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Chapter 2: Problem 77

Let \(f(x)=(3 x+5)\left(2 x^{3}-5 x+4\right)\). Use a graphing utility to graph \(f(x)\) and \(f^{\prime}(x)\) on the same set of coordinate axes. Use TRACE and \(\mathbf{Z O O M}\) to find where \(f^{\prime}(x)=0\).

Short Answer

Expert verified
Use TRACE and ZOOM to find where \(f'(x) = 0\). This occurs where the graph of \(f'(x)\) intersects the x-axis.

Step by step solution

01

- Understand the function

The given function is \(f(x) = (3x + 5)(2x^3 - 5x + 4)\). This needs to be graphed along with its derivative.
02

- Find the derivative

To find \(f'(x)\), use the product rule. The product rule states \( (uv)' = u'v + uv' \). Here, \(u = 3x + 5\) and \(v = 2x^3 - 5x + 4\). Calculate \(u'\) and \(v'\). \(u' = 3\) and \(v' = 6x^2 - 5\)Thus, \(f'(x) = (3)(2x^3 - 5x + 4) + (3x + 5)(6x^2 - 5)\)
03

- Simplify the derivative

Simplify \(f'(x)\): \[ f'(x) = 6x^3 - 15x + 12 + 18x^3 + 30x^2 - 15x - 25 \] Factor and combine like terms: \[ f'(x) = 24x^3 + 30x^2 - 30x - 13 \]
04

- Graph the functions

Use a graphing utility to graph \(f(x)\) and \(f'(x)\) on the same coordinate axes. Input the equations \(f(x) = (3x + 5)(2x^3 - 5x + 4)\) and \(f'(x) = 24x^3 + 30x^2 - 30x - 13\) into the graphing utility.
05

- Use TRACE and ZOOM

Utilize the TRACE and ZOOM features of the graphing utility to examine the graphs and find the point(s) where \(f'(x) = 0\). This occurs at the x-values where the derivative graph intersects the x-axis.

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Key Concepts

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

product rule
To find the derivative of a function that is a product of two other functions, we use the product rule. The product rule states that if you have two functions, say \(u(x)\) and \(v(x)\), their derivative \((uv)'\) is given by:\
\((uv)' = u'v + uv'\).
In our exercise, the function is \(f(x) = (3x + 5)(2x^3 - 5x + 4)\).
Here, \(u = 3x + 5\) and \(v = 2x^3 - 5x + 4\).
By applying the product rule, we first find the derivatives of \(u\) and \(v\).
\(u' = 3\) and \(v' = 6x^2 - 5\).
Next, we multiply each function by the derivative of the other:
  • \(u'v = 3(2x^3 - 5x + 4)\)
  • \(uv' = (3x + 5)(6x^2 - 5)\)
Combining these results gives us: \(f'(x) = 3(2x^3 - 5x + 4) + (3x + 5)(6x^2 - 5)\).
derivative
The derivative of a function provides the rate at which the function is changing at any given point. In our example, we found the derivative of \(f(x) = (3x + 5)(2x^3 - 5x + 4)\) using the product rule.
After applying the product rule, we obtain the derivative function: \(f'(x) = 24x^3 + 30x^2 - 30x - 13\).
Derivatives are very useful in calculus because they help us understand how functions behave. They can show where functions are increasing or decreasing and where they have maximum or minimum values. These insights are essential for curve sketching and optimizing real-world problems.
graphing utility
A graphing utility is a tool, often a calculator or software, that helps in visualizing functions and their derivatives on coordinate planes. In our problem, we utilize the graphing utility to plot the original function \(f(x)\) and its derivative \(f'(x)\).
To do this, you input the equations \(f(x) = (3x + 5)(2x^3 - 5x + 4)\) and \(f'(x) = 24x^3 + 30x^2 - 30x - 13\) into the graphing utility. This allows you to see both graphs on the same set of coordinate axes, which aids in understanding how the function and its derivative relate.
trace and zoom
TRACE and ZOOM are powerful features of graphing utilities that help analyze specific points and behaviors on a graph. TRACE allows you to move along the graph to see the coordinates of different points.
ZOOM helps you to magnify a part of the graph to see details more clearly. For example, to find where \(f'(x) = 0\), you can use TRACE to follow the graph of the derivative until it crosses the x-axis, indicating these points. Utilizing these features helps find where the function has horizontal tangents, which correspond to where the derivative equals zero.
applied calculus
Applied calculus involves using calculus concepts to solve real-world problems. This often includes finding derivatives to understand rates of change, maximizing or minimizing functions, and modeling physical phenomena.
In our exercise, we use applied calculus by finding and interpreting the derivative of a function. By graphing the function and its derivative, we can analyze key behaviors such as where the function increases or decreases and find points where it may have extreme values. Understanding these concepts is essential in many fields, such as physics, engineering, economics, and biology.

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Most popular questions from this chapter

Find the derivative of the given function. $$ f(x)=6 x^{4}-7 x^{3}+2 x+\sqrt{2} $$

The curve \(y^{2}(2-x)=x^{3}\) is called a cissoid. a. Use a graphing utility to sketch the curve. b. Find an equation for the tangent line to the curve at all points where \(x=1\). c. What happens to the curve as \(x\) approaches 2 from the left? d. Does the curve have a tangent line at the origin? If so, what is its equation?

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