Problem 31 - Calculus Early Transcendentals

Question

The given limit represents \(f^{\prime}(a)\) for some function \(f\) and some number \(a .\) Find \(f(x)\) and \(a\) in each case. (a) \(\lim _{\Delta x \rightarrow 0} \frac{\sqrt{1+\Delta x}-1}{\Delta x}\) (b) \(\lim _{x_{1} \rightarrow 3} \frac{x_{1}^{2}-9}{x_{1}-3}\)

Step-by-Step Solution

Answer

(a) \( f(x) = \sqrt{x}, a = 1 \); (b) \( f(x) = x^2, a = 3 \).

Step 1: Recognize the Derivative Form

The limit expressions given are in the form of the definition of a derivative: \\[ f'(a) = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) - f(a)}{\Delta x} \] for part (a) and \\[ f'(a) = \lim_{x_1 \to a} \frac{f(x_1) - f(a)}{x_1 - a} \] for part (b).

Step 2: Solve Part (a)

The expression \\[ \lim _{\Delta x \rightarrow 0} \frac{\sqrt{1+\Delta x}-1}{\Delta x} \] \resembles the derivative of the function \\( f(x) = \sqrt{x} \) evaluated at \\( a = 1 \) because the numerator matches the structure of \\( f(a + \Delta x) - f(a) = \sqrt{1 + \Delta x} - 1. \)

Step 3: Solve Part (b)

The expression \\[ \lim _{x_{1} \rightarrow 3} \frac{x_{1}^{2}-9}{x_{1}-3} \] \involves factoring the numerator to get \\( x_1^2 - 9 = (x_1 - 3)(x_1 + 3) \). This limit matches the structure of the derivative of \\( f(x) = x^2 \) evaluated at \\( a = 3 \), since \\( f(x_1) = x_1^2 \) and \\( f(3) = 9. \)

Step 4: Conclusion

For part (a), \\( f(x) = \sqrt{x} \) and \\( a = 1 \). \( f(x) \) corresponds to the original numerator function of the expression. For part (b), \\( f(x) = x^2 \) and \\( a = 3 \). The process of factoring confirms the derivative's definition.

Key Concepts

Understanding Limit Expressions

A limit expression is a fundamental concept in calculus. It describes how a function behaves as its input approaches a particular point. In many cases, you might deal with limits as variables get infinitesimally close to a value. This is crucial for defining derivatives. The concept becomes more intuitive if you think about the limit as analyzing the behavior of a function's y-values as the x-values get closer to a specific target.

Limit expressions are not just abstract concepts; they have practical applications. For instance,

• The concept of a limit helps determine the slope of the tangent line at a specific point on a curve.
• It is also essential for finding instantaneous rates of change.

Understanding how to evaluate these expressions is foundational for exploring calculus further. These limits form the groundwork for discovering derivatives, as seen in the exercise examples.

The Definition of a Derivative

The derivative of a function at a point provides vital information about the function's behavior. It represents the rate at which the function is changing at that particular point, giving insights into the slope of the tangent to the curve.

The derivative is formally defined using a limit. For a function \(f(x)\) at a point \(a\), the derivative \(f'(a)\) is defined as: \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] This formula captures the idea of finding the slope of the tangent line to \(f(x)\) at point \(a\). The closer \(h\) gets to zero, the closer you get to calculating the exact slope of the tangent line.

Through this process, you transform the idea of rate of change into a precise calculation, which is key to function analysis. Examples like the ones in the exercise show how these limit expressions fit the derivative's structure, hinting at the link between the algebraic manipulation of functions and their geometric and practical interpretations.

Deep Dive into Function Analysis

Function analysis is an essential part of calculus that involves examining a function's properties to understand its behavior. It enables you to identify critical points, analyze growth and decline, and determine concavity.

In calculus, the first step is often finding the derivative, as it plays a pivotal role in assessing:

• The increasing or decreasing nature of a function
• Identifying maxima or minima
• Understanding acceleration in real-world problems

For instance, once you identify a derivative from a limit expression, you gain insights into the function's slope at any given point. This slope helps sketch the graph and predict the function's future values.

For students tackling exercises, the key is breaking down expressions like the ones given and recognizing how they reflect the derivative of specific functions at distinct points. Such analysis guides your comprehension, allowing you to uncover patterns and inner workings within calculus problems.