Problem 31 - Calculus an Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(x)=x^{2}-4 $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = x^{2} - 4 \) using the limit definition is \(2x\).

Step 1: Substitute the function into the limit definition

Substitute \( f(x) = x^{2} - 4 \) into \(\lim_{h\to 0} \frac{f(x+h)- f(x)}{h}\). This gives \(\lim_{h\to 0} \frac{(x+h)^{2}-4 - (x^{2}-4)}{h}\).

Step 2: Simplify the numerators

Simplify the above equation to get rid of the complex fractions. This results in \(\lim_{h\to 0} \frac{x^{2} + 2xh + h^{2}-4 -x^{2} +4 }{h}\). Then, further simplify to: \(\lim_{h\to 0} \frac{2xh + h^{2}}{h}\).

Step 3: Simplify the equation

Finally, simplify the equation by cancelling out the common factor 'h' from the numerator and denominator: \(\lim_{h\to 0} (2x + h)\).

Step 4: Evaluate the limit

As h approaches 0, the equation simplifies to \(2x\).

Key Concepts

Finding Derivatives

Finding derivatives is a core concept in calculus that helps us understand the rate of change of a function regarding its input variable, typically denoted as \( x \). When we find a derivative, we are essentially uncovering how a function behaves as its input changes. A common method of deriving functions is the limit definition of a derivative. This states that if \( f(x) \) is a function, then its derivative \( f'(x) \) can be found using:

• \( f'(x) = \lim_{{h\to 0}} \frac{{f(x+h) - f(x)}}{h} \)

This means we're looking at the ratio of the change in the function to the change in the input as the input change approaches zero. For instance, if \( f(x) = x^2 - 4 \), then using the limit definition helps us find how steep the graph of the function is at any point \( x \). Understanding derivatives takes practice but learning their geometric significance can greatly improve comprehension.

Calculus Problem Solving

When tackling calculus problems, especially those involving derivatives, a structured approach can make things simpler. The example exercise demonstrates this well. Here are some strategies that can help:

• **Substitute**: Begin by substituting your function into the derivative formula. This makes the process less abstract and more about computation.
• **Transform**: Transform expressions, as in expanding \( (x+h)^2 \). This is crucial for dealing with terms fully.
• **Cancel Terms**: Look for terms that drop out when you substitute back into the function. They simplify your expression further.
• **Solve the Limit**: Evaluate the limit by substituting, which gives us the final expression for the derivative.

Using these steps not only helps in arriving at the correct answer, but also in understanding the process and the concept of slope and rates of change.

Simplifying Expressions

Simplifying expressions during calculus problem-solving is key to finding easy and accurate solutions. Notice the simplifications in original exercise:

• **Eliminate Zero Terms:** Identifying terms such as \(-x^2 + x^2 \) which cancel each other out helps reduce complexity.
• **Factor Common Terms:** In the numerator \( 2xh + h^2 \), you can factor out \( h \). Factoring simplifies the expression because the offending \( h \) in the denominator can be canceled.
• **Limit Evaluation**: After simplification, evaluating the limit as \( h \to 0 \) ensures the remaining terms reveal the actual derivative value, like \( 2x \).

This process not only leads to the derivative more smoothly but also ingrains a habit of checking for cancellation. Such checking improves efficiency in problem-solving both in tests and practical calculus applications.