Problem 2 - Contemporary Precalculus

Question

Find the coordinates of the highest or lowest point on the part of the graph of the equation in the given viewing window. Only the range of \(x\) -coordinates for the window are given \(_{i}\) you must choose an appropriate range of \(y\) -coordinates. $$\begin{aligned} &y=2 x^{6}+3 x^{5}+3 x^{3}-2 x^{2} ; \quad \text { lowest point when }\\\ &-3 \leq x \leq 3 \end{aligned}$$

Step-by-Step Solution

Answer

Answer: The lowest point within the given viewing window for the function is approximately at \((-2.5, 0.268)\).

Step 1: Find the first derivative of the function

To find the critical points, we first need to find the first derivative of the function \(y\). The function is given by \(y=2x^6+3x^5+3x^3-2x^2\). The first derivative \(y'\) is: $$y' = \frac{d}{dx}(2x^6+3x^5+3x^3-2x^2)$$ Applying the power rule for differentiation, we get: $$y' = 12x^5 + 15x^4 + 9x^2 - 4x$$

Step 2: Identify critical points

Next, we need to find the critical points of the function by setting \(y'\) to zero: $$12x^5 + 15x^4 + 9x^2 - 4x = 0$$ This polynomial equation may not be easy to factor or solve analytically. So, we use numerical methods (such as calculator, or graphing software) to find the roots of the equation within the given range of x-coordinates (-3 to 3). In this case, we find that there are two critical points in this range, approximately at \(x \approx -2.5\) and \(x \approx 0.37\).

Step 3: Determine appropriate y-coordinate range

Given the range for x-coordinates, we can now plug in those values into the original function to determine the corresponding range of y-coordinates. Remember that we are looking for the lowest point, so we are mainly interested in the y-coordinate range's lower bound. For the critical points \(x \approx -2.5\) and \(x \approx 0.37\), we find: $$y(-2.5) \approx 0.268$$ $$y(0.37) \approx 0.414$$ As we are looking for the lowest point, we can use the lowest y-coordinate we found (\(\approx 0.268\)) as our lower y-bound and choose an appropriate upper y-bound (for example, 10) to ensure that the lowest point of the graph will be within our viewing window. Thus, the y-coordinate range for the viewing window is \(\approx 0.268 \leq y \leq 10\).

Step 4: Identify the lowest point within the viewing window

Following the critical points analysis, we have two candidate lowest points in the viewing window: 1. When \(x \approx -2.5\), \(y(-2.5) \approx 0.268\) 2. When \(x \approx 0.37\), \(y(0.37) \approx 0.414\) Comparing these points, we can see that the lowest point within the given viewing window for the x-coordinates \(-3 \leq x \leq 3\) and our chosen y-coordinate range \(\approx 0.268 \leq y \leq 10\) is point 1 which is approximately at \((-2.5, 0.268)\).

Key Concepts

First Derivative

In precalculus and calculus, the first derivative of a function is a central concept that measures the rate at which the function's value changes. When you plot the graph of a function, the first derivative at any point gives you the slope of the tangent line to the graph at that point. It represents how fast the y-value (output) of the function is changing in relation to the x-value (input).

To obtain the first derivative from a function, we use differentiation rules such as the power rule for differentiation, product rule, quotient rule, or chain rule, depending on the complexity of the function. Understanding and calculating the first derivative is crucial for finding critical points that could represent the highest or lowest points on the graph, which are often points of interest in many scientific and mathematical applications.

Power Rule for Differentiation

The power rule for differentiation is a fundamental tool used to take the derivative of functions of the form \( x^n \), where \( n \) is any real number. According to the power rule, the derivative of \( x^n \) is \( nx^{n-1} \).

Application of the Power Rule

For example, if we have the function \( y=x^3 \), the derivative using the power rule would be \( y'=3x^2 \). This rule significantly simplifies the process of finding derivatives, especially when dealing with polynomials. In our exercise, the power rule is applied to each term of the given polynomial function to find the first derivative, which we then use to locate critical points.

Numerical Methods

Numerical methods refer to techniques used to find approximate solutions to mathematical problems that might be too complex for analytical solutions. In the context of finding critical points, when the derivative of a function is set to zero to find potential maxima or minima, we may end up with an equation that's challenging to solve by hand.

Methods such as the Newton-Raphson algorithm, bisection method, or graphical solutions using software can be employed to find approximate roots of these equations. Students often use graphing calculators or computer programs to find numerical solutions that give the x-values of the critical points, which can then be evaluated to find the function’s lowest or highest points, as seen in our textbook example.

Graph Analysis

Graph analysis is the visual examination of a function's graph to understand its behavior. By analyzing the graph of a function, students can identify regions where the function increases or decreases, locations of local maxima or minima, inflection points, and asymptotic behavior.

When the derivatives are obtained, plotting them can also reveal much about the function, such as the slope at different points along the curve. In our example, after finding the first derivative and determining the critical points, graph analysis allows us to verify the function's behavior around these points and confirm which of them corresponds to the lowest point. By thoroughly understanding graph analysis, students can visually comprehend the calculus concepts that sometimes remain abstract when only handled analytically.