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Chapter 4: Problem 101

101-107. Choose the correct answer: $$ \frac{d}{d x} e^{x}=\quad \text { a. } x e^{x-1} \quad \text { b. } e^{x} \quad \text { c. } 0 $$

Short Answer

Expert verified
The correct answer is b. \( e^{x} \).

Step by step solution

01

Understand the Problem

We are asked to find the derivative of the exponential function \( e^x \). The derivative is represented as \( \frac{d}{dx} e^{x} \).
02

Recall the Derivative Rule

The derivative of \( e^{x} \) with respect to \( x \) is one of the basic rules of differentiation. Specifically, \( \frac{d}{dx} e^x = e^x \).
03

Analyze the Options

We have three options: a. \( x e^{x-1} \), b. \( e^{x} \), c. 0. We need to match our derivative result \( e^{x} \) with the correct option.
04

Choose the Correct Answer

The correct derivative, \( e^{x} \), matches option b. Thus, the correct answer is b. \( e^x \).

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Key Concepts

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

Exponential Function
The exponential function, denoted as \( e^x \), is a mathematical function that describes continuous growth or decay. It is widely used in areas like physics, finance, and biology due to its unique property of self-similarity: the rate of change of \( e^x \) is proportional to its current value. This results in exponential growth where things increase quickly and vastly over time.
  • Unique Nature: The base of the exponential function is the number \( e \), approximately equal to 2.71828. It is an irrational number, much like \( \, \pi \, \), and is fundamental in calculus.
  • Growth Pattern: The function \( e^x \) shows how quantities grow at rates proportional to their current size, which you鈥檒l see in processes like population growth and compound interest.
  • Euler's Formula: This formula connects complex numbers and exponential functions in a profound way: \( e^{ix} = \cos(x) + i\sin(x) \).
Understanding \( e^x \) requires combining algebraic thinking with intuitive insights into growth processes.
Derivative Rules
Differentiation is a central concept in calculus, pivotal for understanding how functions change. When we talk about finding the derivative of a function, we are essentially identifying the function's rate of change.
  • Power Rule: If you have a function \( x^n \), the derivative is \( nx^{n-1} \). This is fundamental for polynomial expressions.
  • Exponential Rule: A very specific and powerful rule is that the derivative of \( e^x \) with respect to \( x \) is \( e^x \) itself. This self-replicating property is unique to exponential functions with the base \( e \).
  • Sum and Difference: For functions consisting of sums or differences, differentiate term by term: \( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \).
  • Product and Quotient Rules: Different rules apply when functions multiply or divide each other: the product rule and quotient rule respectively.
Understanding these basic rules is essential for solving calculus problems involving differentiation, especially when dealing with more complex functions.
Basic Calculus Concepts
Calculus is built upon several foundational ideas that help describe how and why things change in the world around us.
  • Limits: A limit helps us understand what happens to a function as the input approaches a particular point. It lays the groundwork for defining derivatives and integrals.
  • Continuity: This property of functions ensures there are no sudden jumps, breaks, or holes in the graph of a function. For a function to be differentiable, it must first be continuous.
  • Derivatives: As mentioned, derivatives represent rates of change. They're useful in finding slopes of tangent lines, and they provide crucial insights in physics for understanding motion.
  • Integrals: The "opposite" of differentiation, integration is used to calculate areas under curves and accumulate quantities over an interval.
The essence of calculus revolves around these concepts, all aimed at solving problems related to change and accumulation in various scientific and engineering contexts.

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Most popular questions from this chapter

101-107. Choose the correct answer: $$ \frac{d}{d x} \ln x=\quad \text { a. } \frac{x}{1} \quad \text { b. } \frac{1}{x} $$

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.) $$ f(x)=e^{-2 x^{2}} $$

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.

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

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.) $$ f(x)=1-e^{-x^{2} / 2} $$
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