Question

Horizontal tangents The graph of \(y=\cos x \cdot \ln \cos ^{2} x\) has seven horizontal tangent lines on the interval \([0,2 \pi] .\) Find the approximate \(x\) -coordinates of all points at which these tangent lines occur.

Short answer

There are five approximate x-coordinates within the interval [0, 2π] where horizontal tangent lines occur: 1. x ≈ 0 2. x ≈ 1.57 3. x ≈ 3.14 4. x ≈ 4.71 5. x ≈ 6.28

Step-by-step solution

Differentiate the function with respect to x

To find the \(x\)-coordinates of the horizontal tangents, we need to find the first derivative of the function with respect to \(x\). We are given \(y = \cos{x} \cdot \ln{(\cos^2 x)}\). Using the product rule for differentiation, we get: \(y' = (\cos{x})' \cdot \ln{(\cos^2 x)} + \cos{x} \cdot (\ln{(\cos^2 x)})'\) The derivative of \(\cos{x}\) is \(-\sin{x}\), and the derivative of \(\ln{(\cos^2 x)}\) can be found using the chain rule. So, \((\ln{(\cos^2 x)})' = \frac{1}{\cos^2 x} \cdot (\cos^2 x)'\) Since \(\cos^2{x} = (\cos{x})^2\), we can again use the chain rule to find its derivative: \((\cos^2 x)' = 2 (\cos{x})^1 (\cos{x})' = 2\cos{x}(-\sin{x}) = -2\cos{x}\sin{x}\) Now, we can plug these back into the expression for \(y'\) to get: \(y' = (-\sin{x}) \cdot \ln{(\cos^2 x)} + \cos{x} \cdot \frac{-2\cos{x}\sin{x}}{\cos^2 x}\) After simplifications, we get: \(y' = -\sin{x} \ln{(\cos^2 x)} - 2\sin{x}\)

Set the first derivative equal to zero and solve for x

To find the \(x\)-coordinates of the points where the tangent lines are horizontal, we need to set the first derivative equal to zero: \(0 = -\sin{x} \ln{(\cos^2 x)} - 2\sin{x}\) Now, we can factor out the common term, \(\sin{x}\), to get: \(0 = \sin{x}\left(- \ln{(\cos^2 x)} - 2\right)\) Therefore, either \(\sin{x} = 0\) or \(\ln{(\cos^2 x)} = 2\). We know that \(\sin{x} = 0\) at \(x = 0, \pi, 2\pi\). Now, let's solve the other equation: \(\ln{(\cos^2 x)} = 2\) \(\cos^2 x = e^2\) Since \(\cos^2 x\) is always between 0 and 1, there are no solutions to this equation. Thus the only valid points are when \(\sin{x} = 0\).

Check if the solutions are within the interval and approximate the values

We have found the following solutions for \(x\): - \(x = 0\) - \(x = \pi\) - \(x = 2\pi\) All of these solutions are within the interval \([0, 2\pi]\), so the approximate \(x\)-coordinates of the points where the tangent lines are horizontal are: - \(x \approx 0\) - \(x \approx 3.14\) - \(x \approx 6.28\) However, the problem states that there are seven horizontal tangent lines. After reviewing our work, we find that we missed counting the horizontal tangents that occur when \(\cos x=0\). In that case, we have \(\cos x=0\) at \(x = \pi/2\) and \(x= 3\pi/2\). This must happen multiple times within the interval \([0, 2\pi]\). These other values will be \(5\pi/2\) and \(-\pi/2,\) but these values are outside the given interval for this problem. Therefore, the five approximate \(x\)-coordinates of all points at which the horizontal tangent lines occur in the interval \([0, 2\pi]\) are: - \(x \approx 0\) - \(x \approx 1.57\) - \(x \approx 3.14\) - \(x \approx 4.71\) - \(x \approx 6.28\)

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. It's like determining the slope of a curve at a specific point, which helps us understand how the function behaves. In the context of the exercise, differentiation is used to find the horizontal tangents of the graph.

The function we're working with is a product of two parts:

• \(y = \cos{x} \cdot \ln{(\cos^2 x)}\)

To find the horizontal tangents, we need the derivative or the rate of change of this function. Specifically, when the derivative equals zero, the tangent is horizontal. This is because a zero derivative indicates a flat or stationary point on the curve. Differentiation is crucial here because it allows us to identify these specific points by setting the derivative to zero. Thus, finding the derivative of a function helps us spot when our function is climbing or descending, and when it's having those quiet, horizontal moments.

Product Rule

The product rule is a technique used in differentiation when you have to find the derivative of a product of two functions. In our exercise, the product rule is necessary because the original function, \(y = \cos{x} \cdot \ln{(\cos^2 x)}\), is a product of two different functions: the cosine function and the logarithmic function.

By applying the product rule, which states:

• \((fg)' = f'g + fg'\)

we differentiate both parts separately and then combine them. Here's what we do:

• First, differentiate \( \cos{x} \) to get \( -\sin{x} \).
• Next, differentiate \( \ln{(\cos^2 x)} \), which requires the use of another rule, known as the chain rule.

Then, we plug these results back into the product rule. Doing this properly helps us find the full expression for the derivative. This step is essential because it allows precise calculation of the points where the function has horizontal tangents.

Chain Rule

The chain rule is another differentiation tool, perfect for dealing with composite functions—functions nested within one another. It is critically important when the function you're differentiating involves a composition of two or more simpler functions.

In our problem, we encounter \( \ln{(\cos^2 x)} \). It looks simple, but beneath this, there is a function within a function:

• The outer function is the natural logarithm, \( \ln{x} \).
• The inner function is \( \cos^2 x \).

The chain rule helps us differentiate such compositions by taking the derivative of the outer function, evaluating it at the inner function, and then multiplying it by the derivative of the inner function.

The chain rule formula is:

• \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \)

Using the chain rule, we differentiate \( \cos^2 x \) to get \( -2\cos{x}\sin{x} \) and then divide by \( \cos^2 x \), giving us part of the derivative we use in the product rule. The chain rule, together with the product rule, arms us with the full derivative needed to find those all-important horizontal tangents.