Problem 25 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=3 x^{3}+12 x^{2}+15 x $$

Step-by-Step Solution

Answer

The critical numbers are the solution to equation \(9x^{2} + 24x + 15 = 0\). The derivative takes positive values (function is increasing) or negative values (function is decreasing) between these critical numbers. The graph of the function confirms these results.

Step 1: Compute Derivative

First, compute the derivative of the given function. The derivative of \(y=3x^{3}+12x^{2}+15x\) is obtained using power rule of differentiation. The derivative \(y' = 9x^{2} + 24x + 15\).

Step 2: Find Critical points

The critical numbers of the function are obtained by setting the derivative equal to zero. Solve the equation \(9x^{2} + 24x + 15 = 0\). The solutions to this equation will be the critical numbers.

Step 3: Determine intervals

To find whether function is increasing or decreasing, the critical numbers are placed on a number line. The intervals are tested for sign of the derivative. If the derivative is positive in an interval, the function is increasing. If it is negative, it is decreasing.

Step 4: Graphing the function

With the computer or graphing calculator, plot the graph of the function \(y=3x^{3}+12x^{2}+15x\). The graph will confirm visually the intervals where the function is increasing or decreasing and the position of critical points.

Key Concepts

Derivative

A derivative represents the rate at which a function changes. When we talk about a derivative, we're looking for how much a function's output or "y-value" changes as the input or "x-value" changes. It's like the speed at which something is moving. In calculus, finding a derivative gives us a new function, which tells us the slope or steepness of our original function at any point.

Calculating the derivative is the first step when trying to find critical points. For the function \( y = 3x^3 + 12x^2 + 15x \), the process of differentiation helps us identify where this function's slope is zero or undefined. This is important because it helps find potential maxima or minima, or points where the slope changes direction.

• The derivative of \( y = 3x^3 + 12x^2 + 15x \) is calculated to be \( y' = 9x^2 + 24x + 15 \).
• This newly found function \( y' \) helps us examine how the original function behaves in terms of rising or falling trends.

Being able to compute derivatives efficiently is key as they offer us a deep understanding of the behavior of functions.

Power Rule

The power rule is a simple yet powerful tool used to find the derivative of polynomial functions. When you see a function like \( y = 3x^3 + 12x^2 + 15x \), applying the power rule makes differentiation straightforward.

The rule essentially states that if you have a term in the form of \( ax^n \), the derivative is \( n \cdot ax^{n-1} \). In plain terms, you multiply the exponent by the coefficient, and then decrease the exponent by one.

• For the term \( 3x^3 \), its derivative becomes \( 9x^2 \).
• Similarly, \( 12x^2 \) becomes \( 24x \).
• And, the derivative of \( 15x \) is \( 15 \).

Together, these derivatives contribute to forming \( y' = 9x^2 + 24x + 15 \).
The power rule is often one of the first differentiation techniques learned, and it's frequently used, making it a cornerstone concept in calculus.

Graphing Functions

Graphing functions offers a visual understanding of how a function behaves. It helps to illustrate concepts like critical points, and where a function increases or decreases. When dealing with the given function \( y = 3x^3 + 12x^2 + 15x \), sketching or using a graph utility will make theoretical findings clear.

With graphs, equations can transform from abstract ideas to concrete images that can be analyzed for:

• Critical points which are where the slope is zero, might show as peaks or troughs.
• Patterns of rising or falling, indicating intervals of increase or decrease.
• Curves steepness and inflection highlighting acceleration or changing direction.

Using graphing utilities, one can verify theoretical results calculated by hand, providing a confirmation or discovery of subtle patterns.
Remember, while algebraic methods of finding derivatives or critical points are vital, visualization through graphing adds another layer of comprehension.

Increasing and Decreasing Intervals

Once you're equipped with the derivative, analyzing the function's behavior concerning increasing or decreasing intervals is the next logical step. When we identify intervals on which a function is increasing or decreasing, we evaluate the sign of the derivative across these intervals.

Here's how we do it:

• We first find critical numbers by setting the derivative \( y' = 9x^2 + 24x + 15 \) equal to zero and solving for \( x \).
• These critical points partition the number line into intervals.
• By selecting test points within each interval and substituting back into \( y' \), you can determine:
  • If \( y'(x) > 0 \), the entire interval is increasing since the function's slope is upward.
  • If \( y'(x) < 0 \), the interval is decreasing because the slope is downward.

This technique categorizes the behavior of the function across its domain, crucial for understanding not only where peaks and valleys occur but when and how rapidly changes happen.