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Chapter 2: Problem 6

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0\), and \(1 .\) c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{\int(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0)\), and \(f^{\prime}(1)\). How do these slopes compare with those of the lines you drew in part (b)? $$f(x)=-x^{3}$$

Short Answer

Expert verified
Graph the function \(f(x) = -x^3\), draw tangents at specified points, compute \(f'(x) = -3x^2\), and compare tangent slopes.

Step by step solution

01

Graph the Function

Given function is: \[ f(x) = -x^3 \] To graph the function, create a table of values for different values of \(x\) and calculate the corresponding \(f(x)\). Plot these points on the graph and connect them to form the cubic curve.
02

Draw Tangent Lines

Draw tangent lines to the graph at points where the \(x\)-coordinates are \(-2, 0\), and \(1\). Using a ruler, find the slope visually or with estimation.
03

Find the Derivative

To find \( f'(x) \), use the limit definition of the derivative. \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] First, compute \( f(x+h) \): \[ f(x+h) = -(x+h)^3 \] Then, use the formula: \[ f'(x) = \lim_{h \to 0} \frac{-(x+h)^3 + x^3}{h} \] Simplify the expression inside the limit.
04

Simplify the Derivative Expression

Expand \( -(x+h)^3 \): \[ -(x+h)^3 = -[x^3 + 3x^2h + 3xh^2 + h^3] = -x^3 - 3x^2h - 3xh^2 - h^3 \] Substitute into the limit: \[ f'(x) = \lim_{h \to 0} \frac{-x^3 - 3x^2h - 3xh^2 - h^3 + x^3}{h} \] and combine like terms: \[ f'(x) = \lim_{h \to 0} \frac{-3x^2h - 3xh^2 - h^3}{h} \] Factor out \(h\): \[ f'(x) = \lim_{h \to 0} \frac{h(-3x^2 - 3xh - h^2)}{h} \] \[ f'(x) = \lim_{h \to 0} (-3x^2 - 3xh - h^2) \] As \(h\) approaches 0, the remaining terms simplify to: \[ f'(x) = -3x^2 \]
05

Find Specific Derivative Values

Compute the derivative at specific points:\[ f'(-2) = -3(-2)^2 = -3 \times 4 = -12 \] \[ f'(0) = -3(0)^2 = 0 \] \[ f'(1) = -3(1)^2 = -3 \] These values give the slopes of the tangents at \(x = -2, 0, \) and \(1\).
06

Compare the Slopes

Compare the slopes computed: The slopes from part (b) should be visually similar to the computed values. At \(x = -2\) the slope is \(-12\), at \(x = 0\) the slope is \(0\), and at \(x = 1\) the slope is \(-3\).

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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 that provides a visual representation of a mathematical equation. To graph the function \( f(x) = -x^3 \), you need to create a table of values by choosing different values for \( x \) and calculating the corresponding \( f(x) \) values. For example, if \( x \) is -2, then \( f(-2) = -(-2)^3 = -(-8) = 8 \). Repeat this process for other values such as 0, 1, and 2. Plot these points on a coordinate plane and connect them smoothly to reveal the graph. The result is a cubic curve that opens downward because of the negative sign.
Tangent lines
A tangent line to a graph at a particular point is a straight line that touches the curve at that point without crossing it. This line represents the instantaneous rate of change of the function at that specific point. To draw the tangent lines to the graph \( f(x) = -x^3 \) at \( x = -2, 0, \) and \( 1 \), first identify these points on the graph. Using a ruler, draw straight lines that just touch the curve at these points. These are the tangent lines. They visually represent the slope or the derivative at the given points.
Limit definition of derivative
The limit definition of the derivative is a method used to find the derivative of a function at any given point. The derivative \( f'(x) \) of the function \( f(x) \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] For the function \( f(x) = -x^3 \), the steps are:
  • Compute \( f(x+h) \) which gives: \( f(x+h) = -(x+h)^3 \).
  • Use the limit formula: \[ f'(x) = \lim_{h \to 0} \frac{-(x+h)^3 + x^3}{h} \]
  • Simplify the expression inside the limit by expanding \( -(x+h)^3 \) and combining like terms.
  • This process results in \( f'(x) = -3x^2 \), the derivative of our function.
Cubic functions
Cubic functions are polynomial functions of degree three, generally expressed as \( ax^3 + bx^2 + cx + d \). In our example, \( f(x) = -x^3 \) is a cubic function where the coefficient of \( x^3 \) is negative. This impacts the shape of the graph, making it open downwards as opposed to upwards. Key characteristics of cubic functions include:
  • They can have one or three real roots, where the graph intersects the x-axis.
  • They may have inflection points, where the concavity of the graph changes.
  • The end-behavior of the graph depends on the leading coefficient's sign.
Cubic functions have applications in physics, economics, and various fields where modeling of relationships between variables is required.
Slope of tangents
The slope of a tangent to a curve at a given point is the derivative of the function at that point. The process of finding the slope involves:
  • Calculating the derivative, \( f'(x) \), of the function.
  • Substituting the specific points of interest into the derivative.
For our function, \( f(x) = -x^3 \), the derivative \( f'(x) = -3x^2 \) is used to find the slopes.
At \( x = -2 \), \( f'(-2) = -12 \). This means the slope of the tangent line at \( x = -2 \) is -12.
At \( x = 0 \), \( f'(0) = 0 \), indicating a horizontal tangent line.
At \( x = 1 \), \( f'(1) = -3 \). These values reflect how steep or flat the tangents are at the given points.

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Most popular questions from this chapter

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