Question

In Exercises \(19-22,\) (a) find the slope of the curve at \(x=a\) . (b) Writing to Learn Describe what happens to the tangent at \(x=a\) as \(a\) changes. $$y=9-x^{2}$$

Short answer

The slope of the function \(y = 9 - x^2\) at \(x=a\) is \(-2a\). As \(a\) increases, the slope becomes more negative and thus the tangent line gets steeper and slopes downwards, while as \(a\) decreases, the slope becomes less negative (or more positive) and the tangent line gets less steep and eventually slopes upwards.

Step-by-step solution

Differentiate the function

The given function is \(y = 9 - x^2\). The derivative of a constant is 0 and the derivative of \(x^2\) is \(2x\), therefore the derivative of \(y\) with respect to \(x\) (denoted as \(y'\)) is \(0 - 2x\), or \(-2x\).

Evaluate the derivative at \(x=a\)

To find the slope of the curve at \(x=a\), we substitute \(a\) into \(y'\). So \(y'(a) = -2a\). This is the slope of the tangent line at \(x=a\).

Describe the behavior of the tangent line as \(a\) changes

As \(a\) increases, the slope \(y'(a) = -2a\) becomes more negative, meaning the tangent line becomes steeper and slopes downwards. Conversely, as \(a\) decreases, the slope becomes less negative, or more positive, meaning that the tangent line becomes less steep and eventually slopes upwards. At \(a=0\), the slope is \(0\), so the tangent line is horizontal.

Key concepts

Differentiate Functions

Differentiating functions is one of the most fundamental operations in calculus. It involves finding the derivative of a function, which represents the rate at which the function's value changes with respect to a change in its input value. In simpler terms, differentiating gives us the slope of the tangent line to the function's graph at any point.

For example, to differentiate the function given in the exercise, which is a quadratic function, we apply the basic power rule. The power rule states that the derivative of a function in the form of \( x^n \) is \( n*x^{n-1} \). Following this rule, differentiating \( 9 - x^2 \) involves differentiating each term independently—\( 9 \) becoming \( 0 \) and \( -x^2 \) becoming \( -2x \). This results in the derivative \( -2x \), which describes how the function's value changes at any point along its curve.

Tangent Line Behavior

Understanding tangent line behavior is crucial in grasping the concept of derivatives. The tangent line to a curve at a given point is the straight line that just 'touches' the curve at that point. It has the same slope as the curve does at that point, which is the value of the derivative there.

In practical terms, the behavior of the tangent line tells us how the function is changing at that specific location. If the tangent line slopes upwards, the function is increasing; if it slopes downwards, the function is decreasing. When the slope is zero, it means at that point, the function has a horizontal tangent, indicating a local maximum, minimum, or saddle point—depending on the curve's overall shape.

Finding Slope at a Point

Finding the slope of a curve at a particular point is a fundamental application of the derivative. The slope is calculated as the derivative evaluated at the point of interest. For instance, in the textbook exercise, the slope of the curve \( y = 9 - x^2 \) at \( x = a \) is the value of the derivative \( -2x \) when \( x \) is replaced with \( a \)—which yields \( -2a \).

This value is not only abstract; it has real-world implications such as determining speed from a distance-time graph or calculating the efficiency of a changing process. A positive slope indicates increasing values, and a negative slope indicates decreasing values, with the steepness of the slope reflecting the rate of change.

Calculus

Calculus is an extensive field in mathematics that deals with changes. It is divided mainly into two branches: differential calculus, which concerns itself with the concept of a derivative, and integral calculus, which focuses on the concept of an integral as the 'inverse' of differentiation. Together, these disciplines allow us to analyze varying quantities and how they accumulate.

Calculus is essential for solving complex problems in physics, engineering, economics, and beyond. The simple act of finding the slope of a curve at a point with calculus has far-reaching applications—from charting the motion of celestial bodies to modeling the dynamics of stock markets.