Problem 2 - Calculus

Question

Find \((\sin h) / h\) at \(h=0.5\) and 0.1 and .01 . Where does \((\sin h) / h\) go above \(.99 ?\)

Step-by-Step Solution

Answer

\((\sin h)/h > 0.99\) for \(h < 0.1\).

Step 1: Introduction to the Problem

We are tasked to calculate the value of the expression \(\frac{\sin h}{h}\) for \(h\) values of 0.5, 0.1, and 0.01. Additionally, we need to determine the range of \(h\) for which \(\frac{\sin h}{h} > 0.99\).

Step 1: Calculate \(\frac{\sin h}{h}\) for \(h = 0.5\)

First, substitute \(h = 0.5\) into the expression. Calculate \(\sin(0.5)\) using a calculator or table, which is approximately 0.4794. Then, divide by 0.5 to get \(\frac{\sin(0.5)}{0.5} \approx 0.9588\).

Step 2: Calculate \(\frac{\sin h}{h}\) for \(h = 0.1\)

Next, substitute \(h = 0.1\) into our expression. Calculate \(\sin(0.1)\) which is approximately 0.09983. Divide by 0.1 to obtain \(\frac{\sin(0.1)}{0.1} \approx 0.9983\).

Step 3: Calculate \(\frac{\sin h}{h}\) for \(h = 0.01\)

Now, substitute \(h = 0.01\) into the expression. Find \(\sin(0.01)\), which is approximately 0.0099998. Divide by 0.01 to get \(\frac{\sin(0.01)}{0.01} \approx 0.99998\).

Step 4: Analyze Where \(\frac{\sin h}{h} > 0.99\)

From the calculations, we see \(\frac{\sin(0.1)}{0.1} \approx 0.9983\) and \(\frac{\sin(0.01)}{0.01} \approx 0.99998\), both greater than 0.99. These indicate that \(\frac{\sin h}{h} > 0.99\) for \(h < 0.1\). We can infer that for small values of \(h\), particularly \(h < 0.1\), the expression stays above 0.99.

Key Concepts

Trigonometric functions

Trigonometric functions are crucial in mathematics, especially when working with angles and periodic phenomena. The sine function, represented as \( \sin(x) \), is one of these functions. It measures the vertical position of a point on the unit circle as it rotates around the origin.
In context of this problem, we calculate \( \sin(h) \) for specific values of \( h \). For small values of \( h \), \( \sin(h) \) is approximately equal to \( h \), which leads us to the interesting behavior observed in calculating \( \frac{\sin(h)}{h} \).
This approximation is helpful for understanding limits and behavior of functions as values approach zero.

• The sine function is odd, meaning \( \sin(-x) = -\sin(x) \).
• It repeats every \( 2\pi \), or 360 degrees.
• This periodic nature makes it a foundational block in various applications, from physics to engineering.

Calculus problem solving

Calculus problem solving often involves finding limits and understanding how functions behave as certain variables get infinitesimally small or large. In this exercise, we evaluate \( \frac{\sin(h)}{h} \) for small values of \( h \) like 0.5, 0.1, and 0.01.
When \( h \) is close to zero, this function approaches 1, showcasing an important property of calculus known as the limit.
By plugging in smaller values of \( h \):

• \( h=0.5 \) gives us \( \frac{\sin(0.5)}{0.5} \approx 0.9588 \).
• \( h=0.1 \) results in \( \frac{\sin(0.1)}{0.1} \approx 0.9983 \).
• \( h=0.01 \) yields \( \frac{\sin(0.01)}{0.01} \approx 0.99998 \).

All these calculations help develop an intuition for how functions behave, particularly near points of interest like zero within the concept of limits.

Mathematical analysis

Mathematical analysis involves delving into the behavior and properties of functions. By examining \( \frac{\sin(h)}{h} \), students learn about continuity and limits, two foundational concepts.
This function is particularly significant because its limit as \( h \) approaches zero is central in calculus and trigonometry.
For small \( h \), the function approximates one:

• This appears due to the closeness of \( \sin(h) \) and \( h \) for small angles, observable through the angle approximation \( \sin(h) \approx h \). This is why we find \( \frac{\sin(h)}{h} \to 1 \) as \( h \to 0 \).
• Such behavior indicates the function's tendency and is essential for understanding continuity.
• This concept applies when finding derivatives and integrals, making it a foundational principle in further mathematical problem-solving contexts.

The sensitivity of the function to changes in \( h \) shows how minor variations can significantly affect outcomes, teaching valuable lessons in analysis.