Problem 15 - Calculus - AP Edition

Question

A function \(y=f(x)\) and an \(x\) -value \(x_{0}\) are given. (a) Find a formula for the slope of the tangent line to the graph of \(f\) at a general point \(x=x_{0}\). (b) Use the formula obtained in part (a) to find the slope of the tangent line for the given value of \(x_{0}\). $$f(x)=x^{2}-1 ; x_{0}=-1$$

Step-by-Step Solution

Answer

The slope of the tangent line at \(x=-1\) is \(-2\).

Step 1: Identify the Formula for the Slope of the Tangent Line

The formula for the slope of the tangent line to the graph of a function \(f(x)\) at point \(x = x_0\) involves the derivative of the function. The slope is given by \(f'(x)\), the derivative of \(f\) evaluated at \(x_0\).

Step 2: Calculate the Derivative of the Function

First, find the derivative of the function \(f(x) = x^2 - 1\). The derivative, \(f'(x)\), can be calculated using basic differentiation rules:\[f'(x) = \frac{d}{dx}(x^2 - 1) = 2x.\]

Step 3: Evaluate the Derivative at the Given Point

Now, use the derivative to find the slope of the tangent line at \(x_0 = -1\). Evaluate \(f'(x)\) at \(x = -1\):\[f'(-1) = 2(-1) = -2.\]

Step 4: Interpret the Result

Thus, the slope of the tangent line to the graph of \(f(x)\) at the point where \(x = -1\) is \(-2\). This means that the tangent line is decreasing at this point, with a steepness of \(-2\).

Key Concepts

Understanding Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. This special line gives us valuable information about the behavior of the curve near that point. The tangent line's slope is especially important because it tells us how fast the curve is changing right there. It's like peeking into the curve's current trend. Imagine you're hiking along a trail on a hill: when you stop at a point to check the steepness of the trail, you're essentially considering the tangent line at that point.
If the slope is steep, expect a challenging climb; if it's gentle, your walk might be smoother.

In calculus, the concept of a tangent line is crucial as it provides a linear approximation to the curve at the point of tangency. The steeper the tangent line, the faster the change. You can think of it as a snapshot of the curve's direction at that moment.

• Touches the curve at one point
• Locally linear approximation of the curve
• Reflects the rate of change of the curve

Derivatives and Their Calculation

The derivative of a function at a specific point informs us of the slope of the tangent line at that point. In simple terms, the derivative tells us how a function is changing at any given point, much like measuring the speed of a car at a certain moment.
For the function given, \(f(x) = x^2 - 1\), we find its derivative to determine how the slope of the function changes as x changes.

To find the derivative, we rely on differentiation rules. These rules are fundamental in calculus and serve as shortcuts to efficiently calculate derivatives.

• Constant rule: derivative of a constant is 0
• Power rule: derivative of \(x^n\) is \(nx^{n-1}\)

Using the power rule, the derivative of \(f(x) = x^2 - 1\) gives us \(f'(x) = 2x\).
This function, \(f'(x)\), is critical as it directly provides the slope of the function's tangent line at any point x.

The Slope of the Tangent Line

The slope of the tangent line indicates how steep the curve is at a specific point. When you calculate the derivative and evaluate it at a specific point, you get this slope.
For example, in the function \(f(x) = x^2 - 1\), with the given point \(x_0 = -1\), we use the derivative \(f'(x) = 2x\) to find the slope of the tangent line at \(x = -1\).
By evaluating, \(f'(-1) = 2(-1) = -2\), we find that the slope is \(-2\).

This negative slope tells us that at \(x = -1\), the curve is decreasing. The value \(-2\) indicates that for a small increase in x around \(-1\), the function decreases twice as fast. Thus, the tangent line gives us a precise local view of the curve's steepness and direction at that point. Understanding this helps you know exactly how the curve moves, making calculus a powerful tool for understanding change.

• Positive slope means increasing function
• Negative slope means decreasing function
• The absolute value of slope measures steepness