Problem 62 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x} $$

Step-by-Step Solution

Answer

The answer is \(3x^2\).

Step 1: Identify the type of problem

Recognize this problem is related to the definition of a derivative and involves finding the limit of a difference quotient. The function given is \( (x+\Delta x)^{3}-x^{3} \), whose derivative we need to find by applying the limit.

Step 2: Expand the cubic term

Expand the cubic function \( (x+\Delta x)^{3} \) using binomial theorem. We get \( x^3+3x^2\Delta x + 3x \Delta x ^2 + \Delta x^3 \).

Step 3: Simplify the expression

Subtract \( x^3 \) from the expanded cubic and cancel out the \( x^3 \) to get \( 3x^2\Delta x + 3x \Delta x^2 + \Delta x^3 \). Then, divide the entire expression by \( \Delta x \) to get \( 3x^2 + 3x \Delta x + \Delta x^2 \).

Step 4: Apply the limit

Taking the limit of \( 3x^2 + 3x \Delta x + \Delta x^2 \) as \( \Delta x \rightarrow 0 \) yields the simplified derivative of \( 3x^2 \). This is because the terms with \( \Delta x \) will become zero as \( \Delta x \rightarrow 0 \).

Key Concepts

Definition of a Derivative

Understanding the definition of a derivative is crucial in calculus, as it forms the backbone of differential calculus. Simply put, the derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is, in essence, the 'slope' of the curve of the function at a specific point. Mathematically, the derivative of a function at a point x is defined as the limit of the difference quotient as \( \Delta x \rightarrow 0 \) :
\[ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]
Applying this limit allows us to find precisely how the function is behaving around that point, which informs us about its instantaneous rate of change. In the context of the example exercise where we are dealing with a cubic function, this derivative tells us how the output of the function changes for a small change around the input x.

Derivative of a Cubic Function

The process outlined in the solution is an application of finding the derivative of a cubic function. A cubic function is of the form \( f(x) = ax^3 + bx^2 + cx + d \), where a, b, c, and d are constants. The derivative of this function describes how it changes at any given point and is particularly useful in identifying the nature of its graph, such as where it has maxima, minima, or points of inflection. To derive a cubic function, we use the power rule which tells us that the derivative of \( x^n \) with respect to x is \( nx^{n-1} \) for any real number n. The resulting derivative for a cubic function, hence, would be of the form \( f'(x) = 3ax^2 + 2bx + c \), which is a quadratic function. This process is visualized in the provided exercise by expanding \( (x+\Delta x)^3 \) and then applying the limit as \( \Delta x \) approaches zero.

Applying Limits in Calculus

The concept of applying limits in calculus is a foundational one, as limits help us understand the behavior of functions as inputs approach a certain value. Limits can tell us the value that a function approaches as the input gets infinitely close to some number, even if the function is not actually defined at that number. In the step-by-step solution, applying the limit as \( \Delta x \) approaches zero simplifies the expression and removes the terms involving \( \Delta x \) since they become negligible. This is a practical embodiment of the concept of limits—identifying the value a function approaches, as illustrated by the end behavior of the cubic function as we find its derivative. This concept is fundamental when dealing with continuity, rates of change, and other integral aspects of calculus.

Binomial Theorem in Algebra

Encompassed in the process of finding the derivative of the cubic function is the use of the binomial theorem in algebra. The binomial theorem provides a quick way to expand expressions that are raised to a power, such as \( (x + y)^n \). It is represented as:
\[ (x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \]
Here, \( {n \choose k} \) denotes the binomial coefficient, which can be calculated as \( \frac{n!}{k!(n-k)!} \). In our exercise, applying the binomial theorem helps to expand the term \( (x+\Delta x)^3 \) efficiently, which is a necessary step before applying the limit to find the derivative. By understanding and applying the binomial theorem, one can neatly deal with polynomial expansion without having to multiply out terms manually, saving time and reducing the potential for error.