Problem 35 - Calculus Early Transcendentals

Question

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$

Step-by-Step Solution

Answer

\( f(x) = \cos(x) \), \( a = \pi \).

Step 1: Identify the General Form

The given limit \( \lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h} \) is in the form of \( \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \), which represents the derivative of a function \( f(x) \) at a point \( a \).

Step 2: Determine f(x) and a

Given \( f(a+h) = \cos(\pi+h) \) and \( f(a) = -1 \), identify \( f(x) \) and \( a \). Noting that \( \cos(\pi + x) = -\cos(x) \), we determine that \( f(x) = \cos(x) \) and \( a = \pi \) (where \( \cos(\pi) = -1 \)).

Step 3: Verify the Derivative

The derivative of \( f(x) = \cos(x) \) is \( f'(x) = -\sin(x) \). Now check that when \( x = a = \pi \), \( f'(\pi) = -\sin(\pi) = 0 \). The limit simplifies to zero, confirming that we are correct.

Key Concepts

Limits

A limit in mathematics is a way of describing the behavior of a function as its input approaches a particular value. Limits are crucial in defining derivatives, integrals, and continuity. When you see something like \( \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \), it represents the derivative of a function at a specific point. This formula essentially measures the instantaneous rate of change—how much a function's output changes as its input changes infinitesimally. Understanding limits is foundational to calculus. Limits answer how close we can get to a point without actually reaching it, allowing us to handle indeterminate forms or describe functions that approach infinite or undefined values.

Trigonometric Functions

Trigonometric functions are mathematical functions related to the angles of triangles. They are commonly used in various areas of mathematics, particularly in calculus and physics.- **Sine (\( \sin \))**: Measures the y-coordinate of a point on a unit circle.- **Cosine (\( \cos \))**: Measures the x-coordinate of that same point.These functions are periodic, meaning they repeat values in a regular pattern. This periodicity makes them incredibly useful for modeling oscillations, such as sound waves or the motion of pendulums.Trigonometric functions also have unique derivatives which are pivotal in the study of calculus. For example, the derivative of \( \cos(x) \) is \( -\sin(x) \). These derivative relationships help calculate changes in angles or describe the behavior of waves.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any point. It is one of the core concepts in calculus that allows us to understand and describe motion, growth, and decay. When we differentiate a function such as \( f(x) = \cos(x) \), we identify how the function changes with respect to \( x \). The derivative of \( \cos(x) \) is \( -\sin(x) \), meaning that at any point \( x \), the rate of change of the cosine function is determined by the negative sine of that point. Differentiation has practical applications in numerous fields, including physics for expressing velocity and acceleration, in biology for modeling population changes, and in economics for calculating marginal costs.

Cosine Function

The cosine function is one of the primary trigonometric functions and is defined for all angles. It is typically represented as \( \cos(x) \). A point of interest for cosine is its period of \( 2\pi \), meaning it completes one full cycle every \( 2\pi \) radians.- **Properties**: - Peaks at 1, troughs at -1 - It is even, because \( \cos(-x) = \cos(x) \)The cosine function is closely tied to the circle, specifically the x-coordinate of a point on the unit circle. Since \( \cos(x) \) frequently appears in calculus, knowing its derivative is important. The derivative of \( \cos(x) \) is \( -\sin(x) \), which helps when solving problems involving motion or waveforms.In the exercise, the function \( f(x) = \cos(x) \) was evaluated at a point \( a = \pi \), where the value is -1. Understanding these basic properties of cosine aids in solving calculus problems efficiently.