Problem 30 - Calculus

Question

(a) Estimate \(f^{\prime}(0)\) if \(f(x)=\sin x,\) with \(x\) in degrees. (b) In Example 3 on page \(93,\) we found that the derivative of \(\sin x\) at \(x=0\) was \(1 .\) Why do we get a different result here? (This problem shows why radians are almost always used in calculus.)

Step-by-Step Solution

Answer

(a) The derivative is approximately 0.0174. (b) Degrees scale differently, affecting the derivative.

Step 1: Understanding the function

We are given the function \( f(x) = \sin x \), where \( x \) is in degrees. We need to estimate the derivative \( f'(0) \).

Step 2: Converting degrees to radians

When working with derivatives, it's standard practice to work in radians. Since the problem specifies degrees, we first need to convert the angle from degrees to radians using the formula \( x^{\text{(radians)}} = \frac{\pi}{180} \cdot x^{\text{(degrees)}} \).

Step 3: Approximate the derivative using limits

The derivative of a function at a point \( x = 0 \) is given by the limit:\[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} \] Applying the formula for \( f(x) \), we have:\[ f'(0) = \lim_{h \to 0} \frac{\sin(h) - \sin(0)}{h} = \lim_{h \to 0} \frac{\sin(h)}{h} \] We approximate \( \sin(h) \approx h \) when \( h \) is in radians close to 0, so the derivative is approximately \( 1 \) at this point.

Step 4: Calculating in degrees

For \( x \) in degrees, at \( x = 0 \), the derivative becomes:\[ f'(0) = \lim_{h \to 0} \frac{\sin(h \cdot \frac{\pi}{180})}{h \cdot \frac{\pi}{180}} \] This simplifies to \( f'(0) = \frac{\pi}{180} \cdot \cos(0) = \frac{\pi}{180} \), which is approximately \( 0.0174533 \).

Step 5: Explanation for different outcomes

When \( x \) is measured in radians, the derivative of \( \sin x \) at \( x = 0 \) is \( 1 \), because the unit circle is based on radians. However, when measured in degrees, the scaling is different, which changes the derivative. This exercise demonstrates why radians are preferred in calculus calculations, since they maintain the standard trigonometric limits.

Key Concepts

trigonometric_functions

Trigonometric functions are essential tools in calculus and mathematics in general. They relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions you will encounter are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions take an angle as input and produce a ratio as output.
In the context of calculus, trigonometric functions are often used to model periodic phenomena, such as waves. They have derivatives that are widely used in problems involving motion and oscillation.

• **Sine Function**: This function is often graphed as a smooth, wave-like curve. Its derivative is the cosine function, \( \cos x \), when angles are measured in radians.
• **Cosine Function**: Similarly, the cosine function looks like a wave but is offset from the sine curve. Its derivative is \( -\sin x \).
• **Tangent Function**: This function can be graphed with repetition every \( \pi \) radians (or 180 degrees), and it has distinctive asymptotes. Its derivative is \( \sec^2 x \).

Understanding how these functions behave and their derivatives is crucial in calculus because they allow us to calculate rates of change and understand the dynamics of oscillatory systems.

radians_vs_degrees

Radians and degrees are two units to measure angles. Although both serve the same purpose, they are used in different mathematical contexts according to their suitability. In calculus and higher mathematics, radians are typically preferred.
Degrees are common in everyday life, such as with clocks or in navigation, where a full circle is divided into 360 parts. This makes them easier to visualize for such purposes. However, they are less practical in calculus because they don't simplify mathematical relationships, particularly in trigonometry.

• **Radians**: A radian measures the angle using the radius of a circle. Specifically, it's the angle created by bending the radius length along the circumference. Because of this, the mathematical relationship between the radius and arc is direct and natural. One complete rotation around a circle is \( 2\pi \) radians.
• **Conversion**: To switch between the two, remember that \( 180^\circ = \pi \text{ radians}\). This implies that to convert degrees to radians, multiply by \( \frac{\pi}{180} \).

Using radians in calculus simplifies formulas and leads to cleaner computations, as seen in the derivative of trigonometric functions.

limit_definition_of_derivative

The definition of a derivative revolves around the concept of limits. Essentially, the derivative of a function at a particular point reflects how the function's output changes as its input changes infinitesimally.
To mathematically define the derivative at a point \( x \), use the limit:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]This expression gives the slope of the tangent line at the point \( x \), showing how output responds to small input changes. The essence of calculus derivatives lies in this limit process, binding algebraic functions to their geometric interpretations.

• **Practical Use**: The derivative tells us the rate of change of a function. Imagine a car on a road; the instantaneous speed of the car at any given moment (i.e., the speedometer reading) is the derivative of the car's position function.
• **Trigonometric Example**: Consider \( f(x) = \sin x \). To find \( f'(x) \) using limits, you must approach the function's change as the angle \( h \) gets infinitesimally small.

Being comfortable with the limit definition of derivatives helps you tackle complex problems and understand the minute changes in trigonometric or any other functions effectively.