changes in its input. They inform us about the rate of change and the 'slope' of the function at any given point.
In terms of \(\cos x\), its derivative tells us how the function rises or falls as \(x\) increases or decreases.

• The derivative of \(\cos x\) is \(-\sin x\), reflecting how the function decreases at \(x = 0\).
• Polynomials obtained in Taylor series approximations often directly relate to derivatives of the function at a particular point, providing deeper insights into the function's shape.

Finding the limit in the original problem exemplifies how approximating a trigonometric function using derivatives at zero leads to a comprehensible polynomial that simplifies evaluation of limits. By building on derivatives' principles, one can transition from complex functions to easy-to-solve expressions.