Problem 41 - Calculus: Early Transcendentals

Question

Each limit represents the derivative of some function \( f \) at some number \( a \). State such an \( f \) and \( a \) in each case. \( \displaystyle \lim_{h \to 0} \frac{\cos (\pi + h) + 1}{h} \)

Step-by-Step Solution

Answer

The function is \( f(x) = \cos(x) \) and the point is \( a = \pi \).

Step 1: Recognize the Limit Definition of Derivative

To determine the function \( f(x) \) and the point \( a \), recognize that the limit expression \( \lim_{h \to 0} \frac{\cos (\pi + h) + 1}{h} \) fits the general form of a derivative: \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).

Step 2: Manipulate the Trigonometric Function

Use the identity for cosine: \( \cos(\pi + h) = -\cos(h) \). Thus, the expression in the limit becomes \( \lim_{h \to 0} \frac{-\cos(h) + 1}{h} \).

Step 3: Identify the Function \( f(x) \) and Point \( a \)

From the expression, identify \( f(h) = -\cos(h) \) with the point \( a = \pi \). Contacting the formula, \( f(x) = \cos(x) \) and the derivative evaluated at \( a = \pi \) is considered. Therefore, \( f(x) = \cos(x) \).

Step 4: Validate Function and Point

With \( f(x) = \cos(x) \), substitute \( a = \pi \), find \( f(\pi + h) = \cos(\pi + h) \), and \( f(\pi) = -1 \). The expression becomes \( \lim_{h \to 0} \frac{(-\cos(h) + 1)}{h} \), which matches the given limit statement.

Key Concepts

Limit Definition of Derivative

Calculating derivatives using limits can seem intimidating at first, but with understanding, it becomes straightforward. The derivative of a function at a point basically measures how the function changes as its input changes. The foundational formula to grasp is:\[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]This formula is often referred to as the "limit definition of a derivative". It captures the idea that the derivative is the slope of the tangent line at any given point on the curve of a function. Here, \( f(a+h) \) represents the function value slightly ahead of the point \( a \), and \( f(a) \) is the function value at \( a \). The \( h \) is a tiny increment that approaches zero.
This was used in the given exercise to identify \( f(x) = \cos(x) \) and the point \( a = \pi \), thus transforming the limit expression into the recognizable derivative form.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying expressions that involve trigonometric functions like sine and cosine. They allow us to rewrite expressions in ways that make them easier to understand or compute.In our exercise, one critical identity used was:

• \( \cos(\pi + h) = -\cos(h) \)

Using identities can help unravel complex trigonometric expressions, making them more workable. Recognizing and applying these identities correctly is key when working with limits involving trigonometric functions.
So whenever you face an expression with trigonometric terms, pause and recall identities, such as angles summation or double angle formulas, to simplify the terms.

Calculus Problem-Solving

Effective calculus problem-solving often involves breaking down complex problems into more manageable parts. With practice, you'll find certain methods make problems easier to tackle.One such method is the use of identities and limit definitions in simplifying expressions.

• First, recognize the structure of the equation: Identify if it fits known forms like the derivative definition.
• Next, manipulate the expression using algebraic or trigonometric identities to simplify.
• Finally, verify your results by plugging them back into the problem to ensure they align with the original function or expression.

Here, in this exercise, understanding these steps helped to determine and confirm\( f(x) = \cos(x) \) was at point \( a = \pi \), demonstrating the derivative testing by limits. Building this step-by-step problem-solving approach is vital to mastering calculus.