Problem 18 - Essential Calculus Early Transcendentals

Question

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

Step-by-Step Solution

Answer

The absolute maximum is 1 at \(t = 0\), and the absolute minimum is -1 at \(t = -\pi\) and \(t = \pi\).

Step 1: Understand the Function

The function given is \(f(t) = \cos t\), which is a basic cosine function. This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.

Step 2: Determine the Interval

We need to sketch the function over the interval \([-\frac{3\pi}{2}, \frac{3\pi}{2}]\). This range includes more than one full period of the cosine function, which helps in identifying the repeating pattern and where the maxima and minima occur.

Step 3: Sketch the Function

Start by plotting key points of the cosine function. - The function \(f(t) = \cos t\) reaches its maximum value, 1, at \(t = 0, 2\pi, -2\pi\) and so on.- The minimum value, -1, occurs at odd multiples of \(\pi\), such as \(\pi\), \(-\pi\), \(3\pi\), \(-3\pi\), etc.- Zero points are at odd multiples of \(\frac{\pi}{2}\).Use these points to sketch the cosine curve within the specified interval.

Step 4: Identify Maximum and Minimum Values

From the sketch:- The maximum value of the function is \(1\), occurring at \(t = 0\).- The minimum value is \(-1\), which occurs at both \(t = -\pi\) and \(t = \pi\). These points mark the absolute maxima and minima due to symmetry and periodic nature of the cosine function.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos t \), is one of the fundamental trigonometric functions. It relates a real number \( t \), interpreted as an angle measured in radians, to a value between -1 and 1. The graph of \( \cos t \) is a continuous wave-like pattern that oscillates smoothly between these values. This characteristic makes it an excellent function for modeling periodic phenomena like sound waves and rotating wheels.

• The peak of the cosine wave is 1; this is the maximum value the function can output.
• The lowest point on the wave is -1, marking its minimum value.
• At \( t = 0 \), the cosine function starts at its highest point, 1, and completes a full cycle every \( 2\pi \) units.

These traits help students easily predict and plot the behavior of the function over any given interval.

Maximum and Minimum Values

Understanding the maximum and minimum values of the cosine function is key in sketching its graph. These values inform us about the peaks and troughs over specific intervals:

• The absolute maximum value is 1, which occurs at \( t = 0 \), and repeats every \( 2\pi \) along the x-axis.
• The absolute minimum value is -1 at odd multiples of \( \pi \), such as \( t = \pi \) and \( t = -\pi \).

In the interval \([ -\frac{3\pi}{2}, \frac{3\pi}{2}]\), the cosine function reaches its maximum at \( t = 0 \), and its minimum at \( t = \pi \) and \( t = -\pi \). Plotting these values with a smooth transition between them via inflection points like \( t = \pm \frac{\pi}{2} \), creates the recognizable wave pattern.

Periodic Functions

A periodic function is one that repeats its values at regular intervals, known as the period. The cosine function is a classic example with a standard period of \( 2\pi \). This means the pattern of peaks and valleys you see within one cycle will repeat exactly beyond that cycle.

• The periodic nature of \( \cos t \) means any angle \( t \) can be offset by \( 2\pi \), and the function will return to the same value. For example, \( \cos(t) = \cos(t + 2\pi) \).
• This property is useful in physics and engineering, where repetitive motion or vibration needs to be modeled.

Within the interval \([ -\frac{3\pi}{2}, \frac{3\pi}{2}]\), students can observe the repeating pattern of the cosine function as it completes more than one cycle. Recognizing periodicity helps in predicting future values of the function beyond visible points, which is vital for understanding waves and cycles in real-world settings.