Problem 38 - Calculus for Biology and Medicine

Question

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\sin \frac{\pi}{6}}{h}$$ Find \(f\) and \(a\).

Step-by-Step Solution

Answer

The function is \( f(x) = \sin(x) \) and \( a = \frac{\pi}{6} \).

Step 1: Identify the Limit as a Derivative Definition

The given expression \( \lim_{h \to 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\sin \frac{\pi}{6}}{h} \) is the definition of the derivative of a function \( f \) at a point \( a \), where \( f(x) = \sin(x) \) and \( a = \frac{\pi}{6} \). This is the form \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \), which matches the derivative form for the sine function.

Step 2: Determine the Function

In the expression, \( f(x) \) is identified by looking at \( \sin \left(\frac{\pi}{6}+h\right) \). This tells us that \( f(x) = \sin(x) \) since it modifies the input \( x \) of the sine function.

Step 3: Determine the Point 'a'

The point \( a \) can be identified by the part of the expression given by \( \sin \frac{\pi}{6} \), which means that \( a = \frac{\pi}{6} \). This represents where the derivative is being evaluated on the function \( f(x) = \sin(x) \).

Key Concepts

Understanding Limits in Calculus

In calculus, a limit helps us understand the behavior of a function as its input approaches a certain value. When we talk about limits, we're essentially looking at what happens to a function's outcome as we get very close to a specific point.
One of the common uses of limits is in defining derivatives. A derivative represents how a function changes as its input changes. Mathematically, this is represented as:
\[\text{if } f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}\]

This expression finds the slope of the tangent line at a point on the function. Essentially, we're measuring how quickly the function's output changes as we make tiny adjustments to its input near that point.
When evaluating limits, especially in derivatives, it's important to plug in values that approach the target value (in this case, as \( h \to 0 \)), rather than reaching that exact value — this helps us avoid undefined expressions.

Exploring the Sine Function

The sine function is one of the basic trigonometric functions and is robust in phenomena involving wave patterns, circles, and oscillations.
Often represented as \( \sin(x) \), this function outputs the y-coordinate of a point on the unit circle corresponding to a given angle \( x \) in radians.

• The sine of an angle varies between -1 and 1.
• It is periodic, repeating its values every \( 2\pi \) radians.
• The graph of the sine function is a smooth, continuous wave that oscillates above and below the x-axis.

The Role of the Sine Function in Derivatives

When using the sine function to define a derivative, such as in the given exercise, we're examining how the sine function changes at a specific point. The derivative tells us whether the sine function's value is increasing or decreasing at that point and at what rate.

The Point of Evaluation in Derivatives

When determining the derivative at a particular point, known as the 'point of evaluation,' we're examining the exact behavior of the function at that point.
In our given expression, the point of evaluation is crucial for determining where exactly on the function we are calculating the derivative.
For example, if we are finding the derivative where \( a = \frac{\pi}{6} \) on the sine function, we are interested in the rate of change of \( \sin(x) \) when \( x \) is exactly \( \frac{\pi}{6} \).

• At this point, \( \sin(\frac{\pi}{6}) \) equals \( \frac{1}{2} \).
• The derivative gives insight into how rapidly the sine function is rising or falling at \( \frac{\pi}{6} \).

Understanding the point of evaluation helps contextualize the derivative's value, providing insight into the function's behavior at that specific location.