Problem 32 - Calculus Early Transcendentals

Question

a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither. $$f(x)=\sin x \cos x \text { on }[0,2 \pi]$$

Step-by-Step Solution

Answer

Question: Determine the critical points of the function \(f(x) = \sin{x}\cos{x}\) within the interval \([0, 2\pi]\) and identify whether they are local maximums or minimums. Answer: The critical points are as follows: - A local maximum at \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\). - A local minimum at \(x=\frac{3\pi}{4}\) and \(x=\frac{7\pi}{4}\).

Step 1: Take the derivative of the function

To find critical points, we need to find the first derivative of the function. The given function is: $$f(x) = \sin{x} \cos{x}$$ Now, we will apply the product rule to obtain its derivative. The product rule states that if \(f(x) = u(x)v(x)\), then \(f'(x) = u'(x)v(x) + u(x)v'(x)\). Using this rule, we have: $$u(x) = \sin x \Rightarrow u'(x) = \cos x$$ $$v(x) = \cos x \Rightarrow v'(x) = -\sin x$$ Now, we can calculate the derivative: $$f'(x) = \cos{x}\cos{x}-\sin{x}\sin{x} = \cos^2{x}-\sin^2{x}$$

Step 2: Find the critical points

To find the critical points, we need to set \(f'(x)\) to zero and solve for \(x\) within the given interval of \([0,2\pi]\). So, we have: $$\cos^2{x}-\sin^2{x}=0$$ Using the Pythagorean trigonometric identity, \(\sin^2{x}+\cos^2{x}=1\), we can rewrite this as: $$\cos^2{x}=(\sin^2{x})$$ This is true when $$\sin{x} = \cos{x}$$ or $$\sin{x} = -\cos{x}$$ Solving the first equation for angle \(x\) in the interval \([0,2\pi]\), $$\sin{x} = \cos{x}$$ $$\tan{x} = 1$$ The values of \(x\) that satisfy this equation and lie within the given interval are \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\). Now we solve the second equation within the given interval: $$\sin{x} = -\cos{x}$$ $$\tan{x} = -1$$ The values of \(x\) that satisfy this equation and lie within the given interval are \(x=\frac{3\pi}{4}\) and \(x=\frac{7\pi}{4}\). Therefore, the critical points are \(x=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).

Step 3: Use a graphing utility

Now, we will use a graphing utility to determine the nature of the critical points found in step 2. Upon observing the graph, we can determine the nature of the critical points: - At \(x=\frac{\pi}{4}\), we have a local maximum. - At \(x=\frac{3\pi}{4}\), we have a local minimum. - At \(x=\frac{5\pi}{4}\), we have a local maximum. - At \(x=\frac{7\pi}{4}\), we have a local minimum. In conclusion, the critical points are as follows: - A local maximum at \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\). - A local minimum at \(x=\frac{3\pi}{4}\) and \(x=\frac{7\pi}{4}\).

Key Concepts

Derivative

In calculus, the derivative of a function describes how the function's output values change as the input changes. It's like finding the slope of the tangent line to the function at any point, giving us insight into the function's behavior. To find a derivative, we generally differentiate with respect to a variable, often denoted as \(x\). This process involves applying differentiation rules to determine the exact formula representing how the output varies. Derivatives are central to finding critical points, as these are points where the derivative equals zero or is undefined. At these points, a function’s slope is zero, indicating potential maxima, minima, or inflection points. Recognizing these points is crucial for understanding the graph's sketch and behavior.

Product Rule

The product rule is a differentiation technique used when we need to find the derivative of a product of two functions. Remember, the derivative of a product is not simply the product of the derivatives. Instead, we have an elegant rule that states: Given two functions \(u(x)\) and \(v(x)\), the derivative of their product is:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \] In our example function \(f(x) = \sin{x} \cos{x}\), both sine and cosine are differentiable functions. We identify them as \(u(x) = \sin{x}\) and \(v(x) = \cos{x}\), then apply the product rule:

• The derivative of \(u(x)\), \(u'(x) = \cos{x}\).
• The derivative of \(v(x)\), \(v'(x) = -\sin{x}\).

By substituting these into the product rule formula, we derive \(f'(x) = \cos^2{x} - \sin^2{x}\). Thus, the product rule helps us find the derivative when dealing with products of functions.

Trigonometric Identities

Trigonometric identities are fundamental equations in trigonometry that hold true for any angle. They simplify expressions and are essential in calculus for solving equations involving trigonometric functions. One well-known identity applied here is the Pythagorean identity:\[ \sin^2{x} + \cos^2{x} = 1 \] In the context of our critical points exercise, we used identities to transform and simplify expressions. We initially set the derivative equal to zero: \(f'(x) = \cos^2{x} - \sin^2{x} = 0\).Using the identity \(\sin^2{x} + \cos^2{x} = 1\), we rewrite the equation involving \(\cos^2{x}\) and \(\sin^2{x}\) as \(\cos^2{x} = \sin^2{x}\). This implies that either \(\sin{x} = \cos{x}\) or \(\sin{x} = -\cos{x}\), guiding us to the solutions within the specified interval.   Understanding these identities allows us to solve trig equations systematically, uncovering critical points with precision.

Graphing Utility

A graphing utility is a tool, often a calculator or software, that visually represents functions for deeper analysis. By plotting \(f(x) = \sin{x} \cos{x}\) over its domain \([0, 2\pi]\), we can visually verify the nature of critical points found through calculus.After determining critical points, inputting these points into a graph gives a crystal clear picture of whether each represents a local maximum, minimum, or neither. The crests and troughs of the wave-like plot of \(\sin{x} \cos{x}\) effectively illustrate this:

• At \(x = \frac{\pi}{4}\) and \(x = \frac{5\pi}{4}\), the peaks in the graph indicate local maxima.
• At \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), the dips confirm the presence of local minima.

Thus, graphing utilities don't just crunch numbers but allow an intuitive understanding of function dynamics. They confirm analytical work with a visual check, playing a vital role in calculus studies.