Problem 8 - Calculus Single Variable

Question

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ \frac{d \varphi}{d w}\left(\frac{\pi}{4}\right), \varphi(\omega)=2 \sin (\omega)+5 $$

Step-by-Step Solution

Answer

The derivative evaluated at \( \frac{\pi}{4} \) is \( \sqrt{2} \).

Step 1: Identify the Function and Its Components

The function given is \( \varphi(\omega) = 2\sin(\omega) + 5 \). This is a sum of two functions: a sinusoidal part \( 2\sin(\omega) \) and a constant part \( 5 \).

Step 2: Apply the Derivative Rules

To find the derivative of \( \varphi(\omega) \), we need to differentiate both terms separately. The derivative of \( 2\sin(\omega) \) is \( 2\cos(\omega) \) since the derivative of \( \sin(\omega) \) is \( \cos(\omega) \), and constants are multiplied as is. The derivative of a constant \( 5 \) is \( 0 \).

Step 3: Combine the Derivatives

Using the results from Step 2, the derivative of the total function \( \varphi' (\omega) \) is the sum of the derivatives of the individual terms: \( \varphi'(\omega) = 2\cos(\omega) + 0 = 2\cos(\omega) \).

Step 4: Evaluate the Derivative at \( \omega = \frac{\pi}{4} \)

Substitute \( \omega = \frac{\pi}{4} \) into the derivative \( \varphi'(\omega) = 2\cos(\omega) \) to find \( \varphi'(\frac{\pi}{4}) = 2\cos(\frac{\pi}{4}) \).

Step 5: Compute the Trigonometric Value

The value of \( \cos(\frac{\pi}{4}) \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \varphi'(\frac{\pi}{4}) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions play a crucial role in many areas of mathematics, especially in calculus. They are used to model periodic phenomena such as sound waves, light waves, and more. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions relate the angles of a triangle to ratios of its sides.

• **Sine Function**: \( \sin(\omega) \) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine Function**: \( \cos(\omega) \) provides the ratio of the adjacent side to the hypotenuse.
• **Tangent Function**: \( \tan(\omega) \) is the ratio of sine to cosine, or opposite to adjacent side.

Trigonometric functions repeat their values in regular intervals, making them useful for modeling cycles and oscillations. For example, the sine function varies smoothly and periodically between -1 and 1, giving it a wave-like pattern.

Derivative Rules

Derivative rules are formulas that help us find the rate at which a function is changing at any point. In calculus, differentiation is the process of calculating a derivative. For functions involving addition and subtraction, you can differentiate each term separately. Here are some basic rules:

• **Power Rule**: For \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
• **Constant Rule**: The derivative of a constant is zero. If \( f(x) = c \), a constant, then \( f'(x) = 0 \).
• **Sum and Difference Rule**: The derivative of a sum/difference is the sum/difference of the derivatives. For \( f(x) + g(x) \), the derivative is \( f'(x) + g'(x) \).
• **Trigonometric Derivatives**: \( \frac{d}{dx} \sin(x) = \cos(x) \) and \( \frac{d}{dx} \cos(x) = -\sin(x) \).

These rules make it easier to differentiate complex functions by breaking them into simpler parts.

Sinusoidal Functions

Sinusoidal functions are a type of trigonometric function that describe oscillating phenomena. They are represented by the sine and cosine functions. Characteristics of sinusoidal functions include amplitude, period, phase shift, and vertical shift.

• **Amplitude**: The maximum distance from the function's midline to its peak. For \( 2\sin(\omega) \), the amplitude is 2.
• **Period**: The length of one complete cycle of the wave. The natural period of \( \sin(\omega) \) and \( \cos(\omega) \) is \( 2\pi \).
• **Phase Shift**: This is the horizontal shift along the \( \omega \)-axis. It determines where the cycle starts.
• **Vertical Shift**: A constant added to the function shifts it up or down. \( 5 \) in \( 2\sin(\omega) + 5 \) is a vertical shift.

In the exercise, \( \varphi(\omega) = 2\sin(\omega) + 5 \) illustrates a shifted cosine wave by 5 units up and a stretch in the amplitude.

Constant Functions

Constant functions are the simplest types of functions where the output value does not change, regardless of the input. Their graph is a horizontal line.

• For any constant function \( f(x) = c \), all output values are equal to \( c \).
• **Derivative of Constants**: The derivative of a constant function is always zero because the rate of change is zero. If a function doesn’t change, then its slope is flat.
• **Role in Derivatives**: In \( \varphi(\omega) = 2\sin(\omega) + 5 \), the \( 5 \) is a constant. When you differentiate, it becomes 0 and does not affect the change rate of the rest of the function.

Understanding constant functions aids in identifying shifts in graphs and simplifies derivative calculations by removing unchanging parts.