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Chapter 2: Problem 33

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$$

Short Answer

Expert verified
The function is \(f(x) = 2^x\) and the point \(a = 5\).

Step by step solution

01

Identify the Derivative Form

We start by recognizing the format of the limit. The expression \(\lim _{x \rightarrow a} \frac{f(x) - f(a)}{x-a}\) is the definition of the derivative of a function \(f(x)\) at \(x = a\).
02

Determine Function and Point a

Given the expression \(\lim _{x \rightarrow 5} \frac{2^{x} - 32}{x - 5}\), we compare it to the derivative definition. Here, \(f(x) = 2^x\) and the point \(a\) is 5 because \(2^5 = 32\).
03

Verify the Expression

Substitute \(f(a) = 2^5 = 32\) back into the limit expression to confirm: \(\lim _{x \rightarrow 5} \frac{2^{x} - 32}{x-5}\). This matches the derivative form at \(x = a = 5\).
04

State the Function and the Point

The function \(f(x)\) is \(2^x\) and the point \(a\) where the derivative is being taken is 5.

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Key Concepts

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

Limit Definition of Derivative
The limit definition of the derivative is a fundamental concept in calculus. It defines the derivative as the limit of the difference quotient as the variable approaches a specific value. This means the derivative is the slope of the tangent line to the curve of a function at a specific point. The formula used is: \[ \lim_{x \to a} \frac{f(x) - f(a)}{x-a} \] In this expression:
  • \( f(x) \) represents the function
  • \( f(a) \) is the function value at a specific point \( a \)
  • \( x \) approaches \( a \)
By using the limit definition, you are effectively finding how fast the function's value is changing at that point, which is the essence of the derivative.
Exponential Functions
Exponential functions are powerful mathematical expressions, represented by \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent. These functions are known for their rapid growth or decay, based on the value of the base \( a \). For example, when \( a > 1 \), the function grows exponentially. In the exercise, the function involved was \( 2^x \), an exponential function. It鈥檚 essential to understand that:
  • Exponential functions have a constant base raised to the power of a variable.
  • They appear frequently in contexts of growth, such as populations or investments.
  • These functions are smooth and continuous, making them suitable for derivative calculations using limits.
Knowing this helps in quickly identifying and working with exponential functions in derivative problems.
Calculus
Calculus is a vast field of mathematics focused on change and motion. It primarily consists of two main branches: differential calculus and integral calculus. The derivative, a key calculus concept, measures how a function value changes as its input changes. In our context, differential calculus helps us find the derivative of a function, essential for understanding rates of change. Using the limit definition of the derivative, we can determine the slope of a tangent line to any function's curve, revealing important insights about the function's behavior. Key aspects of branch calculus include:
  • Understanding how functions change over time or various conditions.
  • The ability to compute derivatives, which are crucial for optimization problems.
  • Developing a deep comprehension of mathematical models in physical sciences and engineering.
Function Notation
Function notation is a way to represent functions in a clear and concise manner. A function is often written as \( f(x) \), where \( f \) names the function and \( x \) represents the input variable. This notation is crucial for expressing mathematical ideas effectively. For instance, in the given exercise, \( f(x) = 2^x \) denotes an exponential function. The notation easily allows:
  • Clear identification of the function's formula, i.e., how to compute \( f(x) \) for each \( x \).
  • Simplification in expressing operations, such as differentiation or integration.
  • An organized framework for expressing complex mathematical relationships and their manipulations.
Using function notation, we can write and solve problems more succinctly, aiding in comprehension and manipulation of functions.

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Most popular questions from this chapter

Make a careful sketch of the graph of \(f\) and below it sketch the graph of \(f^{\prime}\) in the same manner as in Exercises \(4-11\) . Can you guess a formula for \(f^{\prime}(x)\) from its graph? \(f(x)=e^{x}\)

The cost of producing x ounces of gold from a new gold mine is \(C=f(x)\) dollars. (a) What is the meaning of the derivative \(f^{\prime}(x) ?\) What are its units? (b) What does the statement \(f^{\prime}(800)=17\) mean? (c) Do you think the values of \(f^{\prime}(x)\) will increase or decrease in the short term? What about the long term? Explain.

Find a formula for a function that has vertical asymptotes \(x=1\) and \(x=3\) and horizontal asymptote \(y=1\)

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{t \rightarrow 1} \frac{t^{4}+t-2}{t-1}$$

(a) Use numerical and graphical evidence to guess the value of the limit $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{\sqrt{x}-1}$$ (b) How close to 1 does \(x\) have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?
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