Problem 42 - Calculus: Early Transcendentals

Question

Describe the concavity of the functions below. $$ y=\sin x+\cos x $$

Step-by-Step Solution

Answer

The function is concave down on intervals \((0, \pi) + 2k\pi\) and concave up on intervals \((\pi, 2\pi) + 2k\pi\) for integer \( k \).

Step 1: Find the First Derivative

To determine the concavity, we first need the second derivative. Start by finding the first derivative of the function \( y = \sin x + \cos x \). The derivative \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \). Therefore, the first derivative is: \[ y' = \cos x - \sin x \]

Step 2: Find the Second Derivative

Next, differentiate the first derivative to find the second derivative. Using \( \frac{d}{dx}(\cos x) = -\sin x \) and \( \frac{d}{dx}(-\sin x) = -\cos x \), the second derivative is: \[ y'' = -\sin x - \cos x \]

Step 3: Analyze the Second Derivative

To analyze the concavity, inspect the sign of the second derivative. The function \( y'' = -\sin x - \cos x \) can be rewritten using a trigonometric identity: \( \sin x + \cos x = \sqrt{2} \sin(x + \frac{\pi}{4}) \). Hence, \( y'' = -\sqrt{2} \sin(x + \frac{\pi}{4}) \). The concavity depends on whether this is positive or negative.

Step 4: Determine Intervals of Concavity

The function \( -\sqrt{2} \sin(x + \frac{\pi}{4}) \) is negative when \( \sin(x + \frac{\pi}{4}) > 0 \) and positive when \( \sin(x + \frac{\pi}{4}) < 0 \). Hence, \( y \) is concave down when \( x + \frac{\pi}{4} \in (0, \pi) + 2k\pi \) and concave up when \( x + \frac{\pi}{4} \in (\pi, 2\pi) + 2k\pi \) for integer \( k \).

Key Concepts

First Derivative

When we think about derivatives, the first derivative \(y'\) informs us about the slope of the function at any point. Basically, it tells us how the function is changing—it represents the rate of change. For our function \(y = \sin x + \cos x\), the first derivative is \(\cos x - \sin x\). This derivative is computed by applying standard differentiation rules to each part of the function.

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).

By combining these, we can express how rapidly \(y\) changes as we move along the \(x\)-axis. If \(y'\) is positive, the function \(y\) is rising; if negative, \(y\) is falling. Understanding the first derivative is a crucial step in our exploration of a function's behavior.

Second Derivative

The second derivative \(y''\) digs deeper because it helps us to understand the concavity of a function. While the first derivative focuses on the slope, the second derivative tells us how this slope is changing. It paints a picture of whether the curve is bending upwards (concave up) or downwards (concave down).
For the function \(y = \sin x + \cos x\) that translates to the second derivative \(-\sin x - \cos x\).

• Using trigonometric identities, this can be rewritten as \(-\sqrt{2} \sin(x + \frac{\pi}{4})\).

The sign of this second derivative is what gives us insight into the concavity of the function.

• If \(y'' > 0\), the function is concave up, resembling a cup that holds water.
• If \(y'' < 0\), the function is concave down, like an umbrella.

In our step-by-step solution, we explored where \(y'' = -\sqrt{2} \sin(x + \frac{\pi}{4})\) is positive or negative to understand the intervals of concavity.

Trigonometric Functions

Trigonometric functions like \(\sin x\) and \(\cos x\) are fundamental in math, especially in analyzing wave-like behaviors or periodic phenomena. They cycle through values in repetitive patterns, often noted in their unit circle representation. In the function we're analyzing \(y = \sin x + \cos x\), these trigonometric components oscillate between \(-1\) and \(1\).

• The combined behavior of \(\sin x\) and \(\cos x\) helps define a unique wave.
• These waves repeat their values over fixed intervals, called periods.

When exploring derivatives, understanding these periodic changes is critical. Derivatives of trigonometric functions, like the ones we found earlier, \(\cos x\) and \(-\sin x\), also follow patterns but shifted in phase or amplitude. Such understanding is pivotal when it comes to examining changes and anticipating how the function behaves over intervals, which includes determining concavity through second derivatives. Understanding these basics provides a solid foundation to further grapple with trigonometric identities and calculus concepts.