Problem 61 - Calculus of a Single Variable

Question

Consider the function \(f(x)=3 \cos ^{2}\left(\frac{\pi x}{2}\right)\) (a) Use a graphing utility to graph \(f\) and \(f^{\prime}\) (b) Is \(f\) a continuous function? Is \(f^{\prime}\) a continuous function? (c) Does Rolle's Theorem apply on the interval \([-1,1] ?\) Does it apply on the interval \([1,2] ?\) Explain. (d) Evaluate, if possible, \(\lim _{x \rightarrow 3^{-}} f^{\prime}(x)\) and \(\lim _{x \rightarrow 3^{+}} f^{\prime}(x)\)

Step-by-Step Solution

Answer

The derivative of the given function is \(f'(x) = -\frac{3\pi}{2}\cos\left(\frac{\pi x}{2}\right)\sin\left(\frac{\pi x}{2}\right)\). Both the function and its derivative are continuous. Rolle's Theorem applies on the interval \([-1, 1]\) but does not apply on the interval \([1, 2]\). The left and right hand limits of \(f'(x)\) as \(x \rightarrow 3\) are 0.

Step 1: Find the Derivative

The derivative of the function \(f(x)=3 \cos ^{2}\left(\frac{\pi x}{2}\right)\) can be found using the chain rule: \(\frac{d}{dx}\cos^2(u)=2\cos(u)(-\sin(u))\cdot u'\). For our function, \(u=\frac{\pi x}{2}\), so \(u'=\frac{\pi}{2}\). Hence, \(f'(x) = -\frac{3\pi}{2}\cos\left(\frac{\pi x}{2}\right)\sin\left(\frac{\pi x}{2}\right)\).

Step 2: Determine Continuity of the Function and its Derivative

Since both the function and its derivative are combinations of basic trigonometric functions, and trigonometric functions are continuous for all real numbers, both \(f(x)\) and \(f'(x)\) are continuous for all real numbers.

Step 3: Apply Rolle's Theorem

Rolle's Theorem is applicable to a function on an interval \(I=[a, b]\) if the function is continuous on \(I\) and differentiable on \(I')=(a, b)\), and if \(f(a)=f(b)\). On range \([-1,1]\), \(f(-1)=3\cos^2(-\pi/2)=0\) and \(f(1)=3\cos^2(\pi/2)=0\), so Rolle's Theorem applies. On interval \([1,2]\), \(f(1)=3\cos^2(\pi/2)=0\) and \(f(2)=3\cos^2(\pi)=3\), so Rolle's Theorem does not apply because \(f(1)\neq f(2)\).

Step 4: Evaluate the Limits

To evaluate, \(\lim _{x \rightarrow 3^{-}} f^{\prime}(x)\) and \(\lim _{x \rightarrow 3^{+}} f^{\prime}(x)\), substitute \(x = 3\) into \(f'(x)\). As \(f'(x)\) is continuous, its limits from left and right will exist and be equal to the function's value at this point. But do not forget that the arguments of trigonometric functions are periodic: \(f'(3) = -\frac{3\pi}{2}\cos\left(\frac{3\pi}{2}\right)\sin\left(\frac{3\pi}{2}\right)=0\), thus both limits equal 0.

Key Concepts

Derivative

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. Essentially, a derivative represents the slope of the tangent line to the graph of the function at any given point. For the function we are examining, we use the chain rule to find its derivative.
The chain rule is used when we have a composite function, which is a function of another function. In our case, the function is given as \(f(x) = 3 \cos^2\left(\frac{\pi x}{2}\right)\). Thus, we consider \(u = \frac{\pi x}{2}\) to aid in the differentiation step.

• The derivative of \(\cos^2(u)\) consists of applying the outer derivative \(2\cos(u)(-\sin(u))\) and then differentiating \(u\) itself.
• This results in \(f'(x) = -\frac{3\pi}{2}\cos\left(\frac{\pi x}{2}\right)\sin\left(\frac{\pi x}{2}\right)\).

The negative sign comes from the derivative of cosine, which is negative sine. The multiplication by \(\frac{\pi}{2}\) comes from the derivative of \(\frac{\pi x}{2}\).
The derivative tells us how the function behaves at each point, which can help us analyze the behavior like increase or decrease tendencies of the function.

Continuity

Continuity of a function means that small changes in the input result in small changes in the output. More formally, a function \(f\) is continuous at a point \(c\) if the limit of the function as it approaches \(c\) from any direction equals the function's value at \(c\).
In simpler terms, you can draw the function without lifting your pencil from the paper.
Given the function \(f(x) = 3 \cos^2\left(\frac{\pi x}{2}\right)\) and its derivative, both are combinations of trigonometric functions. Trigonometric functions such as cosine and sine are continuous for all real numbers. Thus:

• The function \(f(x)\) is continuous because cosine squared is continuous.
• The derivative \(f'(x)\) is also continuous because it, too, is constructed from sine and cosine, both of which are inherently continuous.

Continuity is a key condition for many important theorems in calculus, including Rolle's Theorem, which requires a function to be continuous on a given interval.

Trigonometric functions

Trigonometric functions are a set of functions that include sine, cosine, and tangent, among others. These functions stem from ratios of sides of a right triangle and have numerous applications in periodic phenomena.
In the context of our example, the function uses the cosine function:

• The cosine function, often written as \(\cos\), is a smooth, periodic function oscillating between -1 and 1.
• When we see \(\cos^2\), this implies squaring the cosine function, which makes the output non-negative, as any negative cosine value squared becomes positive.

The angles for trigonometric functions are typically measured in radians. In our specific function, we have \(\frac{\pi x}{2}\), causing the periodicity to depend on \(x\).
These trigonometric identities allow us to understand transformations and compositions in calculus to explore function patterns and predict future values.

Limits

Limits are a fundamental concept that help us understand the behavior of a function as the input approaches a particular value. A limit of a function \(f(x)\) as \(x\) approaches \(c\) is a way of determining what value \(f(x)\) approaches as \(x\) gets arbitrarily close to \(c\).
For the exercise, we evaluated the limits of the derivative function \(f'(x)\) as \(x\) approaches 3 from both the left \(3^-\) and the right \(3^+\):

• The limit from the left \(\lim_{x \rightarrow 3^-} f'(x)\) indicates analyzing the function's trend just before reaching \(x = 3\).
• The limit from the right \(\lim_{x \rightarrow 3^+} f'(x)\) focuses on the trend just after passing \(x = 3\).
• Because \(f(x)\) is based on continuous trigonometric functions, both these limits equate to the function's actual value at \(x = 3\), which is 0 in this case.

Evaluating limits can help us determine values at points of discontinuity or investigate the instantaneous rate of change.