Problem 35 - Precalculus Mathematics for Calculus

Question

(a) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$\begin{array}{l}f(x)=3 \cos x+1, g(x)=\cos x-1 \\ [-2 \pi, 2 \pi] \text { by }[-2.5,4.5] \end{array}$$

Step-by-Step Solution

Answer

Intersection points are \((\pi, -2)\) and \((-\pi, -2)\).

Step 1: Graph the Functions

First, graph the functions \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \). Set the viewing rectangle to \([-2\pi, 2\pi]\) on the x-axis and \([-2.5, 4.5]\) on the y-axis. Look for the points where the graphs of \( f \) and \( g \) intersect.

Step 2: Identify Intersection Points Graphically

Use the graph from Step 1 to find the approximate x-values of intersection points by observing where the graphs of \( f(x) \) and \( g(x) \) overlap. The intersection points found graphically need to be rounded to two decimal places.

Step 3: Set Equations Equal for Algebraic Intersection

To find the intersection points algebraically, set \( f(x) \) equal to \( g(x) \). This gives us the equation: \[ 3 \cos x + 1 = \cos x - 1 \].

Step 4: Solve the Algebraic Equation

Solve the equation \( 3 \cos x + 1 = \cos x - 1 \). Subtract \( \cos x \) from both sides to get:\[ 2 \cos x + 1 = -1 \].Next, subtract 1 from both sides:\[ 2 \cos x = -2 \].Divide by 2:\[ \cos x = -1 \].

Step 5: Find x-values with \( \cos x = -1 \)

Analyze where \( \cos x = -1 \) within the interval \([-2\pi, 2\pi]\). This occurs at \( x = \pi \) and \( x = -\pi \).

Step 6: Final Verification

Verify these intersection points by substituting \( x = \pi \) and \( x = -\pi \) back into the original functions and confirm they result in the same y-values. Both functions yield \( y = -2 \) at these x-values, confirming the points of intersection are correct.

Key Concepts

Intersection Points

When analyzing graphs of functions, one task is to identify where they intersect, known as intersection points. Understanding intersection points helps you determine where two functions have the same value.
To find these points for functions like \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \), you must observe where their graphs overlap. First, plot both functions over the set interval, in this case \([-2\pi, 2\pi]\) on the x-axis and \([-2.5, 4.5]\) on the y-axis.
When observing the intersections graphically, it’s essential to remember:

• Properly setting the viewing window allows for accurate identification of overlaps.
• Points need to be precisely read from the graph, rounded to two decimal places for clarity.

This graphical approach gives an intuitive understanding but rounding means slight inaccuracies.To ensure precision, algebraic solutions often complement this graphical method.

Cosine Function

Trigonometric functions, like the cosine function, are periodic and fundamental in analyzing cycles. The cosine function, represented as \( \cos x \), repeats every \( 2\pi \) radians (or 360 degrees).This periodicity means certain values recur at consistent intervals over its domain.
For the cosine function, critical values include:

• \( \cos 0 = 1 \), aligning with the maximum point.
• \( \cos \frac{\pi}{2} = 0 \), marking a zero-crossing.
• \( \cos \pi = -1 \), indicating the minimum point.
• \( \cos \frac{3\pi}{2} = 0 \), another zero-crossing.

When considering functions such as \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \), keep in mind the vertical shifts and stretches. This transforms how the cosine wave appears and where intersection points might occur within the given viewing rectangle.Such understanding helps break down and predict the behavior of modified trigonometric functions.

Algebraic Solutions

Finding algebraic solutions to term functions is a rigorous way of confirming intersections calculated graphically. For the functions \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \), setting them equal helps work out exact intersection locations on the x-axis analytically.
Here is the process to find these algebraic solutions:

• Begin by setting the equations equal: \( 3 \cos x + 1 = \cos x - 1 \).
• Simplify by subtracting \( \cos x \) from both sides: \( 2 \cos x + 1 = -1 \).
• Further simplify by subtracting 1: \( 2 \cos x = -2 \).
• Divide by 2: \( \cos x = -1 \).

The solutions for \( x \) where \( \cos x = -1 \) within the interval \([-2\pi, 2\pi]\) are crucial:

• They occur at \( x = \pi \) and \( x = -\pi \).

Substituting these back into the original functions as a verification step ensures their consistency, solidifying their status as accurate intersection points.