Problem 54 - Contemporary Precalculus

Question

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$f(t)=\cos t \text { from } t=\pi / 4 \text { to } t=\pi / 3$$

Step-by-Step Solution

Answer

Answer: The average rate of change of the function \(f(t)=\cos t\) over the interval \(\frac{\pi}{4} \le t \le \frac{\pi}{3}\) is \(\frac{6(1-\sqrt{2})}{\pi}\).

Step 1: Evaluate the function at the endpoints of the interval

To find the function's values at the given endpoints, substitute \(\pi/4\) and \(\pi/3\) into the function and calculate the cosine. $$f(\frac{\pi}{4})= \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$f(\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}$$

Step 2: Find the difference in function values

Now, we should find the difference between the values of the function at the given endpoints of the interval. $$f(\frac{\pi}{3}) - f(\frac{\pi}{4}) = \frac{1}{2} - \frac{\sqrt{2}}{2}$$

Step 3: Find the length of the interval

We need to find the length of the interval, which is the difference between the given values of \(t\). $$\Delta t = \frac{\pi}{3} - \frac{\pi}{4} = \frac{\pi}{12}$$

Step 4: Calculate the average rate of change

Now that we have the difference in function values and the length of the interval, we can compute the average rate of change by dividing the difference in function values by the length of the interval. $$\text{Average Rate of Change} = \frac{f(\frac{\pi}{3}) - f(\frac{\pi}{4})}{\Delta t} = \frac{\frac{1}{2} - \frac{\sqrt{2}}{2}}{\frac{\pi}{12}}$$

Step 5: Simplify the result

Finally, simplify the expression for the average rate of change. $$\text{Average Rate of Change} = \frac{\frac{1-\sqrt{2}}{2}}{\frac{\pi}{12}} = \frac{1-\sqrt{2}}{2} \cdot \frac{12}{\pi} = \frac{6(1-\sqrt{2})}{\pi}$$ The average rate of change of the function \(f(t)=\cos t\) over the interval \(t=\frac{\pi}{4}\) to \(t=\frac{\pi}{3}\) is \(\frac{6(1-\sqrt{2})}{\pi}\).

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided into two main parts: differential calculus and integral calculus. Differential calculus concerns instantaneous rates of change and the slopes of curves, while integral calculus concerns the accumulation of quantities and the areas under and between curves.

In the context of the exercise, we deal with differential calculus, specifically with the concept of the average rate of change, which is a measure of how much a function changes over a specific interval. We calculate this by finding the difference in function values at the endpoints of the interval and dividing it by the length of that interval.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in mathematics, with applications ranging from geometry to periodic phenomena in engineering and science. The cosine function, which appears in our exercise, describes the ratio of the adjacent side to the hypotenuse of a right-angled triangle. More broadly, it is a periodic function that varies smoothly between -1 and 1, often used to model oscillatory behavior.

Function Intervals

Intervals in mathematics define a range of values along the domain of a function. A function's interval can be all the real numbers or just a portion of them, and it is generally expressed in terms of its endpoints.

In calculus, the concept of an interval is crucial when discussing the average rate of change because it defines the 'window' over which we are considering the function's behavior. For instance, the interval from \(\pi/4\) to \(\pi/3\) specifies that we are looking at the function only in that specific part of its domain.

Cosine Function

The cosine function is one of the basic trigonometric functions characterized by its wavelike graph and symmetry across the y-axis. It is denoted as \(\cos(x)\) where x is the angle in radians. The cosine of an angle in a right triangle is the length of the adjacent side divided by the length of the hypotenuse.

In our exercise, we work with the cosine function in the context of radians and intervals, which is a common application in calculus. For example, \(\cos(\pi/4)\) and \(\cos(\pi/3)\) evaluate to \(\sqrt{2}/2\) and \(1/2\), respectively, based on known values from the unit circle.