Problem 43 - Precalculus

Question

In each part, determine where \(f\) is differentiable. (a) \(f(x)=\sin x\) (b) \(f(x)=\cos x\) (c) \(f(x)=\tan x\) (d) \(f(x)=\cot x\) (e) \(f(x)=\sec x\) (f) \(f(x)=\csc x\) (g) \(f(x)=\frac{1}{1+\cos x}\) (h) \(f(x)=\frac{1}{\sin x \cos x}\) (i) \(f(x)=\frac{\cos x}{2-\sin x}\)

Step-by-Step Solution

Answer

(a) All; (b) All; (c) Except \(x=\frac{\pi}{2}+k\pi\); (d) Except \(x=k\pi\); (e) Except \(x=\frac{\pi}{2}+k\pi\); (f) Except \(x=k\pi\); (g) Except \(x=\pi+2k\pi\); (h) Except \(x=k\pi, \frac{\pi}{2}+k\pi\); (i) All.

Step 1: Analyze Differentiability of Sine Function

The sine function, given by \( f(x) = \sin x \), is a standard trigonometric function and is differentiable everywhere on the real line. There are no discontinuities or undefined points for \( \sin x \). Therefore, the function \( f(x) = \sin x \) is differentiable for all \( x \in \mathbb{R} \).

Step 2: Analyze Differentiability of Cosine Function

Similarly, the cosine function \( f(x) = \cos x \) is also a standard trigonometric function. It is continuous and differentiable throughout its domain, which covers the entire real line. Hence, \( f(x) = \cos x \) is differentiable for all \( x \in \mathbb{R} \).

Step 3: Analyze Differentiability of Tangent Function

The tangent function \( f(x) = \tan x \) is given by \( \frac{\sin x}{\cos x} \). The function is not defined where \( \cos x = 0 \). These points are when \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. At these points, tangents have vertical asymptotes resulting in non-differentiable points. Thus, \( f(x) = \tan x \) is differentiable for \( x \) values except \( x = \frac{\pi}{2} + k\pi \).

Step 4: Analyze Differentiability of Cotangent Function

The cotangent function \( f(x) = \cot x = \frac{\cos x}{\sin x} \) is undefined where \( \sin x = 0 \). These points occur at \( x = k\pi \). At these points, the function has discontinuities. Therefore, \( f(x) = \cot x \) is differentiable for all \( x eq k\pi \).

Step 5: Analyze Differentiability of Secant Function

The secant function \( f(x) = \sec x = \frac{1}{\cos x} \) is undefined where \( \cos x = 0 \). Such values occur at \( x = \frac{\pi}{2} + k\pi \). At these points, the function has vertical asymptotes and is not differentiable. Thus, \( f(x) = \sec x \) is differentiable for \( x \) such that \( \cos x eq 0 \).

Step 6: Analyze Differentiability of Cosecant Function

The cosecant function \( f(x) = \csc x = \frac{1}{\sin x} \) is undefined where \( \sin x = 0 \). This occurs at the points \( x = k\pi \). At these points, the function has discontinuities and vertical asymptotes. Therefore, \( f(x) = \csc x \) is differentiable for all \( x eq k\pi \).

Step 7: Analyze Differentiability of \\frac{1}{1+\\cos x}

The function \( f(x) = \frac{1}{1+\cos x} \) is undefined when \( 1 + \cos x = 0 \). This happens when \( \cos x = -1 \), which occurs at \( x = \pi + 2k\pi \). At these points, the function has discontinuities. Thus, \( \frac{1}{1+\cos x} \) is differentiable for \( x eq \pi + 2k\pi \).

Step 8: Analyze Differentiability of \\frac{1}{\\sin x \\cos x}

The function \( \frac{1}{\sin x \cos x} \) is undefined where either \( \sin x = 0 \) or \( \cos x = 0 \). This occurs at \( x = k\pi \) and \( x = \frac{\pi}{2} + k\pi \). At these points, the function is discontinuous. Thus, \( \frac{1}{\sin x \cos x} \) is differentiable for \( x eq k\pi \) and \( x eq \frac{\pi}{2} + k\pi \).

Step 9: Analyze Differentiability of \\frac{\\cos x}{2-\\sin x}

The function \( \frac{\cos x}{2-\sin x} \) is undefined where \( 2 - \sin x = 0 \), which happens when \( \sin x = 2 \). However, since the range of \( \sin x \) is between \(-1\) and \(1\), \( \sin x = 2 \) is never true. The function is well-defined and differentiable everywhere, as long as \( \sin x eq 2 \), which is always the case. Hence, \( \frac{\cos x}{2-\sin x} \) is differentiable for all \( x \in \mathbb{R} \).

Key Concepts

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are key components in mathematics. They often appear in various scenarios, from simple geometric calculations to complex waveforms.

Here's a quick overview of the basic trigonometric functions:

• Sine (\(\sin x\)): Gives the y-coordinate on the unit circle. It's a smooth and continuous function, differentiable everywhere on the real line.
• Cosine (\(\cos x\)): Gives the x-coordinate on the unit circle. Like sine, it's also continuous and differentiable on the entire real line.
• Tangent (\(\tan x\)): Defined as \(\frac{\sin x}{\cos x}\), it has periodic undefined points due to division by zero, where \(\cos x = 0\).

These functions are foundational in both calculus and trigonometry, and understanding their differentiability is crucial for solving related problems.

Undefined Points

Certain trigonometric functions become undefined at specific points. This occurs when the function's denominator equals zero, leading to division by zero.

For instance:

• Tangent (\(\tan x\)): Undefined where \(\cos x = 0\), or \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer.
• Cotangent (\(\cot x\)): Undefined where \(\sin x = 0\), at \(x = k\pi\).
• Secant (\(\sec x\)): Undefined where \(\cos x = 0\)
• Cosecant (\(\csc x\)): Undefined where \(\sin x = 0\)

When a function is undefined at a point, it often leads to discontinuities or asymptotes, making those points key in analyzing differentiability.

Vertical Asymptotes

Vertical asymptotes occur in functions where they approach infinity as they near certain points. For trigonometric functions, these are spots where the function becomes undefined, causing it to "spike" infinitely high or low.

Functions like:

• Tangent (\(\tan x\)) and secant (\(\sec x\)): Both have vertical asymptotes where \(\cos x = 0\)
• Cotangent (\(\cot x\)) and cosecant (\(\csc x\)): Both become undefined at points where \(\sin x = 0\)

These asymptotes are crucial for identifying areas where the function isn't smooth and hence not differentiable.

Continuous Functions

Continuity is a fundamental concept in calculus. A function is continuous if there are no breaks, jumps, or holes in its graph.

Continuous trigonometric functions include:

• Sine (\(\sin x\)): Always continuous, providing a smooth waveform.
• Cosine (\(\cos x\)): Also continuous, mirroring the properties of the sine function.

Discontinuities often occur in functions that experience vertical asymptotes, where division by zero leads to undefined points.

Continuous functions without such interruptions are fully differentiable across their domains.