Problem 19 - Precalculus: Mathematics for Calculus

Question

Find the derivative of the function at the given number. $$f(x)=1-3 x^{2}, \quad \text { at } 2$$

Step-by-Step Solution

Answer

The derivative of the function at \( x = 2 \) is \( -12 \).

Step 1: Identify the Function

We are given the function \( f(x) = 1 - 3x^2 \). We need to find its derivative at the point where \( x = 2 \).

Step 2: Apply the Power Rule

To find the derivative \( f'(x) \) of the function \( f(x) = 1 - 3x^2 \), use the power rule, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \).

Step 3: Differentiate Each Term

Differentiate each term of \( f(x) \). The derivative of the constant, 1, is 0. Using the power rule, the derivative of \( -3x^2 \) is \( -6x \). Thus, \( f'(x) = 0 - 6x \) or \( f'(x) = -6x \).

Step 4: Evaluate the Derivative at x = 2

Substitute \( x = 2 \) into the derivative \( f'(x) = -6x \) to find \( f'(2) \). This gives \( f'(2) = -6 \times 2 \).

Step 5: Calculate the Result

Completing the multiplication, we find \( f'(2) = -12 \). This is the derivative of the function at \( x = 2 \).

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function changes at any given point. It's like the mathematical version of measuring how fast something is moving when you look at the speedometer of a car. Derivatives are a fundamental tool in calculus as they can tell us how a function behaves. For any function, if you want to find out its rate of change, you're essentially looking to calculate its derivative. For instance, if you have a function that describes an object's position over time, its derivative will give you the object's velocity.

Mastering the Power Rule

The power rule is one of the most important tools you will encounter when working with derivatives. It offers a straightforward way to differentiate functions where the variable is raised to a power. In simple terms, if you have a function that looks like this:

• \(f(x) = x^n\)

then, according to the power rule, its derivative \(f'(x)\) is:

• \(f'(x) = nx^{n-1}\)

This rule makes it easy to compute derivatives of polynomial terms. For example, if you have \(x^2\), using the power rule gives you \(2x\). Knowing this can save a lot of time and simplifies the differentiation process.

Evaluating Derivatives at a Specific Point

Once you have found the derivative of a function, the next step is often to evaluate it at a specific point—exactly what you would do if you want to know the rate of change at that precise location. For the function \(f(x) = 1 - 3x^2\), after differentiating, you find that the derivative is \(-6x\). When you want to find the derivative at, say, \(x = 2\), you simply substitute \(2\) into the derivative. This means you calculate \(-6 imes 2\), which results in \(-12\). This value tells you the rate at which the original function is changing when \(x = 2\). It's like checking the speedometer at an exact moment in time.

Connecting to Precalculus Mathematics

Before jumping into derivatives, a solid understanding of precalculus mathematics is beneficial. Precalculus provides the necessary background in algebra and functions, which is crucial for studying calculus. Understanding functions, their graphs, and behavior is key, as calculus builds upon these concepts. When you deal with derivatives, you're often working with polynomial expressions.

• Precalculus helps with understanding how functions behave without derivatives, thus making sense of what calculus builds upon.
• Recognizing how terms of an equation contribute to its overall structure helps when you start differentiating.

Hence, bridging precalculus and calculus is essential for mastering more complex calculus concepts.