Problem 3 - Calculus Concepts

Question

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=3 x^{2}\) is \(f^{\prime}(x)=6 x\).

Step-by-Step Solution

Answer

The derivative of the function is correctly confirmed as \(f'(x) = 6x\).

Step 1: Understand the Limit Definition of Derivative

The limit definition of a derivative for a function \(f(x)\) is given by \(f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\). This formula will allow us to find the derivative of \(f(x)\) by considering the instantaneous rate of change as \(h\) approaches zero.

Step 2: Substitute the Function into the Definition

Substitute \(f(x) = 3x^2\) into the limit definition: \(f'(x) = \lim_{{h \to 0}} \frac{{3(x+h)^2 - 3x^2}}{h}\). The goal is to simplify this expression and compute the limit.

Step 3: Expand \((x+h)^2\)

Expand the term \((x+h)^2\):\[\(f'(x) = \lim_{{h \to 0}} \frac{{3(x^2 + 2xh + h^2) - 3x^2}}{h}\).\] Simplifying this expression will allow for easier computation of the limit.

Step 4: Simplify the Expression

Simplify the expression in the limit: \(f'(x) = \lim_{{h \to 0}} \frac{{3x^2 + 6xh + 3h^2 - 3x^2}}{h}\). Cancel out the \(3x^2\) terms: \(f'(x) = \lim_{{h \to 0}} \frac{{6xh + 3h^2}}{h}\).

Step 5: Factor Out \(h\) from the Numerator

Factor \(h\) from the numerator: \(f'(x) = \lim_{{h \to 0}} \frac{{h(6x + 3h)}}{h}\). This allows \(h\) to cancel, simplifying the expression further.

Step 6: Cancel the \(h\) and Compute the Limit

Cancel the \(h\) in the numerator and denominator: \(f'(x) = \lim_{{h \to 0}} (6x + 3h)\). Now, you can directly compute the limit by letting \(h\) approach zero: \(f'(x) = 6x\).

Step 7: Conclusion

Thus, the derivative using the limit definition is confirmed to be \(f'(x) = 6x\), verifying the statement is correct.

Key Concepts

Instantaneous Rate of Change

The instantaneous rate of change of a function at a given point provides insight into how the function behaves at that exact location. You could imagine it as the speedometer of a car, showing how fast it's going at a particular moment.
In calculus, this concept is central because it helps us find the derivative of a function, which represents this rate of change at every possible point along the graph.
For a polynomial function like \(f(x) = 3x^2\), understanding the rate at which it changes at any point \(x\) means looking at how each small change in \(x\) affects the value of \(f(x)\).
As the small change \(h\) approaches zero, the calculation of this rate becomes precise, offering a clear picture of the function's behavior at every instant.

Derivative Calculation

Calculating the derivative involves a very precise method called the limit definition of the derivative. This method is key to representing the instantaneous rate of change mathematically.
To get the derivative of a function \(f(x)\), you apply the formula:

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

This approach allows you to understand what happens to the function as the increment \(h\) becomes very small, practically shrinking to nothing.
In practical steps, you substitute the function into this formula and simplify until you are left with an expression free of \(h\), which represents the derivative \(f'(x)\).

Algebraic Method

The algebraic method used in calculating derivatives involves several handy algebraic manipulations to clean up the expressions.
For example, with the function \(f(x) = 3x^2\), you substitute it into the limit definition and apply some algebra:

• Start with \(f'(x) = \lim_{{h \to 0}} \frac{{3(x+h)^2 - 3x^2}}{h}\).
• Expand \((x+h)^2\) to get \(3(x^2 + 2xh + h^2)\).
• Subtract \(3x^2\) from \(3(x^2 + 2xh + h^2)\) to simplify.
• Factor out \(h\) from the result to further simplify.
• This facilitates the cancellation of \(h\) in the fraction, preparing you for the limit calculation.

This method not only confirms theoretical results but also enhances your problem-solving skills in calculus.

Polynomial Functions

Polynomial functions, such as \(f(x) = 3x^2\), are among the most straightforward when it comes to differentiating. These functions are expressed in terms of powers of \(x\), such as \(x^2\) in our example.
A significant characteristic of polynomial functions is that their derivatives are also polynomials. This feature is incredibly valuable because it keeps the process manageable and predictable.
For a polynomial \(f(x) = 3x^2\):

• The derivative, \(f'(x) = 6x\), is a consequence of applying the power rule and the constant rule through the limit definition.
• As you differentiate, the degree of each term in the polynomial decreases by one, simplifying the function further.

Understanding how to differentiate polynomial functions is foundational, forming a basis for entering more complex calculus topics.