Problem 54 - Calculus - AP Edition

Question

Find the limit by interpreting the expression as an appropriate derivative. (a) \(\lim _{\Delta x \rightarrow 0} \frac{\ln \left(e^{2}+\Delta x\right)-2}{\Delta x}\) (b) \(\lim _{w \rightarrow 1} \frac{\ln w}{w-1}\)

Step-by-Step Solution

Answer

(a) \( \frac{1}{e^2} \), (b) 1.

Step 1: Recognize the Expression as a Derivative

For part (a), the limit expression \( \lim _{\Delta x \rightarrow 0} \frac{\ln \left(e^{2}+\Delta x\right)-2}{\Delta x} \) resembles the definition of a derivative. Here, it looks like \( \lim _{h \to 0} \frac{f(x+h) - f(x)}{h} \) where \( f(x) = \ln(x) \). Therefore, the expression can be interpreted as the derivative of \( f(x) = \ln(x) \) at \( x = e^2 \).

Step 2: Compute the Derivative for Part (a)

The derivative of \( f(x) = \ln(x) \) is \( f'(x) = \frac{1}{x} \). Therefore, at \( x = e^2 \), the derivative is \( f'(e^2) = \frac{1}{e^2} \). So, the limit is \( \frac{1}{e^2} \).

Step 3: Recognize the Expression as a Derivative for Part (b)

For part (b), the limit expression \( \lim _{w \rightarrow 1} \frac{\ln w}{w-1} \) can similarly be recognized as a derivative at a point. It resembles \( \lim _{h \to 0} \frac{f(x+h) - f(x)}{h} \) where \( f(x) = \ln(x) \) evaluated at \( x = 1 \).

Step 4: Compute the Derivative for Part (b)

Since we know \( f'(x) = \frac{1}{x} \), at \( x = 1 \), the derivative is \( f'(1) = \frac{1}{1} = 1 \). Thus, the limit is 1.

Key Concepts

Derivative Definition

In calculus, a derivative is a way to show how a function changes as its input changes. It is often seen as a rate of change or the slope of the function's graph at a particular point. The formal definition involves a limit, usually written as:\[\lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]This expression is called the derivative of the function \(f\) with respect to \(x\). The derivative measures how much \(f(x)\) changes with a very small change in \(x\), making it foundational in understanding trends and motion in mathematical functions.Key points to remember:

• The derivative represents the instantaneous rate of change.
• It is the slope of the tangent line to the curve of \(f(x)\) at any point \(x\).
• In our example, recognizing limits as derivatives simplifies the computation process.

This fundamental concept is used in all branches of calculus and applies to understanding how functions behave at specific points.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \(e\), where \(e \approx 2.71828\). It is one of the key functions used in calculus due to its unique properties in growth and decay models. The natural logarithm answers the question: "To what power do we need to raise \(e\) to get \(x\)?"Important properties include:

• \( \ln(e) = 1 \) because \(e^1 = e\).
• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• \( \ln(xy) = \ln(x) + \ln(y) \), showing its logarithmic identity property.
• \( \ln(1) = 0 \) because any number raised to the power of 0 is 1.

In the given exercise, recognizing that the derivative of \( \ln(x) \) is \( \frac{1}{x} \) helps us take limits effectively by interpreting them as derivatives. This is useful for simplifying tricky calculus problems.

Limit Computation

Limit computation is a central part of calculus and involves finding the value that a function approaches as the input approaches a certain point. Limits are used to define the derivatives and integrals, among other things.Some steps to compute limits effectively:

• First, try direct substitution if the function is continuous at the point.
• If you encounter an indeterminate form like \(\frac{0}{0}\), apply algebraic manipulation such as factoring or rationalizing.
• Sometimes, limits are best recognized as derivatives, as seen in the exercise problems. This is especially handy when the expression fits into the format of the derivative definition.
• Using L'Hôpital’s Rule is another method when applicable, which involves the derivatives of the numerator and denominator.

The limits in the problem serve as practical examples of how limits can be computed by recognizing them in the form of derivatives, streamlining the problem-solving process.

Differential Calculus

Differential calculus is the field of mathematics focused on the concept of the derivative. It deals with the study of how functions change, known as rate of change, slopes, and curves. Core concepts include:

• The derivative, which is the main tool in differential calculus, and shows how functions change at any given point.
• Differentiation is the process of finding a derivative, which can be done through rules like the power, product, quotient, and chain rules.
• In real-world scenarios, derivatives tell us about velocities, accelerations, and other rates of change.
• Differential calculus helps in optimization problems, finding maximum and minimum points on graphs.

For the given exercise, interpreting a limit as a derivative is a perfect example of employing differential calculus. It allowed us to find the limit by recognizing the underlying derivative problem, demonstrating the practical aspects of differential calculus.