Problem 36 - Essential Calculus

Question

In Exercises \(35-38,\) use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. $$ e \approx 1+1+\frac{1^{2}}{2 !}+\frac{1^{3}}{3 !}+\frac{1^{4}}{4 !}+\frac{1^{5}}{5 !} $$

Step-by-Step Solution

Answer

The approximate error would be less than or equal to \(\frac{e}{6!}\). The actual error is given by the absolute value of the difference between the actual value of \(e\) and the approximated value. These are compared to validate the Taylor theorem error bound.

Step 1: Calculate the Taylor series approximation

The approximation for \(e\) using Taylor series up to the fifth term is: \(e \approx 1+1+\frac{1^{2}}{2 !}+\frac{1^{3}}{3 !}+\frac{1^{4}}{4 !}+\frac{1^{5}}{5 !}\). Calculate this value for further error analysis.

Step 2: Compute the upper bound of the error using Taylor's Theorem

The remainder term \(R_n(x)\) according to Taylor's theorem is given by \( \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\). Here, as the function is \(e^x\), \(f^{(n+1)}(c)\) equals to \(e^c\) and, \(x=a=1\). So, the error is bounded by \(\frac{e}{(n+1)!}\). It is required to use the next term \(n=6\) in the series to get the error which would be \(\frac{e}{6!}\). So calculate this error bound.

Step 3: Calculate the actual error

The actual error is given by the difference between the actual value and the approximated value, i.e., \(|e - approximation|\). Calculate the actual value of \(e\) using standard methods and then find the difference with the approximation from step 1.

Step 4: Compare the approximate and exact error

Compare the approximate error obtained in step 2 with the actual error derived in step 3. This step verifies the correctness of the error upper bound estimation.

Key Concepts

Error Approximation

Error approximation refers to how closely a mathematical approximation represents the actual value.
This concept becomes crucial when dealing with Taylor series, which provides an approximate value of a function using a polynomial.
The exact error is the absolute difference between the approximated value and the true value. In Taylor's Theorem, the error term is often called the remainder or the error bound.

• The error term, denoted as \( R_n(x) \), indicates how much the approximation deviates from the true value.
• Taylor's Theorem allows for predicting an upper bound to this error without computing the actual value.
• This bounded error is useful in practical applications where calculating the exact error is costly or impossible.

Example: Approximating \( e \) using a polynomial of degree 5, as in our exercise, yields an error bounded by the next term in the series: \( \frac{e}{6!} \). This means that the deviation between our approximation and the true value of \( e \) cannot be greater than this bound.

Taylor Series

Taylor series is a representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point.
It allows complex functions to be expressed in a polynomial form, making them easier to work with for calculations and approximations.
Specifically, Taylor series for a function \( f(x) \) about a point \( a \) is given by:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \]

• This series can approximate functions like \( e^x \) by taking a finite number of terms.
• The more terms used, the more accurate the approximation becomes.
• The sum truncates when fewer terms are used, leading to an approximation instead of an exact value.

• In our example, up to the fifth term, \( e \approx 1 + 1 + \frac{1^2}{2!} + \frac{1^3}{3!} + \frac{1^4}{4!} + \frac{1^5}{5!} \), was used to estimate \( e \).

This method illustrates how Taylor series can convert a complex, non-polynomial function into a simpler polynomial form.

Exponential Function e

The exponential function \( e^x \) is a fundamental mathematical function, critical in calculus and mathematical analysis.
The number \( e \) (approximately 2.71828) itself is a mathematical constant that is the base of the natural logarithm.
Calculating \( e \) can be done efficiently through its Taylor series expansion.

• Using the Taylor series for \( e^x \) at \( x = 1 \) gives a powerful tool to approximate \( e \).
• The series starts at \( e^0 = 1 \) and builds on this by adding terms that involve factorials in the denominators.
• The polynomial format lets us use additions and multiplications, which are simpler than exponential functions.

In situations where computational simplicity is desired, using such expansions allows for manageable calculations even when precision isn’t perfectly exact. This is why Taylor series are significant not only in pure mathematics but also in fields requiring precise scientific computations.