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Chapter 2: Problem 5

Draw the graph of \(y=f(x)=x^{3}\) between \(x=0\) and \(x=1.5\). Find the slope of the chord between (a) \(x=1\) and \(x=1.1,\) (b) \(x=1\) and \(x=1.001,\) (c) \(x=1\) and \(x=1.00001\). Then use algebra to find a simple formula for the slope of the chord between 1 and \(1+\Delta x\). (Use the expansion \(\left.(A+B)^{3}=A^{3}+3 A^{2} B+3 A B^{2}+B^{3} .\right)\) Determine what happens as \(\Delta x\) approaches \(0,\) and in your graph of \(y=x^{3}\) draw the straight line through the point (1,1) whose slope is equal to the value you just found. \(\Rightarrow\)

Short Answer

Expert verified
The slope approaches 3 as \( \Delta x \to 0 \); the tangent at \( x = 1 \) has a slope of 3.

Step by step solution

01

Graph the Function

Plot the graph of the function \( y = f(x) = x^3 \) from \( x = 0 \) to \( x = 1.5 \). This is a simple cubic function where \( y \) increases as \( x \) increases. Notice the curve's shape as it rises sharply beyond \( x = 1 \).
02

Slope of the Chord for Case (a)

To find the slope of the chord between \( x = 1 \) and \( x = 1.1 \), use the formula for the slope of a line between two points: \[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]Here, \( x_1 = 1 \) and \( x_2 = 1.1 \). First, find \( f(1) = 1^3 = 1 \) and \( f(1.1) = (1.1)^3 = 1.331 \). Thus, the slope is:\[ m = \frac{1.331 - 1}{1.1 - 1} = 3.31 \]
03

Slope of the Chord for Case (b)

Use the same slope formula for \( x = 1 \) and \( x = 1.001 \). Compute \( f(1.001) = (1.001)^3 \). By expanding, we find \( (1.001)^3 = 1 + 3\cdot0.001+3(0.001)^2 + (0.001)^3 \approx 1.003003 \). Thus, the slope is:\[ m = \frac{1.003003 - 1}{1.001 - 1} = 3.003 \]
04

Slope of the Chord for Case (c)

Repeat the procedure for \( x = 1 \) and \( x = 1.00001 \). Compute \( f(1.00001) = (1.00001)^3 \approx 1 + 3\cdot0.00001 = 1.00003 \). Thus, the slope is:\[ m = \frac{1.00003 - 1}{1.00001 - 1} = 3.0003 \]
05

General Formula for the Slope of the Chord

Let \( \Delta x \) be the change from \( x = 1 \), so the slope between 1 and \( 1+\Delta x \) is:\[ m = \frac{(1 + \Delta x)^3 - 1^3}{\Delta x} = \frac{1 + 3\Delta x + 3(\Delta x)^2 + (\Delta x)^3 - 1}{\Delta x} = \frac{3\Delta x + 3(\Delta x)^2 + (\Delta x)^3}{\Delta x} = 3 + 3\Delta x + (\Delta x)^2 \]
06

Analyze the Limit as \( \Delta x \to 0 \)

As \( \Delta x \to 0 \), the slope \( m = 3 + 3\Delta x + (\Delta x)^2 \to 3 \). This confirms that the instantaneous slope (or derivative) of \( y = x^3 \) at \( x = 1 \) is 3.
07

Draw the Tangent Line

With the instant slope calculated as 3, draw a straight line through the point \( (1, 1) \) on the graph of \( y = x^3 \) with a slope of 3. This line is the tangent to the curve at \( x = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing functions
Graphing functions is a fundamental concept in calculus and represents the process of drawing the curve of a function on a set of axes. For the function given, \( y = f(x) = x^3 \), it involves plotting the values of \( y \) for each corresponding value of \( x \). This involves:
  • Choosing the range of \( x \): From \( x = 0 \) to \( x = 1.5 \) in this case.
  • Calculating \( y \) for various \( x \) and plotting them.
  • Connecting these points to form a smooth curve.
The graph of \( y = x^3 \) is a cubic curve, showing how \( y \) increases sharply with increasing \( x \). Understanding the shape and behavior of graphs are essential in visualizing the solutions and properties of a function.
Cubic functions
Cubic functions are polynomial functions of the form \( f(x) = ax^3 + bx^2 + cx + d \). In the given exercise, \( f(x) = x^3 \), a simple cubic function where the coefficients are \( a = 1, b = 0, c = 0, \) and \( d = 0 \).
Cubic functions are characterized by their unique graph shape:
  • They typically have one or two bends or inflection points.
  • The end behavior of the function, meaning how \( y \) behaves as \( x \) reaches very large positive or negative numbers, generally extends infinitely in both directions.
  • In our simple case of \( f(x) = x^3 \), the graph will rise indefinitely and sharply after \( x = 1 \).
These features make cubic functions interesting and important in mathematical studies.
Tangent lines
Tangent lines are lines that touch a curve at exactly one point and their slope represents the instantaneous rate of change of the function at that point. For the function \( y = x^3 \) at \( x = 1 \), the tangent line has a slope of 3.
Here's how to relate a tangent line to a function's graph:
  • Find where the line only just touches the curve, without intersecting it.
  • Determine the slope at that point, which matches the slope of the tangent line.
  • Draw the line using the calculated slope.
Tangent lines are fundamentally important in calculus as they help find the derivative of a function, indicating how a function is changing at a specific point.
Derivatives
Derivatives represent a key concept in calculus, defining the rate of change of a function with respect to a variable. Essentially, the derivative tells us how the function behaves as \( x \) changes. For \( y = x^3 \), its derivative, calculated from \( dy/dx = 3x^2 \), gives us the function鈥檚 velocity.
To gain insight:
  • Compute the derivative by applying derivative rules.
  • Use it to determine rates of change and slopes of lines touching the curve.
  • For \( x = 1 \), the derivative \( y' = 3x^2 \) evaluates to 3.
Understanding how to find and interpret derivatives is pivotal for grasping the mathematical description of changes and trends in functions.
Slope of a line
The slope of a line is a measure of its steepness and direction. For a line between two points on a curve, the slope is determined by the difference in \( y \)-values divided by the difference in \( x \)-values. \( m = \frac{f(x_2)-f(x_1)}{x_2-x_1} \).
This concept is used to find the chord slope and extends to the derivative as \( \Delta x \to 0 \), leading to the slope of the tangent.
  • A slope of 0 means a horizontal line.
  • A positive slope means an upward direction as you move right.
  • A negative slope means a downward direction.
In this exercise, it showcases the transition from a chord to a tangent, summarizing the movement within calculus from the incremental changes to instantaneous.

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