Problem 14 - Calculus Early Transcendentals

Question

Make a careful sketch of the graph of \(f\) and below it sketch the graph of \(f^{\prime}\) in the same manner as in Exercises \(4-11\) . Can you guess a formula for \(f^{\prime}(x)\) from its graph? \(f(x)=\sin x\)

Step-by-Step Solution

Answer

The derivative of \( f(x) = \sin x \) is \( f'(x) = \cos x \).

Step 1: Understand the Function

The given function is \( f(x) = \sin x \). This is a well-known trigonometric function that represents a wave-like pattern oscillating between -1 and 1.

Step 2: Determine the Characteristics of the Graph

The graph of \( f(x) = \sin x \) is a continuous wave starting at the origin (0,0). It has a period of \( 2\pi \), an amplitude of 1, and it crosses the x-axis at multiples of \( \pi \). Key points to note are at \( x = 0, \pi, 2\pi \), etc.,

Step 3: Sketch the Graph of \( f(x) \)

Draw the x and y axes. Mark key points where the sine function reaches its maximum (\( \frac{\pi}{2} \)) and minimum (\( \frac{3\pi}{2} \)). Attach these with smooth curves to illustrate a wave that completes one cycle every \( 2\pi \).

Step 4: Identify the Derivative \( f'(x) \)

The derivative of \( f(x) = \sin x \) is known to be \( f'(x) = \cos x \). This derivative indicates the slope of the tangent to the sine wave at any given point.

Step 5: Sketch the Graph of \( f'(x) \)

On the same x-axis beneath the \( f(x) \), sketch the graph of \( f'(x) = \cos x \). Begin by identifying key points at \( x = 0, \pi, \) and \( 2\pi \), where \( \cos x \) is 1, -1, and 1, respectively. Connect these points with smooth curves to complete the wave.

Step 6: Analyze the Derivative Graph

Observe that the graph of \( f'(x) \) is a cosine wave, phase-shifted by \( \frac{\pi}{2} \) to the left compared to \( f(x) \). It suggests that the derivative of \( \sin x \) perfectly aligns with the characteristics of \( \cos x \).

Key Concepts

Trigonometric Functions

Trigonometric functions are some of the most fundamental building blocks in calculus and mathematics as a whole. These functions, such as sine, cosine, and tangent, describe the relationships between angles and lengths in right triangles. They are periodic, which means they repeat their values at regular intervals. This is a key feature in modeling waves and oscillations, making them critical in areas ranging from engineering to physics.

For the function given in the exercise,

• \( \sin x \) oscillates between -1 and 1.
• Its graph is sinusoidal, with a period of \( 2\pi \).
• At \( x = 0, \pi, \text{and} \ 2\pi \), the sine function crosses the x-axis.

Starting at the origin, the sine wave peaks at \( \frac{\pi}{2} \) and reaches a trough at \( \frac{3\pi}{2} \). Understanding this wave pattern is crucial when transitioning to concepts like derivatives.

Graph of Derivatives

The graph of a derivative gives us insightful information about the rate of change of a function's output with respect to its input. Here, for \( f(x) = \sin x \), the derivative \( f'(x) = \cos x \) shows how the slope of the sine wave varies.

The sketch of the derivative graph, \( f'(x) = \cos x \), is also a wave:

• Its frequency is the same as \( \sin x \), which is one full oscillation per \( 2\pi \).
• However, \( \cos x \) is a phase-shifted version of \( \sin x \), moving \( \frac{\pi}{2} \) to the left.
• Key points on the cosine wave include \( x = 0 \), where \( f'(x) \) reaches its maximum value of 1.

By sketching \( f'(x) \) beneath \( f(x) \), you can observe these patterns readily. This helps you understand not just where \( f(x) \) increases or decreases, but also the rate of change at which these variations occur. This transition from one graph to the other visually reinforces the idea that a derivative represents instantaneous rate of change.

Trigonometric Derivatives

Trigonometric derivatives are pivotal in calculus because they describe how trigonometric functions change. For \( f(x) = \sin x \), the known derivative \( f'(x) = \cos x \) emerges naturally from observing the behavior of \( \sin x \) and its slopes.

Here's what to remember about trigonometric derivatives:

• \( \frac{d}{dx} (\sin x) = \cos x \)
• The derivative explains how "fast" the sine function is moving up or down at any point.
• At points where \( \sin x \) is maximized or minimized, the derivative \( \cos x \) hits zero, showing no change in the slope (these are the top and bottom of the wave crests).

Understanding these derivatives means recognizing how sine and cosine are interrelated through differentiation. This relation is pivotal when handling various problems in calculus, particularly those involving motion, waves, and oscillations.