Problem 13 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the derivative of the function \( f(x) = 1 - 3x^2 \) and evaluate this derivative at \( x = 2 \). The derivative represents the slope of the tangent to the curve at any point \( x \).

Step 2: Differentiate the Function

To find the derivative, use the power rule, which states that the derivative of \( ax^n \) is \( nax^{n-1} \). Applying this to \( f(x) = 1 - 3x^2 \), we find that the derivative \( f'(x) = 0 - 3 \cdot 2x^{2-1} = -6x \).

Step 3: Evaluate the Derivative at the Given Point

Substitute \( x = 2 \) into the derivative \( f'(x) = -6x \). This gives \( f'(2) = -6 \cdot 2 = -12 \).

Step 4: Conclusion

The derivative of \( f(x) = 1 - 3x^2 \) at the point where \( x = 2 \) is \( -12 \). This means the slope of the tangent to the curve at \( x = 2 \) is \(-12\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool in calculus used for finding the derivative of functions of the form \( ax^n \). The rule states that the derivative of such a term is given by multiplying the exponent \( n \) by the coefficient \( a \), and then reducing the exponent by one. In simpler terms, for any term \( ax^n \), the derivative is \( nax^{n-1} \).

For example, if you need to find the derivative of \( 3x^2 \), you apply the power rule as follows:

• Identify the exponent \( n = 2 \) and the coefficient \( a = 3 \).
• Multiply them together to get \( 2 \times 3 = 6 \).
• Reduce the exponent by one, resulting in \( x^{2-1} = x^1 \).

Thus, the derivative is \( 6x \). This straightforward rule greatly simplifies the process of differentiation, making it easier to find the slope of the curve at any given point.

Evaluating Derivatives at a Specific Point

Once you have derived the general derivative of a function, the next step often involves evaluating this derivative at a specific point. This is done by substituting the given point into the derivative function. The value you obtain from this substitution tells you the slope of the tangent line to the function at that specific point.

For instance, consider the function \( f(x) = 1 - 3x^2 \), whose derivative is \( f'(x) = -6x \). To find the slope of the tangent at \( x = 2 \), substitute \( x = 2 \) into \( f'(x) \):

• \( f'(2) = -6 \times 2 \)
• Which results in \( -12 \).

This result of \(-12\) indicates that the tangent line at the point \( x = 2 \) has a steep downward slope, reflecting the curve's direction and behavior at that point.

Comprehending the Slope of Tangent

The slope of the tangent line to a curve at a given point describes how steep the curve is at that location. In calculus, the slope of the tangent is equivalent to the derivative of the function at that specific point. Understanding this concept is essential, as it gives insight into the function's rate of change at any point along its curve.

When you calculate the derivative, you are essentially finding a function that gives the slope of the tangent line at every possible point \( x \) on the original function. For our function \( f(x) = 1 - 3x^2 \), the derivative \( f'(x) = -6x \) implies that:

• When \( x = 2 \), the slope of the tangent line is \(-12\), indicating a steep decline.
• If \( x \) were negative, the slope would be positive, signaling an upward trend.

Thus, the slope of the tangent not only shows how steep a curve is but also reveals the directional nature of the curve, whether it's increasing or decreasing.