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Chapter 3: Problem 70

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$

Short Answer

Expert verified
Function: \(f(x) = \cos(x) + 1\), Point: \(a = \pi\).

Step by step solution

01

Recognize the Limit Definition of the Derivative

The limit given, \( \lim_{h\rightarrow 0} \frac{\cos(\pi+h)+1}{h}\), represents the definition of the derivative of a function \( f(x) \) at a point \( a \). The general formula for the derivative using limits is \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
02

Identify the Function and Point

From the limit expression \( \frac{\cos(\pi+h)+1}{h} \), we can see that the term \( \cos(\pi + h) \) suggests a function involving cosine. Set \( f(x) = \cos(x) + 1 \). Therefore, the point \( a \) where the derivative is evaluated must be where \( x \) in \( \cos(x) + 1 \) is replaced by \( \pi \).
03

Verify the Function and Point

For \( f(x) = \cos(x) + 1 \), let's verify: \( f(\pi + h) = \cos(\pi + h) + 1 \). The derivative expression becomes \( \lim_{h\rightarrow 0} \frac{f(\pi + h) - f(\pi)}{h} \). We know \( f(\pi) = \cos(\pi) + 1 = -1 + 1 = 0 \), therefore, \( \frac{\cos(\pi + h) + 1 - 0}{h} \) matches the original limit.

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Key Concepts

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

Derivative
Derivatives are vital in calculus because they quantify how a function changes at any given point. In simple terms, a derivative measures the slope of a function's graph at a particular point. This slope indicates the rate of change, which can be used for understanding growth patterns, motion, and various other time-dependent processes in mathematics and science.
Proficiency in derivatives is crucial for solving complex calculus problems and modeling real-world situations. The derivative of a function is often denoted by \( f'(x) \) or \( \frac{dy}{dx} \), and its computation can vary based on the context.
Limit Definition
The limit definition of a derivative is a foundational concept in calculus. It provides a way to precisely calculate the derivative of a function. This is achieved by analyzing how the function behaves as it approaches a particular point.
The formal definition is: \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]Here鈥檚 how it works:
  • \( h \) is a small increment approaching zero.
  • \( f(a+h) - f(a) \) represents the change in the function鈥檚 value over the interval \( h \).
  • The entire expression gives the instantaneous rate of change, or the derivative at \( x = a \).
Understanding this definition is key to understanding other derivative concepts and solving derivative problems effectively.
Trigonometric Functions
Trigonometric functions like sine and cosine frequently appear in calculus due to their repetitive and wave-like characteristics. These functions are often used to model oscillations and other cyclical phenomena.
The cosine function is defined as the horizontal coordinate of a point on the unit circle. For example, \( \cos(\pi + h) \) indicates the cosine of an angle formed by adding a small increment \( h \) to the angle \( \pi \).
Importantly, trigonometric functions have powerful derivative properties:
  • The derivative of \( \cos(x) \) is \( -\sin(x) \).
  • These derivatives are useful for resolving rates of change in cyclic systems.
Mastering these functions and their derivatives is crucial for tackling complex calculus tasks and understanding periodic behaviors in physics and engineering.
Problem Solving Steps
Approaching calculus problems effectively requires a systematic method of solving. Detailed problem-solving steps guide you to break down complex tasks into manageable parts. Here鈥檚 a general approach:
  1. **Identify the Problem**: Understand what is being asked. For example, you need to determine which function a given limit represents.
  2. **Apply the Limit Definition**: Use the definition to rewrite expressions. Determine how the function behaves as the change \( h \) approaches zero.
  3. **Substitute and Simplify**: Plug in known values or expressions to simplify the limit equation further. Simplifying is key to revealing recognizable patterns.
  4. **Verify Results**: Double-check calculations to ensure accuracy. Verification with the original problem can confirm if the conclusions make sense.
Developing strong problem-solving skills in calculus allows you to approach any exercise systematically, ensuring you reach accurate answers confidently.

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Most popular questions from this chapter

IT] For the following position functions \(y=s(t), \quad\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\mathrm{i}) h=0.1, \quad\) (ii) \(h=0.01\) (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and c. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=\frac{1}{3} t+5 $$

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