Question

Use a CAS to do Problems 71 and \(72 .\) Draw the graphs of \(f(x)=x^{3}-4 x^{2}+3\) and its derivative \(f^{\prime}(x)\) on the interval \([-2,5]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)<0 ?\) (b) Where on this interval is \(f(x)\) decreasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Short answer

(a) \(f'(x) < 0\) on \((0, \frac{8}{3})\). (b) \(f(x)\) is decreasing on \((0, \frac{8}{3})\). (c) Functions decrease where their derivative is negative.

Step-by-step solution

Find the Derivative

The function given is \( f(x) = x^3 - 4x^2 + 3 \). To find the derivative \( f'(x) \), we apply the power rule. For \( x^3 \), the derivative is \( 3x^2 \), and for \( -4x^2 \), the derivative is \( -8x \). The derivative of the constant \( 3 \) is \( 0 \). Thus, \( f'(x) = 3x^2 - 8x \).

Graph the Functions

Using a Computer Algebra System (CAS), plot the original function \( f(x) = x^3 - 4x^2 + 3 \) and its derivative \( f'(x) = 3x^2 - 8x \) on the interval \([-2, 5]\). The graph will show the behavior of each function across this range.

Determine Where \(f'(x)

\(f'(x)\) is negative where the graph of the derivative lies below the x-axis. We need to find the intervals of \(x\) for which \(3x^2 - 8x < 0\). Solving \(3x^2 - 8x = 0\): Factor out an \(x\):\[ x(3x - 8) = 0 \] which gives \(x = 0\) and \(x = \frac{8}{3}\). Testing intervals around these roots, \(f'(x) < 0 \) between \(0 < x < \frac{8}{3}\).

Determine Where \(f(x)\) is Decreasing

By definition, \(f(x)\) is decreasing where \(f'(x) < 0\). Based on Step 3, \(f(x)\) is decreasing on the interval \(0 < x < \frac{8}{3}\).

Make a Conjecture and Experiment

The decreasing interval of \(f(x)\) matches where \(f'(x) < 0\). Thus, the conjecture is that a function \(g(x)\) is decreasing wherever its derivative \(g'(x)\) is negative. Experiment with various functions like \(g(x) = x^2 - 5x + 6\) or \(g(x) = \sin(x)\) over different intervals to support this conjecture. In each case, find the derivative, plot it, and verify if the descending sections of the original function align with where its derivative is negative.

Key concepts

Understanding Derivatives

Finding a derivative is like finding the rate at which a function changes. When you have a function like \( f(x) = x^3 - 4x^2 + 3 \), the derivative \( f'(x) \) tells you how steep the curve is at any point \( x \). To compute the derivative, we use rules such as the power rule. The power rule is simple: if you have a term like \( x^n \), the derivative is \( nx^{n-1} \). So, for our original function, the derivative becomes \( 3x^2 - 8x \). This derivative function helps us understand the behavior of the curve of \( f(x) \). When the derivative is positive, the function rises, and when it's negative, the function falls.

Decreasing Functions

Decreasing functions are where the curve slopes downwards. The key to identifying decreasing sections of a function is through the derivative. If the derivative \( f'(x) \) is less than zero, then \( f(x) \) is a decreasing function at that interval. For example, in our exercise, \( f(x) = x^3 - 4x^2 + 3 \) is decreasing on the interval \( 0 < x < \frac{8}{3} \), since in this interval the derivative \( 3x^2 - 8x \) is negative. Studying how the function's slope, given by its derivative, changes helps in understanding where the function's journey goes downhill. Practically, it means whatever quantity \( f(x) \) represents, it is reducing within that interval.

Leveraging Computer Algebra Systems (CAS)

A Computer Algebra System (CAS) is an invaluable tool for solving calculus problems. It allows us to compute derivatives, solve equations, and even plot graphs without manual errors. In our exercise, a CAS helps to find the derivative \( f'(x) = 3x^2 - 8x \) accurately and quickly. More than just finding derivatives, CAS can also aid in experimenting with different functions and problems to explore conjectures.By using a CAS, you can verify if a function is decreasing, where, and why. It's excellent for visualizing complex operations, making it much easier to understand how mathematical models behave over various intervals.

Graphing Functions

Graphing functions provides a visual representation of both the original function and its derivative. By plotting \( f(x) \) and \( f'(x) \) on the same axes, you gain a clearer understanding of how the function behaves over an interval. Visualizing these graphs highlights where the function increases or decreases.For \( f(x) = x^3 - 4x^2 + 3 \), the graph shows the nature of its changes on the interval \([-2,5]\). The plot illustrates the intersection points as well as the curves' peaks and valleys. By analyzing these visual clues, you can quickly identify where the derivative is positive or negative, providing a strong basis to make and test conjectures.