Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Linear approximations provide particularly good approximations near points of inflection. Using a graphing calculator, investigate this behavior in Problems 34-36. Graph \(y=\cos x\) and its linear approximation \(L(x)=-x+\pi / 2\) at \(x=\pi / 2 .\)

Step-by-Step Solution

Answer

Both graphs intersect at \( x = \frac{\pi}{2} \), where the linear approximation closely follows \( y = \cos x \).

Step 1: Understanding the Function and Linear Approximation

The function we are investigating is \( y = \cos x \), and the linear approximation provided is \( L(x) = -x + \frac{\pi}{2} \). This approximation is taken at the point \( x = \frac{\pi}{2} \), which is the point of inflection for the function \( \cos x \). At this point, the derivative of \( \cos x \) is zero, indicating a change in concavity.

Step 2: Using the Graphing Calculator

To visualize both \( \cos x \) and its linear approximation, graph the function \( y = \cos x \) along with the straight line \( L(x) = -x + \frac{\pi}{2} \) using a graphing calculator. Set the viewing window around \( x = \frac{\pi}{2} \) (for example, from \( 0 \) to \( \pi \)) to clearly see the behavior near the point of inflection.

Step 3: Examine the Graph

Once both graphs are plotted, observe the point where \( x = \frac{\pi}{2} \). The curve of \( y = \cos x \) should intersect the line \( L(x) = -x + \frac{\pi}{2} \) at this point. Notice how closely \( L(x) \) approximates \( \cos x \) around this region.

Step 4: Evaluate the Approximation Quality

The linear approximation \( L(x) = -x + \frac{\pi}{2} \) is expected to be a good approximation near \( x = \frac{\pi}{2} \). In this region, \( L(x) \) almost overlaps with \( \cos x \), showing the accuracy of linear approximations near points of inflection. Confirm this by examining the numerical values of both functions when \( x \) is near \( \frac{\pi}{2} \).

Key Concepts

Point of Inflection

A point of inflection is where a curve changes its concavity. In simpler terms, it's where the curve transitions from being concave upward (shaped like a smile) to concave downward (shaped like a frown), or vice versa. For the cosine function, there is a key point of inflection at \( x = \frac{\pi}{2} \). At this specific point for the function \( y = \cos x \), the derivative is zero. This indicates that the slope of the tangent line is neither increasing nor decreasing at this juncture. Instead, it is flat, signaling a shift in the curve's direction. Using this property, linear approximations at points of inflection often provide very accurate estimates of the function's value. The closeness of the linear approximation to the actual curve around these points highlights the mathematical significance of the point of inflection.

Cosine Function

The cosine function, represented as \( y = \cos x \), is one of the fundamental trigonometric functions. It describes how the cosine of an angle changes as that angle sweeps around from 0 to \( 2\pi \). This function is periodic, meaning it repeats its values in a regular pattern. The graph of the cosine function undulates between the values of -1 and 1, forming a wave-like shape. Key features of the graph include maximum points at intervals of \( 2\pi \), minimum points at odd multiples of \( \pi \), and points of inflection within each cycle.In relation to linear approximations, at points like \( x = \frac{\pi}{2} \), the function experiences a change in concavity. Utilizing this susceptibility to alteration in direction, linear functions can effectively approximate the behavior of the cosine function near these points.

Graphing Calculator

A graphing calculator is a practical tool that allows you to visualize mathematical functions and their transformations. This handheld device facilitates a deeper understanding of complex graphs by providing interactive plotting and analysis capabilities. For exploring the properties of a function such as \( y = \cos x \), a graphing calculator can illustrate differentiation between the original function and its linear approximation. By adjusting the scale and focus to magnify the area around the point of approximation, in this case \( x = \frac{\pi}{2} \), the calculator becomes essential for observing that fine detail.While using a graphing calculator:

• Input the function \( y = \cos x \) and the linear approximation \( L(x) = -x + \frac{\pi}{2} \).
• Set the window settings to frame the desired portion of the graph, ensuring the nuances near the point of inflection are visible.
• Use the tool to investigate other potential points of inflection or analyze how adjustments in the function parameters affect the approximation.

Overall, graphing calculators are an indispensable aid in the study of functions, assisting in making abstract concepts tangible and easier to grasp.