Problem 53 - Thomas Calculus

Question

Centered difference quotients The centered difference quotient $$ \frac{f(x+h)-f(x-h)}{2 h} $$ is used to approximate \(f^{\prime}(x)\) in numerical work because \((1)\) its limit as \(h \rightarrow 0\) equals \(f^{\prime}(x)\) when \(f^{\prime}(x)\) exists, and \((2)\) it usually gives a better approximation of \(f^{\prime}(x)\) for a given value of \(h\) than Fermat's difference quotient $$ \frac{f(x+h)-f(x)}{h} $$ See the accompanying figure. a. To see how rapidly the centered difference quotient for \(f(x)=\sin x\) converges to \(f^{\prime}(x)=\cos x,\) graph \(y=\cos x\) together with $$ y=\frac{\sin (x+h)-\sin (x-h)}{2 h} $$ over the interval \([-\pi, 2 \pi]\) for \(h=1,0.5,\) and \(0.3 .\) Compare the results with those obtained in Exercise 51 for the same values of \(h .\) b. To see how rapidly the centered difference quotient for \(f(x)=\cos x\) converges to \(f^{\prime}(x)=-\sin x,\) graph \(y=-\sin x\) together with $$ y=\frac{\cos (x+h)-\cos (x-h)}{2 h} $$ over the interval \([-\pi, 2 \pi]\) for \(h=1,0.5,\) and \(0.3 .\) Compare the results with those obtained in Exercise 52 for the same values of \(h .\)

Step-by-Step Solution

Answer

Graph both functions with approximations for decreasing \( h \) to visualize convergence to their derivatives.

Step 1: Understand the Definitions

We have two different difference quotients to analyze: 1) the centered difference quotient \( \frac{f(x+h)-f(x-h)}{2 h} \) and 2) Fermat's difference quotient \( \frac{f(x+h)-f(x)}{h} \). We will focus on the centered difference quotient, because it provides a better approximation of \( f^{\prime}(x) \) for a given \( h \).

Step 2: Identify the Functions and Their Derivatives

For part (a), we are given \( f(x) = \sin x \) whose derivative is \( f^{\prime}(x) = \cos x \). For part (b), we have \( f(x) = \cos x \) with a derivative \( f^{\prime}(x) = -\sin x \).

Step 3: Formulate the Centered Difference Quotients

For part (a), the centered difference quotient is \( y = \frac{\sin(x+h) - \sin(x-h)}{2h} \). For part (b), it is \( y = \frac{\cos(x+h) - \cos(x-h)}{2h} \). These expressions approximate the derivatives \( \cos x \) and \( -\sin x \) respectively.

Step 4: Graph the Functions Over the Interval

Using a graphing utility or software, plot the graphs from \(-\pi\) to \(2\pi\). For part (a), graph both \( y = \cos x \) and the centered difference quotient for \( h = 1, 0.5, \) and \( 0.3 \). Similarly, for part (b), plot \( y = -\sin x \) with the centered difference quotient for the same \( h \) values.

Step 5: Analyze the Graphs

Observe how closely the centered difference quotient approximations match their respective derivative functions as \( h \) decreases. Typically, as \( h \) becomes smaller, the approximation becomes more accurate. Compare these observations to the exercises mentioned (51 and 52) and note the differences in convergence rates.

Key Concepts

Centered Difference Quotient

The centered difference quotient is a tool used in numerical differentiation to approximate the derivative of a function. It is expressed as \(\frac{f(x+h)-f(x-h)}{2h} \).
This method considers the difference between points symmetrically placed around \(x\). The benefit of this approach is its ability to provide a more accurate estimate of the derivative than other methods.

• The formula converges more rapidly to the true derivative as \(h\) approaches zero.
• It reduces error by averaging changes from both sides of \(x\), thus balancing any asymmetrical disturbances.

By graphing functions such as \(\sin x\) and \(\cos x\), alongside their derivatives, we can visualize how effectively the centered difference quotient approximates the derivative for different values of \(h\). This makes it preferable for applications requiring numerical differentiation.

Fermat's Difference Quotient

Named after the mathematician Pierre de Fermat, Fermat's difference quotient is another method to approximate derivatives. It is given by the formula\(\frac{f(x+h)-f(x)}{h}\).
While this approach calculates the slope of the tangent line to the curve at \(x\), it does so by measuring the slope between \(x\) and a nearby point \(x+h\).

• This results in a straightforward computation that serves as the foundation for defining the derivative in calculus.
• However, the method is prone to larger errors compared to the centered difference quotient. These errors arise due to the lack of symmetry in its calculation.

In practice, when numerical precision is crucial, it's often substituted by the centered difference for better convergence properties.

Trigonometric Functions

Trigonometric functions, like \(\sin x\) and \(\cos x\), play a significant role in mathematics, especially in describing waves and oscillatory motions. In the context of numerical differentiation, these functions serve as excellent examples because of their well-defined derivatives.

• The derivative of \(\sin x\) is \(\cos x\), while \(\cos x\) has a derivative of \(-\sin x\).
• They exhibit periodic behavior, which makes them ideal candidates for studying the effectiveness of difference quotients over an interval like \([-\pi, 2\pi]\).

Graphing these functions and their difference quotient approximations allows us to visibly compare how closely these numerical methods match the actual derivatives. This comparison is valuable for understanding the precision of different approximation techniques.

Approximation of Derivatives

In numerical analysis, the approximation of derivatives is crucial, especially for functions that are difficult to differentiate analytically. The use of difference quotients helps mathematicians and scientists estimate the rate of change of such functions.

• The centered difference quotient tends to provide superior approximations due to its balanced approach.
• As parameter \(h\) becomes smaller, the approximations generally become more accurate, highlighting the importance of choosing appropriate step sizes in numerical calculations.

While exact derivatives are sometimes not obtainable, numerical approximations offer significant insight into the behavior of complex functions. They allow for practical computation in various fields such as engineering, physics, and finance, where understanding change is pivotal to problem-solving.