Problem 21 - Calculus

Question

Find the average slope of \(y=x^{2}\) between \(x=x_{1}\) and \(x=x_{2} .\) What does this average approach as \(x_{2}\) approaches \(x_{1} ?\)

Step-by-Step Solution

Answer

The average slope approaches \(2x_1\) as \(x_2\) approaches \(x_1\), equal to the derivative at \(x_1\).

Step 1: Define the Average Slope

The average slope of a function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula for the slope of a secant line:\[ m_{avg} = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \(y_1 = x_1^2\) and \(y_2 = x_2^2\).

Step 2: Substitute the Function Values

Substitute the given function values into the average slope formula:\[m_{avg} = \frac{x_2^2 - x_1^2}{x_2 - x_1} \]

Step 3: Simplify the Expression

Apply the difference of squares formula to simplify the numerator:\[m_{avg} = \frac{(x_2 - x_1)(x_2 + x_1)}{x_2 - x_1} \]Cancel the \((x_2 - x_1)\) terms from the numerator and the denominator:\[m_{avg} = x_2 + x_1\]

Step 4: Consider the Behavior as \(x_2\) Approaches \(x_1\)

As \(x_2\) approaches \(x_1\), the value of the average slope expression \(x_2 + x_1\) approaches \(2x_1\). This is equivalent to the derivative of \(y = x^2\) at \(x = x_1\).

Step 5: Interpret the Result

The average slope between two points \(x_1\) and \(x_2\) on the curve \(y = x^2\) becomes the instantaneous rate of change at \(x = x_1\) as \(x_2\) approaches \(x_1\). This is represented by the derivative \(y' = 2x\).

Key Concepts

Secant Line

The secant line is a straight line that connects two points on a curve. It provides an average rate of change (or average slope) between those two points, which is calculated using the formula for slope:

• \[ m_{avg} = \frac{y_2 - y_1}{x_2 - x_1} \]

For the function \(y = x^2\), this translates to connecting points \((x_1, x_1^2)\) and \((x_2, x_2^2)\). The secant line thus gives us an idea of how steeply the curve changes between these points. As you think about it, imagine this secant line as a bridge between two hills, showing the gradient or slope needed to connect them.

As the two points get closer together, the secant line begins to resemble the tangent line, which touches the curve at a single point and represents the slope at precisely that point.

Difference of Squares Formula

The difference of squares formula is a valuable tool for simplifying expressions and finding solutions. It states that for any two numbers, \(a\) and \(b\), their difference of squares can be expressed as:

• \[ a^2 - b^2 = (a - b)(a + b) \]

In the exercise, the difference of squares formula helps to simplify the expression:

• \[ x_2^2 - x_1^2 = (x_2 - x_1)(x_2 + x_1) \]

This simplification is crucial because it allows us to cancel out the \((x_2 - x_1)\) terms, making calculations much easier and leading us to the average slope \(x_2 + x_1\).

The difference of squares is not only applicable in this context but also widely used in various mathematical scenarios, making complex algebraic expressions more manageable, revealing hidden factors and facilitating easier computations.

Instantaneous Rate of Change

Instantaneous rate of change is essentially the slope of the tangent line at a single point on a curve. It tells you how fast something is changing at an exact moment, contrasted with the average rate, which considers changes over an interval. When the two points used to calculate a secant line get infinitely close, the average rate of change approaches the instantaneous rate of change.

This concept is key to understanding calculus because it links directly to derivatives. As you bring \(x_2\) closer to \(x_1\), in our exercise on the function \(y = x^2\), the average slope \(x_2 + x_1\) zeroes in on \(2x_1\), which shows the instantaneous change at \(x_1\).

• This shift from a secant line to an instantaneous change is pivotal in understanding the behavior of curves and dynamics of rates—whether it’s velocity in physics or changes in economic trends.

Derivative

A derivative represents what happens when we calculate the instantaneous rate of change of a function. It's the fundamental idea in calculus and shows how a function changes at any point. For any function \(y = f(x)\), the derivative \(f'(x)\) is the limit of the function's average rate of change as the interval approaches zero.

In the exercise, as \(x_2\) approaches \(x_1\), the average slope of \(y = x^2\) approached \(2x_1\). This limit and resultant expression \(2x_1\) is the derivative of the function at that point.

• The general derivative of \(y = x^2\) is \(y' = 2x\), which means at any point \(x\), the function changes at a rate of \(2x\).

Understanding derivatives helps to predict and model virtually anything that changes, paving the way for solving real-world problems such as optimizing processes or predicting outcomes efficiently.