Problem 38 - Calculus - AP Edition

Question

Let \(f(x)=\sin x .\) Estimate \(f^{\prime}(\pi / 4)\) by (a) using a graphing utility to zoom in at an appropriate point until the graph looks like a straight line, and then estimating the slope (b) using a calculating utility to estimate the limit in Formula (13) by making a table of values for a succession of values of \(w\) approaching \(\pi / 4\)

Step-by-Step Solution

Answer

The derivative \( f'(\pi/4) \approx 0.7071 \).

Step 1: Understanding the function

The given function is \( f(x) = \sin x \). We need to find the derivative \( f'(x) \) at \( x = \frac{\pi}{4} \). \( f'(x) \) gives us the slope of the tangent to the curve at this point.

Step 2: Using a Graphing Utility

Plot the function \( f(x) = \sin x \) using a graphing calculator or software. Focus on \( x = \frac{\pi}{4} \). Zoom into \( \frac{\pi}{4} \) until the curve appears as a straight line. The slope of this line is approximately \( f'(\frac{\pi}{4}) \). Through visual inspection or a digital tool, estimate this slope.

Step 3: Calculating the Derivative Using the Limit Definition

The derivative at a point \( x \), \( f'(x) \), is defined as \( \lim_{w \to x} \frac{f(w) - f(x)}{w - x} \). For \( f'(\frac{\pi}{4}) \), substitute \( x = \frac{\pi}{4} \) and calculate \( \lim_{w \to \frac{\pi}{4}} \frac{f(w) - \sin(\frac{\pi}{4})}{w - \frac{\pi}{4}} \).

Step 4: Creating a Table of Values

Construct a table of values by choosing \( w \) values approaching \( \frac{\pi}{4} \) from both sides (e.g., \( 0.75, 0.8, 0.79, 0.78, 0.77 \) etc.). Calculate \( \frac{f(w) - \sin(\frac{\pi}{4})}{w - \frac{\pi}{4}} \) for each \( w \).

Step 5: Estimating the Limit from the Table

Examine the calculated values from the table, for \( w \to \frac{\pi}{4} \). The values should approach a specific number, which is an estimate for \( f'(\frac{\pi}{4}) \).

Key Concepts

Limit Definition of Derivative

To understand the limit definition of a derivative, think of it as a method for finding the slope of the tangent line to a curve at a specific point. Imagine you are looking at a very zoomed-in section of the graph of a function. At this point, the curve may start to look like a straight line. The goal is to find the slope of this line, which represents how steep the curve is at that point. This is what the derivative tells us.

Mathematically, the derivative of a function \( f(x) \) at a point \( x \) is defined by the limit:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a small increment that approaches zero. Essentially, we check how much \( f(x) \) changes as \( x \) changes by a tiny amount \( h \). Although in real life, calculating this manually could be tricky, graphing utilities can provide a visual confirmation of how the limit behaves.

Slope of Tangent Line

The slope of a tangent line at a point on a curve is one of the most intuitive ways to understand derivatives. If you were to draw a line that just "kisses" a function at one point and doesn't cross it, that's your tangent line. Its slope describes the rate at which \( f(x) \) is changing at that specific point on the curve.

To find this slope for the function \( f(x) = \sin x \) at \( x = \frac{\pi}{4} \), one approach is to use graphing utilities. Zooming in extremely close to \( x = \frac{\pi}{4} \) on the sine curve can simplify the curve to an apparent straight line; thus the slope can be visually estimated. Technologically advanced calculators or software will allow you to approximate this slope simply by focusing intently on the desired point.

Numerical Approximation

Sometimes, finding an exact derivative analytically can be complex or, given software capabilities, a quick numerical approximation better serves our needs. When using numerical approximation, the idea is to get close to the point of interest, such as \( x = \frac{\pi}{4} \), by evaluating the function at values surrounding this point.

Here's how you can do it: make a table of values where you compute \( \frac{f(w) - f(x)}{w - x} \) for values of \( w \) approaching \( \frac{\pi}{4} \) from both directions. For example, try \( w = 0.75, 0.76, 0.77, ... \) and similarly for values slightly above. Finding the derivative this way is like playing "hot and cold"—as \( w \) gets closer to \( \frac{\pi}{4} \), the computed slope gets "hotter," i.e., more accurate.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics and have unique properties that influence their derivatives. The derivative of a trigonometric function not only represents the rate of change but can also hint at the oscillatory nature of these functions.

For instance, for \( f(x) = \sin x \), its derivative \( f'(x) = \cos x \) tells us how the sine curve changes at each point \( x \). At \( x = \frac{\pi}{4} \), \( f'(\frac{\pi}{4}) \), which equals \( \cos(\frac{\pi}{4}) \), gives us the instantaneous rate of change or "slope." Trigonometric derivatives often result in periodic changes that relate closely to the original function's period and frequency, reflecting their cyclical patterns. Understanding these principles helps us navigate and anticipate behaviors not just in statistics, but in wave behavior across physics and engineering.