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Chapter 4: Problem 42

For each function, calculate "in your head" the relative rate of change. $$ f(x)=e^{x} $$

Short Answer

Expert verified
The relative rate of change is 1.

Step by step solution

01

Define Relative Rate of Change

The relative rate of change of a function \( f(x) \) is defined as the derivative of the function divided by the function itself. It is given by the formula \( \frac{f'(x)}{f(x)} \).
02

Differentiate the Function

For the function \( f(x) = e^x \), we find its derivative. The derivative \( f'(x) \) is \( e^x \), as the derivative of \( e^x \) with respect to \( x \) is simply \( e^x \).
03

Calculate the Relative Rate of Change

Substitute \( f(x) = e^x \) and \( f'(x) = e^x \) into the formula for the relative rate of change to get \( \frac{e^x}{e^x} = 1 \).
04

Conclusion

The relative rate of change for \( f(x) = e^x \) is constant and equal to 1 at any point \( x \).

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Key Concepts

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

Exponential Function
An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a constant and \( a > 0 \). In this specific case, \( a \) is the mathematical constant \( e \), approximately equal to 2.718.
The function \( f(x) = e^x \) is unique because it is its own derivative. This means that as you increase \( x \), the rate of growth of the function is proportional to the value of the function itself.
Exponential functions like \( e^x \) are commonly used in various fields including biology, economics, and physics to model growth processes.
Calculus Derivative
The derivative of a function is a key concept in calculus that represents the rate at which a function is changing at any given point. It is denoted as \( f'(x) \).
For the function \( f(x) = e^x \), the derivative is particularly simple: it is \( e^x \) itself. This results from the unique property of the exponential function, where the rate of change is always proportional and equal to its current value.
Understanding derivatives allows you to find the slope of the tangent line to the curve of the function at any point, which is crucial for analyzing motion, optimizing functions, and modeling change in systems.
Function Differentiation
Function differentiation is the process of finding the derivative of a function. This process helps determine how a function behaves as its input changes. In mathematical notation, to differentiate a function \( f(x) \), we denote it as \( f'(x) \).
In our example, when differentiating \( f(x) = e^x \), we performed the differentiation process to find \( f'(x) = e^x \). Differentiation here was straightforward due to the nature of the exponential function, which remains unchanged.
The ability to differentiate functions is fundamental for solving many real-world problems where understanding the rate of change is essential, from calculating speed to predicting financial trends.

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Most popular questions from this chapter

GENERAL: Temperature A covered cup of coffee at 200 degrees, if left in a 70 -degree room, will cool to \(T(t)=70+130 e^{-2.5 t}\) degrees in \(t\) hours. Find the rate of change of the temperature: a. at time \(t=0\). b. after 1 hour.

\(45-50 .\) For each function, find the indicated expressions. $$ f(x)=x^{4} \ln x, \quad \text { find } \quad \text { a. } f^{\prime}(x) \quad \text { b. } f^{\prime}(1) $$

1-44. Find the derivative of each function. $$ f(z)=\frac{12}{1+2 e^{-z}} $$

In each pair of equations, one is true and one is false. Choose the correct one. \(\frac{\ln x}{\ln y}=\ln (x-y)\) or \(\ln \frac{x}{y}=\ln x-\ln y\)

69-72. Use your graphing calculator to graph each function on a window that includes all relative extreme points and inflection points, and give the coordinates of these points (rounded to two decimal places). [Hint: Use NDERIV once or twice with ZERO.] (Answers may vary depending on the graphing window chosen.) $$ f(x)=e^{x}+e^{-x} $$
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