Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=\sin x ; a=0$$

Step-by-Step Solution

Answer

The estimated derivative \( f'(0) \) is approximately 1.

Step 1: Understand the Derivative Definition

The derivative of a function at a point, often represented as \( f'(a) \), provides the slope of the tangent line to the function at that point. It can be estimated using the limit definition of a derivative: \[ f'(a) \approx \frac{f(a+h) - f(a)}{h} \] for a very small value of \( h \).

Step 2: Choose a Small Value for h

To accurately estimate the derivative, pick a very small value for \( h \). In this calculation, let's choose \( h = 0.001 \), which is small enough to give a good approximation.

Step 3: Calculate f(a) and f(a+h)

First, calculate \( f(a) \) by substituting \( a = 0 \) into the function: \( f(0) = \sin(0) = 0 \). Next, calculate \( f(a+h) \) using \( a+h = 0+0.001 = 0.001 \): \( f(0.001) = \sin(0.001) \). Use a calculator to find \( \sin(0.001) \approx 0.001 \).

Step 4: Substitute Values into Derivative Formula

With \( f(0) = 0 \) and \( f(0.001) \approx 0.001 \), substitute into the derivative formula: \[ f'(0) \approx \frac{\sin(0.001) - \sin(0)}{0.001} = \frac{0.001 - 0}{0.001} \].

Step 5: Calculate the Derivative

Perform the calculation from the previous step: \[ f'(0) \approx \frac{0.001}{0.001} = 1 \]. So, the estimated value for \( f'(0) \) is \( 1 \).

Key Concepts

Limit Definition of a Derivative

Understanding the concept of derivative estimation begins with the limit definition of a derivative. This method provides a powerful way to calculate the slope of a function at a point. Derived from the foundation of limits, the formula is:

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

This equation helps us find the derivative by taking the limit as the segment \( h \) approaches zero. Essentially, it measures how \( f(x) \) changes over an infinitesimally small interval, thereby determining the function's instantaneous rate of change. Practice choosing small values for \( h \) using a calculator. It will direct you towards an accurate approximation of a derivative. For this exercise, \( h = 0.001 \) was selected to ensure precision.

Slope of Tangent Line

The concept of the slope of a tangent line naturally ties into derivatives. At any point on a curve, the tangent line is the line that just 'touches' the curve at that point without intersecting it, like a gentle tap.
The slope of the tangent line is exactly what the derivative of the function at that point tells us. It demonstrates how steep the curve is at that specific point, providing crucial information about the behavior of the function almost instantly.

• A positive slope means the function is rising at that point.
• A negative slope indicates a downward trend.
• A zero slope suggests a flat tangent, indicating a potential local extremum.

In this example, the derivative \( f'(0) \) was calculated to be "1". This means the slope of the tangent line to \( f(x) = \sin x \) at \( x=0 \) is exactly 1—indicating a moderate positive slope.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent, play a pivotal role in derivative calculation. These functions are periodic, which means they repeat values in regular intervals, offering unique behavior suitable for illustrating derivatives.
In our scenario, we are examining the sine function \( f(x) = \sin x \). The sine function is smooth and continuous, inherently making it a great candidate for derivative analysis using the limit definition.
For any given angle \( x \), the sine function outputs the height of the corresponding point on a unit circle. In our example:

• \( f(0) = \sin(0) = 0 \)
• \( f(0.001) = \sin(0.001) \approx 0.001 \)

It's crucial to note that the derivative of \( \sin x \) at \( x=0 \) should be 1, aligning well with the calculation where it indicates a linear rise in the angle close to zero. This linear approximation in small angles makes calculations very precise with minimal error.