Problem 24 - Calculus

Question

Draw the graph of \(y=\sin x+\sqrt{3} \cos x\). Its maximum value is \(y=\underline{\text { at } x=}\) The slope at that point is

Step-by-Step Solution

Answer

Maximum value is 2 at \(x=\frac{\pi}{6}\) with slope 0.

Step 1: Recognize the Trigonometric Identity

We start by expressing the given function in a standard trigonometric form. The function is \( y = \sin x + \sqrt{3} \cos x \). We notice this can be simplified using a trigonometric identity into a single sinusoidal function. This can be done by expressing it as \( R \sin(x + \alpha) \).

Step 2: Determine the Amplitude

We find \( R \) using the identity \( R = \sqrt{a^2 + b^2} \) where \( a = 1 \) (coefficient of \( \sin x \)) and \( b = \sqrt{3} \) (coefficient of \( \cos x \)). So, \( R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \).

Step 3: Find the Phase Angle α

The angle \( \alpha \) can be found using \( \tan \alpha = \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3} \). This implies \( \alpha = \frac{\pi}{3} \) radians or 60 degrees, because \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

Step 4: Rewrite the Function

The function can now be expressed as \( y = 2 \sin(x + \frac{\pi}{3}) \). This is a standard sinusoidal function with amplitude 2. Thus, the maximum value of \( y \) is 2.

Step 5: Determine the Value of x for Maximum y Value

For \( y = 2 \sin(x + \frac{\pi}{3}) \) to achieve its maximum value of 2, the argument of \( \sin \) must be \( \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer. This gives the equation \( x + \frac{\pi}{3} = \frac{\pi}{2} + 2k\pi \), which simplifies to \( x = \frac{\pi}{6} + 2k\pi \).

Step 6: Find the Slope at the Maximum Point

The slope of the function is given by the derivative. The derivative of \( y = 2 \sin(x + \frac{\pi}{3}) \) is \( \frac{dy}{dx} = 2 \cos(x + \frac{\pi}{3}) \). At the maximum point \( x = \frac{\pi}{6} + 2k\pi \), the derivative becomes \( 2 \cos\left(\frac{\pi}{6} + \frac{\pi}{3}\right) = 2 \cos\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0 \).

Key Concepts

Amplitude

Amplitude is a key idea when dealing with sinusoidal functions like sine and cosine waves. It identifies how far the wave extends above and below its center position, also known as its equilibrium. In the context of the function \(y = \sin x + \sqrt{3} \cos x\), we simplify it to the form \(R \sin(x + \alpha)\). Doing this lets us clearly find the amplitude, which is \( R \).
We calculate \( R \) with the formula \( R = \sqrt{a^2 + b^2} \).
This formula combines the coefficients \(a = 1\) and \(b = \sqrt{3}\). So, \( R = \sqrt{1^2 + (\sqrt{3})^2} = 2\).

• The amplitude indicates the peak height of the wave, or the largest value \(y\) reaches, which here is 2.
• A larger amplitude means a taller wave; a smaller amplitude means a shorter one.

This simple property helps predict maximum and minimum values in oscillating systems.

Phase Angle

Phase angle \(\alpha\) shifts the trigonometric function along the x-axis, affecting its starting position. In the function \(y = 2 \sin(x + \frac{\pi}{3})\), \(\alpha\) represents this shift. Identifying \(\alpha\) is essential for understanding where exactly the wave begins its cycle on the graph.
To find it, use \( \tan \alpha = \frac{b}{a} = \sqrt{3}\). This leads to \(\alpha = \frac{\pi}{3}\), or 60 degrees:

• A positive \(\alpha\) shifts the wave left
• A negative \(\alpha\) moves it right

The phase angle ensures comprehending how the sinusoidal graph aligns with specific values.

Derivative

The concept of the derivative is fundamental in understanding how functions change. It tells us the slope or rate of change of the function at any given point. For the function \(y = 2 \sin(x + \frac{\pi}{3})\), finding its derivative gives us insights into its behavior.
The derivative of \(y\) is \(\frac{dy}{dx} = 2 \cos(x + \frac{\pi}{3})\). This result stems from differentiating the sine function:

• The derivative at a specific point indicates how steep the graph is at that point.
• The value of \(\cos\) in the derived function unveils whether the slope at a location is rising, falling, or flat.

By understanding derivatives, we can analyze the function’s turning points and overall trend.

Slope

Slope is a measure of the steepness or incline of a line. In trigonometric functions represented by derivatives, it defines the incline or decline at a given point.
At the maximum value of the function \(y = 2 \sin(x + \frac{\pi}{3})\), we calculate the slope using its derivative: \(2 \cos(x + \frac{\pi}{3})\). At \(x = \frac{\pi}{6}\), the slope evaluates to \(2 \cdot 0 = 0\).

• A slope of 0 indicates that the curve is flat at that point; it's neither increasing nor decreasing.
• Slope analysis helps identify peaks, troughs, and inflection points in a graph.

Understanding slope makes it possible to predict how graphs move and where key features are located.