Problem 31 - Calculus

Question

Use a fourth-degree Taylor approximation for \(e^{h},\) for \(h\) near \(0,\) to evaluate the following limits. Would your answer be different if you used a Taylor polynomial of higher degree? (a) \(\lim _{h \rightarrow 0} \frac{e^{h}-1-h}{h^{2}}\) (b) \(\lim _{h \rightarrow 0} \frac{e^{h}-1-h-\frac{h^{2}}{2}}{h^{3}}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{2}\); (b) \(\frac{1}{6}\). The answer would be the same with a higher-degree series.

Step 1: Recall Taylor Series Expansion

The Taylor series for a function about 0 is given by: \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4 + \ldots \). For \(e^h\), the derivatives at 0 are all 1, so the expansion is \(e^h = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \frac{h^4}{4!} + \ldots\).

Step 2: Simplify Expression for (a) Using the Taylor Series

Substitute the series expansion into \( e^h - 1 - h \):\[ e^h - 1 - h = \left(1 + h + \frac{h^2}{2} + \frac{h^3}{6} + \frac{h^4}{24}\right) - 1 - h = \frac{h^2}{2} + \frac{h^3}{6} + \frac{h^4}{24} \]The expression becomes \( \frac{\frac{h^2}{2} + \frac{h^3}{6} + \frac{h^4}{24}}{h^2} \).

Step 3: Evaluate the Limit for (a)

Divide each term by \( h^2 \):\[ \frac{\frac{h^2}{2} + \frac{h^3}{6} + \frac{h^4}{24}}{h^2} = \frac{1}{2} + \frac{h}{6} + \frac{h^2}{24} \]As \( h \to 0 \), the higher-order terms vanish, so the limit is \( \frac{1}{2} \).

Step 4: Simplify Expression for (b) Using the Taylor Series

Substitute the series expansion into \( e^h - 1 - h - \frac{h^2}{2} \):\[ e^h - 1 - h - \frac{h^2}{2} = \left(1 + h + \frac{h^2}{2} + \frac{h^3}{6} + \frac{h^4}{24}\right) - 1 - h - \frac{h^2}{2} = \frac{h^3}{6} + \frac{h^4}{24} \]The expression becomes \( \frac{\frac{h^3}{6} + \frac{h^4}{24}}{h^3} \).

Step 5: Evaluate the Limit for (b)

Divide each term by \( h^3 \):\[ \frac{\frac{h^3}{6} + \frac{h^4}{24}}{h^3} = \frac{1}{6} + \frac{h}{24} \]As \( h \to 0 \), the higher-order terms vanish, so the limit is \( \frac{1}{6} \).

Step 6: Consider Higher Degree Taylor Series Impact

Using a Taylor series with a degree higher than 4 would not change the results for the given limits, as higher-degree terms vanish when divided by the respective powers of \( h \) as \( h \to 0 \). The contributions of higher-degree terms become negligible in the limit.

Key Concepts

Taylor approximation

The Taylor approximation is a powerful tool in calculus for estimating the values of functions that might be too complex to compute directly. Essentially, it involves using a polynomial to approximate the function. This is done by matching the function's value and several of its derivatives at a particular point, most commonly 0 for simplicity, which is then called a Maclaurin series.

• The Taylor series for a function is expressed as an infinite sum of terms, each involving a derivative of the function evaluated at a specific point, divided by a factorial term.
• The approximation's accuracy depends on the number of terms used; more terms mean higher accuracy.
• For the exponential function, the series converges quickly, providing very good approximations even with a few terms.

To demonstrate, consider the Taylor series for the function of the exponential function, it can be written as:\[ e^h = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \frac{h^4}{4!} + \ldots \] where all derivatives of \( e^h \) are 1 at \( h = 0 \). This makes the calculations straightforward.

Limits in Calculus

Limits are a cornerstone concept in calculus, providing a method to determine how a function behaves as its input approaches a specific value. Limits are essential for understanding continuous functions and are the foundational elements for defining derivatives and integrals.

• In the context of Taylor approximations, limits help determine the behavior of the function as an independent variable becomes infinitely small or large.
• This is particularly useful when examining the differences between a function and its Taylor approximation, as the small parameters in the expansion diminish.

For example, in the given exercise, the limits \( \lim_{h \rightarrow 0} \frac{e^h - 1 - h}{h^2} \) and \( \lim_{h \rightarrow 0} \frac{e^h - 1 - h - \frac{h^2}{2}}{h^3} \) involve examining how these expressions behave when \( h \) approaches zero. By using Taylor approximations to simplify these expressions, calculating the limits becomes manageable, as we focus only on the remaining significant terms.

Exponential function

The exponential function, denoted as \( e^x \), is one of the most important functions in mathematics. It is the function that describes continuous growth or decay and is defined as the limit of \( (1 + \frac{x}{n})^n \) as \( n \to \infty \). This function has unique properties that make it central in calculus.

• The derivative of the exponential function is the function itself, i.e., \( \frac{d}{dx} e^x = e^x \), which simplifies calculations significantly.
• At \( x = 0 \), \( e^0 = 1 \), which provides a convenient base point for Taylor expansions.

In the context of Taylor series, the exponential function is particularly appealing because its derivatives at zero are all 1, simplifying the calculation of the series terms considerably. This leads to the well-known expansion \( e^h = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \ldots \), which captures the behavior of the function very effectively near \( h = 0 \).

Polynomial approximation

Polynomial approximation involves representing a complex function with a polynomial, making it easier to analyze and compute. The most common form of polynomial approximation in calculus is the Taylor series, where a function is approximated by a series of its derivatives at a specific point.

• The Taylor polynomial creates a tangent approximation that locally closely matches the function.
• The degree of the polynomial indicates the number of derivatives included and thus influences the approximation's accuracy.
• Lower degree polynomials provide rough approximations, while higher degree polynomials improve precision.

In practical terms, for the function \( e^h \), a fourth-degree Taylor polynomial would include terms up to \( \frac{h^4}{24} \). This level of approximation is generally sufficient for many applications, as higher-order terms contribute less significantly, especially in limit problems where \( h \to 0 \). Hence, polynomial approximation through the Taylor series is a versatile and valuable technique in calculus.