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Chapter 3: Problem 8

Find formulas for \(d y\) and \(\Delta y\) at a general point \(x\). $$y=\sin x$$

Short Answer

Expert verified
\( dy = \cos x \cdot dx \) and \( \Delta y = \sin(x+\Delta x) - \sin x \).

Step by step solution

01

Understand the Problem

We are given the function \( y = \sin x \) and are asked to find the formulas for \( dy \) (the differential) and \( \Delta y \) (the change in \( y \)) at a general point \( x \).
02

Find the Derivative \( \frac{dy}{dx} \)

First, we need to find the derivative of \( y = \sin x \). The derivative is given by \( \frac{d}{dx} (\sin x) = \cos x \).
03

Determine the Differential \( dy \)

The differential \( dy \) is given by \( dy = \frac{dy}{dx} \cdot dx \). Since \( \frac{dy}{dx} = \cos x \), this implies \( dy = \cos x \cdot dx \).
04

Express the Change in \( \Delta y \)

The change in \( y \), denoted as \( \Delta y \), over a small interval \( \Delta x \) is given by the difference \( \sin(x + \Delta x) - \sin x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that relate a function with its derivatives. In simple terms, these equations involve rates at which things change, and they describe a wide range of phenomena in the real world, from physics to biology. In the context of the given exercise, we are interested in the basic idea of differentials.
  • The differential, represented as \( dy \), gives an infinitesimal (very tiny) change in the function \( y \) with respect to a small change in \( x \).
  • For a function \( y = f(x) \), the differential is defined as \( dy = \frac{dy}{dx} \cdot dx \), where \( \frac{dy}{dx} \) is the derivative of the function.
This tells us how much \( y \) changes when \( x \) changes by a very small amount. By understanding this, we can better grasp the behavior and properties of the function.
Derivative of Trigonometric Functions
When dealing with calculus, one of the most crucial skills is finding the derivative of various functions, including trigonometric ones such as sine and cosine. In this exercise, we deal with the sine function.
  • Given the function \( y = \sin x \), its derivative \( \frac{dy}{dx} \) is computed using standard derivative rules for trigonometric functions.
  • The derivative of \( \sin x \) is \( \cos x \).
This derivative tells us the slope of the tangent to the curve \( y = \sin x \) at any point \( x \). It's vital because it illustrates how the sine function changes as \( x \) varies. Understanding these derivatives allows us to analyze and predict the behavior of trigonometric functions over their domains.
Mathematical Notation
Mathematical notation is a structured language used to describe mathematical concepts precisely and unambiguously. In calculus, the notations for derivatives and differentials are particularly fundamental.
  • In the exercise, \( dy \) denotes the differential of \( y \), and \( \Delta y \) represents the change in the function's value over a small interval \( \Delta x \).
  • The notation \( \frac{dy}{dx} \) indicates the derivative of \( y \) with respect to \( x \), showing how \( y \) changes as \( x \) changes.
This concise system allows mathematicians and students to convey complex relationships and operations without ambiguity. By becoming fluent in this language, one can communicate and solve mathematical problems effectively.

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Most popular questions from this chapter

Find \(d y /\left.d x\right|_{x=-2},\) given that \(y=(5 / x)+1\)

find \(f^{\prime}(x)\) $$f(x)=\csc \left(x^{3}\right)$$

In Exercises find \(f^{\prime}(x)\). $$f(x)=\frac{\cot x}{1+\csc x}$$

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find \(f^{\prime}(x)\) $$f(x)=\tan ^{4}\left(x^{3}\right)$$
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