Problem 60 - Thomas Calculus

Question

a. Series for sinh \(^{-1} x\) Find the first four nonzero terms of the Taylor series for \begin{equation} \sinh ^{-1} x=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} \end{equation} b. Use the first three terms of the series in part (a) to estimate sinh \(^{-1} 0.25 .\) Give an upper bound for the magnitude of the estimation error.

Step-by-Step Solution

Answer

a. \( x - \frac{x^3}{6} + \frac{3x^5}{40} - \frac{5x^7}{112} \) b. \( \approx 0.24776 \), Error \(< 0.0000122\)

Step 1: Express Function as a Derivative

We start by recognizing that \( f(x) = \sinh^{-1} x \) is defined as an integral: \( f(x) = \int_{0}^{x} \frac{dt}{\sqrt{1+t^{2}}} \). To find the Taylor series, we need to differentiate this to obtain: \( f'(x) = \frac{1}{\sqrt{1+x^2}} \).

Step 2: Find Taylor Series Expansion of \( \frac{1}{\sqrt{1+x^2}} \)

The Taylor series of \( \frac{1}{1 + x^2} \) is \( 1 - x^2 + x^4 - x^6 + \cdots \). Taking the square root modifies this, so start with: \( \frac{1}{\sqrt{1+x^2}} = 1 - \frac{x^2}{2} + \frac{3x^4}{8} - \frac{5x^6}{16} + \cdots \).

Step 3: Integrate Term by Term

To find the series for \( \sinh^{-1} x \), integrate the terms obtained from \( \frac{1}{\sqrt{1+x^2}} \). The integration gives us: \( x - \frac{x^3}{6} + \frac{3x^5}{40} + \cdots \).

Step 4: First Four Nonzero Terms

Thus, the first four nonzero terms of the Taylor series for \( \sinh^{-1} x \) are: \[ x - \frac{x^3}{6} + \frac{3x^5}{40} - \frac{5x^7}{112} \].

Step 5: Estimate \( \sinh^{-1} 0.25 \) Using First Three Terms

We approximate \( \sinh^{-1} 0.25 \) using the first three terms: \[ 0.25 - \frac{(0.25)^3}{6} + \frac{3(0.25)^5}{40} \]. Calculating each term gives: \( 0.25 - 0.0026041667 + 0.0003662109 \). Thus, the approximation is \( 0.2477620442 \).

Step 6: Calculate Error Bound

The error is at most the next term in the series. So, the error term is \( \frac{5(0.25)^7}{112} \), which is about \( 0.0000122070 \). This means the magnitude of the error is less than \( 0.0000122070 \).

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are analogs of inverse trigonometric functions but for hyperbolic functions. These functions, such as sinh, cosh, and tanh, have their inverses defined in terms of logarithms and algebraic expressions. For instance, \( \sinh^{-1} x \) represents the inverse of the hyperbolic sine function. It's noteworthy that these functions are different from regular trigonometric inverses, and they offer unique properties due to their definition involving a hyperbola rather than a circle.
The function \( \sinh^{-1} x \) is particularly interesting because it is often represented by an integral, \( \sinh^{-1} x = \int_{0}^{x} \frac{dt}{\sqrt{1+t^{2}}} \). This form is important in calculus as it provides a way to express and calculate this function using integral calculus, showcasing its smooth transition between algebraic and calculus concepts. When solving problems involving inverse hyperbolic functions, Taylor series can be quite useful, as they allow us to approximate these functions around a specific point.

Error Estimation

Error estimation is a crucial part of numerical analysis where we assess how accurate our approximations are. When we use a Taylor series to estimate a function like \( \sinh^{-1} x \), it is important to gauge how far off our approximation could be from the true value.
To estimate the error, we consider the next term in the series that we did not include in our approximation. For example, the error term for estimating \( \sinh^{-1} 0.25 \) involves calculating the magnitude of the next term in the Taylor series. This error term can be expressed and calculated as \( \frac{5(0.25)^7}{112} \), providing a precise upper bound to help us understand the possible deviation from the actual value.

• Error estimation is vital because it assures us of the reliability of our approximation.
• Knowing the error helps us decide whether our approximation is sufficient for practical purposes or if further refinement is needed.

By being aware of the error, students can better grasp the balance between approximation and precision in practical applications of calculus.

Integral Calculus

Integral calculus is a branch of calculus concerned with accumulation of quantities and finding areas under curves. It complements differential calculus, and together they make calculus extraordinarily powerful in analyzing change and motion.
In the context of inverse hyperbolic functions, integral calculus shows its strength by providing ways to express these inverses, like \( \sinh^{-1} x \), in integral form. Such expressions allow computation via definite and indefinite integrals, enabling approximation through techniques like series expansion.
For example, by knowing that \( \sinh^{-1} x = \int_{0}^{x} \frac{dt}{\sqrt{1+t^{2}}} \), you can recognize that integral calculus provides a complete framework to evaluate such functions via integration. This represents a practical application of the fundamental theorem of calculus, linking antiderivatives and definite integrals.

• This integration approach is useful not only for theoretical purposes but also for practical computations in physics and engineering.
• Learning integral calculus aids in understanding a plethora of scientific and mathematical phenomena.

Applying integral techniques to evaluate inverse hyperbolic functions reinforces the integral calculus tools students need for broader problem-solving efforts.