Problem 54 - Thomas Calculus

Question

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}. \) $$ f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi $$

Step-by-Step Solution

Answer

Local max at \( x = \frac{3\pi}{4} \), local min at \( x = \frac{7\pi}{4} \).

Step 1: Find the Derivative

To find the local extrema, we first need to find the derivative of the function. The given function is \( f(x) = \sin x - \cos x \). Taking the derivative, we have: \[ f'(x) = \cos x + \sin x \] This derivative will help us find the critical points.

Step 2: Solve for Critical Points

To find the critical points, we set the derivative \( f'(x) = 0 \): \[ \cos x + \sin x = 0 \] This can be rewritten as \[ \cos x = -\sin x \].Dividing both sides by \( \sin x \), we get: \[ \tan x = -1 \] The solutions to \( \tan x = -1 \) in \( 0 \leq x \leq 2\pi \) are \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \), since the tangent function has a period of \( \pi \).

Step 3: Evaluate at Interval Endpoints

The given interval is \( 0 \leq x \leq 2\pi \). We need to evaluate \( f(x) \) at the endpoints \( x = 0 \) and \( x = 2\pi \), as well as the critical points \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \):- At \( x = 0 \): \( f(0) = \sin 0 - \cos 0 = -1 \)- At \( x = \frac{3\pi}{4} \): \( f\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \) - At \( x = \frac{7\pi}{4} \): \( f\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2} \) - At \( x = 2\pi \): \( f(2\pi) = \sin 2\pi - \cos 2\pi = -1 \)

Step 4: Identify Local Extrema

Based on evaluations:- Local maximum: \( f(x) = \sqrt{2} \) at \( x = \frac{3\pi}{4} \)- Local minimum: \( f(x) = -\sqrt{2} \) at \( x = \frac{7\pi}{4} \)Endpoints \( x = 0 \) and \( x = 2\pi \) are not extrema since they have equal function values that are not lesser or greater than those at critical points.

Step 5: Graph Function and Derivative

When graphing \( f(x) = \sin x - \cos x \) and \( f'(x) = \cos x + \sin x \), observe:- \( f'(x) > 0 \) implies the function is increasing, which occurs between \( x = 0 \) to \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \) to \( x = 2\pi \).- \( f'(x) < 0 \) implies the function is decreasing, which occurs between \( x = \frac{3\pi}{4} \) to \( x = \frac{7\pi}{4} \). Graphs confirm this behavior, with peaks occurring at points where \( f'(x) \) changes sign.

Key Concepts

Local Extrema

In calculus, local extrema are points on a function's graph where the function reaches a local maximum or minimum. A local maximum is a point where the function's value is higher than all nearby points, while a local minimum is where it is lower. When solving for local extrema, we find the derivative and determine where it equals zero to find critical points. Then we analyze these points to see if they are indeed local maxima or minima.

• Local Maximum: Highest value of function in the immediate area.
• Local Minimum: Lowest value of function in the immediate area.

In the example function \( f(x) = \sin x - \cos x \), local extrema involve evaluating the derivative and checking function values at critical points. For this specific function, a local maximum occurs at \( x = \frac{3\pi}{4} \) with a value of \( \sqrt{2} \), and a local minimum occurs at \( x = \frac{7\pi}{4} \), having a value of \( -\sqrt{2} \). These are the crucial spots where the behavior of \( f(x) \) as a sinusoidal function changes locally.

Critical Points

Critical points are where the derivative of a function is zero or undefined. These points indicate where a function could change from increasing to decreasing or vice versa. They help identify possible local extrema.

To find the critical points of \( f(x) = \sin x - \cos x \) in the given range, \( 0 \leq x \leq 2\pi \), we calculated the derivative \( f'(x) = \cos x + \sin x \) and set it to zero: \( \cos x + \sin x = 0 \). Simplifying leads to \( \tan x = -1 \), giving us critical points at \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \).

These critical points are crucial because they mark potential locations where the graph of the function might have local maxima or minima.

Derivative

The derivative of a function is a key concept in calculus used to determine the function's rate of change. It is symbolized by \( f'(x) \) if \( f(x) \) is the original function.

• Indicates slope of the tangent line to the function at any point.
• Used to find critical points where potential extrema occur.

For the function \( f(x) = \sin x - \cos x \), the derivative \( f'(x) = \cos x + \sin x \) helps us explore the function's increasing or decreasing behavior.

By finding where \( f'(x) = 0 \) or changes sign, we can determine intervals of increase or decrease. This provides insight into where the graph has peaks (local maxima) and troughs (local minima). Thus, derivatives play a pivotal role in sketching the behavior of graphs.

Tangent Function

The tangent function is one part of trigonometry and also indicates the slope of a line touching just one point on a graph, also known as the tangent line. In calculus, the tangent line is derived using the derivative.

When we take \( f'(x) \) and evaluate critical points where \( f'(x) = 0 \), the slope of the tangent line is zero, indicating a flat (horizontal) tangent that usually occurs at local maxima or minima.

• Tangent function: \( \tan x \)
• Slope of tangent line at critical points reveals changes in direction of graph.

In the given function, evaluating \( \tan x = -1 \) allowed us to find critical points. Each critical point potentially indicates a spot for an extremum due to a change in the slope generated by the tangent line flatness at those spots.

Graphing Functions

Graphing a function and its derivative provides visual insight into the behavior of the function across an interval.

• Graphs show increasing/decreasing trends, peaks, and valleys.
• The points where the derivative crosses the x-axis indicate where peaks and valleys occur.

In this exercise, graphing \( f(x) = \sin x - \cos x \) and its derivative \( f'(x) = \cos x + \sin x \) together reveals key aspects:

- Positive \( f'(x) \) suggests the function is rising.- Negative \( f'(x) \) suggests it is falling.
By observing these changes, we identify where the function changes direction, marking local maxima at \( \frac{3\pi}{4} \) and minima at \( \frac{7\pi}{4} \). Graphs make these changes also visually evident, helping to confirm calculated analytic solutions.