Problem 81 - Calculus Single Variable

Question

Plot the given region. Plot the parabola with Cartesian equation \(y=2+x+x^{2}\) in the viewing window \([0,2] \times[0,8] .\) Add the plots of \(y=2 x\) and \(y=4 x\) to the window. Intuition suggests that there is a value of \(m\) between 2 and 4 such that the graph of \(y=m x\) has exactly one point of intersection with the parabola. Use a graphical method to approximate \(m\) to one decimal place. (In Chapter \(3,\) calculus will enable you to determine \(m\) exactly.

Step-by-Step Solution

Answer

The approximate value of \(m\) is 3.0.

Step 1: Plot the Parabola

Start by plotting the parabola given by the equation \(y = x^2 + x + 2\) within the specified window \([0, 2] \times [0, 8]\). This can be done by substituting values of \(x\) between 0 and 2 into the equation to get corresponding \(y\) values.

Step 2: Plot the Linear Functions

Next, plot the two linear functions: \(y = 2x\) and \(y = 4x\) in the same window as the parabola. These lines are straightforward to plot because they are straight lines through the origin with slopes of 2 and 4, respectively.

Step 3: Analyze Graph and Intuition

Visualize the graph to see where the straight lines intersect the parabola. Notice that both lines \(y = 2x\) and \(y = 4x\) intersect the parabola more than once.

Step 4: Approximate the Slope

To find the slope \(m\) where \(y = mx\) intersects the parabola exactly once, gradually change the slope from 2 towards 4. Plot lines with slopes such as \(y = 2.5x\), \(y = 3x\), and so on, observing where each line only touches the parabola at one point.

Step 5: Find the Value of m

The graphical method will show that the correct approximation of \(m\) is when the line appears to just touch (tangent to) the parabola, which can be iterated graphically to approximate \(m = 3.0\) exactly at one touching point.

Key Concepts

Parabola

A parabola is a curve that represents a quadratic function and has a distinctive "U" or "∩" shape. The standard form of a parabolic equation is typically given by \( y = ax^2 + bx + c \). In our example, the equation for the parabola is \( y = x^2 + x + 2 \). This equation is part of the family of quadratic functions where the parabola opens upwards due to the positive coefficient of \( x^2 \), which is 1 in this case.

• The vertex of the parabola is the turning point of the curve. For our equation, it is not explicitly calculated here, but it can be found using vertex formulas.
• The parabola is symmetric around a vertical line that passes through its vertex, known as the axis of symmetry.
• In the given Cartesian plane, the parabola is plotted from \( x = 0 \) to \( x = 2 \) which helps visualize its intersection with other functions.

Understanding parabolas is crucial because they appear in many real-world scenarios like projectile motion and optimization problems.

Linear Functions

Linear functions are the simplest type of functions and are graphed as straight lines. In their standard form, a linear function can be expressed as \( y = mx + c \), where \( m \) denotes the slope, and \( c \) is the y-intercept. In this exercise, we work with two linear equations: \( y = 2x \) and \( y = 4x \), where both pass through the origin meaning that their y-intercepts are zero.

• The slope \( m \) of a line indicates its steepness. A higher absolute value of \( m \) means a steeper line.
• The lines \( y = 2x \) and \( y = 4x \) show different steepness, with \( y = 4x \) rising faster than \( y = 2x \).
• Linear functions are easy to graph, simply take a couple of values for \( x \) and compute \( y \) using the function, then connect the dots with a ruler.

Understanding linear functions is essential as they represent constant rates of change in contexts like speed and direction.

Intersection Points

An intersection point of two graphs is where the graphs meet or cross each other in the Cartesian plane. To find intersection points, the equations of the graphs must be equal at that point. For our exercise, we are concerned with intersection points between the parabola \( y = x^2 + x + 2 \) and the lines \( y = 2x \), \( y = 4x \), and other lines drawn between these.

• Set the equations equal to solve for \( x \). For example, find where \( x^2 + x + 2 = 2x \).
• Substituting the \( x \) value back into either equation will give the corresponding \( y \) value.
• These coordinates \((x, y)\) are the exact points where the two graphs intersect.

Identifying intersection points is a common technique used in solving simultaneous equations and in determining solutions to systems of equations.

Graphical Method

A graphical method is a visual approach to finding solutions by plotting graphs, usually on a Cartesian coordinate system. This method is used to approximate or identify the precise behavior of two or more functions graphically.

• Start by plotting the graphs of interest using specific equations over a chosen range.
• For our problem, visualize both the parabola \( y = x^2 + x + 2 \) and various lines \( y = mx \).
• By adjusting \( m \), the goal is to find a line that just touches the parabola at exactly one point, which occurs when the line is tangent to the curve.

This trial-and-error approach using visuals is particularly effective for approximating the value of \( m \) without complex calculations. It helps to intuitively understand where curves intersect and explore the overall shape and relation of different functions on a graph. This graphical method provides a tangible insight into mathematical concepts, making abstract ideas more concrete.