/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 121 Given \(e^{x} \geq 1\) for \(x \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 121

Given \(e^{x} \geq 1\) for \(x \geq 0\), it follows that \(\int_{0}^{x} e^{t} d t \geq \int_{0}^{x} 1 d t .\) Perform this integration to derive the inequality \(e^{x} \geq 1+x\) for \(x \geq 0\)

Short Answer

Expert verified
By integrating each side of the given inequality and rearranging, we obtain the inequality \(e^{x} \geq 1 + x\) for \(x \geq 0\).

Step by step solution

01

Integrate the left side of the inequality

The integration is done for the function \(e^{t}\) from 0 to \(x\). The integral of \(e^{t}\) with respect to \(t\) is \(e^{t}\). Because the limits of the integral are from 0 to \(x\), the definite integral becomes \(e^{x} - e^{0}\). Here \(e^{0} = 1\), so the result of the integral from 0 to \(x\) of \(e^{t} dt\) is \(e^{x} - 1\).
02

Integrate the right side of the inequality

On the right side of the inequality, we need to integrate the function \(1\) with respect to \(t\) from \(0\) to \(x\). The integral of \(1\) with respect to \(t\) is just \(t\). By substituting the limits of the integral, we get \(x - 0 = x\). Therefore, the result of the integral from 0 to \(x\) of \(1 dt\) is \(x\).
03

Combine the results and derive the inequality

After evaluating both integrals, we substitute the results back into the inequality to get \(e^{x} - 1 \geq x\). This can be rearranged into \(e^{x} \geq 1 + x\), which is the desired result.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Exponential Function
An exponential function is a powerful mathematical concept used across various fields. This function is represented as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. It's a constant that serves as the base of natural logarithms.

The exponential function has unique properties:
  • Continuous Growth: It represents continuous exponential growth or decay.
  • Derivative: The derivative of \( e^x \) is \( e^x \) itself, meaning the rate of change is proportional to its current value.
  • Euler's Identity: The function is foundational in Euler's formula \( e^{ix} = \cos x + i \sin x \), revealing deep connections in mathematics.
In the context of inequalities like \( e^x \geq 1+x \), it demonstrates how the function grows faster than linear terms. Understanding these properties can simplify complex calculus problems.
Definite Integral
A definite integral is a calculation that finds the area under a curve between two specific points. In mathematical terms, it is represented as \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the function to be integrated, and \( a \) and \( b \) are the limits of integration.

Key aspects to know about definite integrals include:
  • Limits of Integration: These are the starting and ending points on the x-axis, defining the region of interest.
  • Geometric Interpretation: It can be visually understood as the area under the curve of \( f(x) \) from \( a \) to \( b \).
  • Evaluation: Use the Fundamental Theorem of Calculus, which relates the definite integral of a function to its antiderivative.
In our exercise, integrating \( e^t \) and \( 1 \) yields areas representing \( e^x - 1 \) and \( x \) respectively, leading us to the inequality \( e^x \geq 1+x \). Understanding these concepts helps translate graphical interpretations into precise mathematical statements.
Mathematical Proof
A mathematical proof is a logical argument establishing the truth of a mathematical statement. It employs a sequence of statements, each building logically from the previous ones, to arrive at a conclusion.

Types of Proofs Include:
  • Direct Proof: Begins with known truths and proceeds through logical steps to prove the statement directly.
  • Indirect Proof: Assumes the negation of the statement and shows that this leads to a contradiction.
  • Proof by Induction: Proves a base case and then shows that if one case holds, the next case also holds.
In our exercise, we use a direct approach, integrating and simplifying to prove the inequality \( e^x \geq 1+x \). This clear and structured method ensures the statement holds universally for \( x \geq 0 \). Understanding how proofs validate mathematical ideas is crucial for deeper insights into the discipline.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Is the converse of the second part of Theorem \(5.7\) true? That is, if a function is one-to-one (and therefore has an inverse function), then must the function be strictly monotonic? If so, prove it. If not, give a counterexample.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the inverse function of \(f\) exists, then the \(y\) -intercept of \(f\) is an \(x\) -intercept of \(f^{-1}\).

An object is projected upward from ground level with an initial velocity of 500 feet per second. In this exercise, the goal is to analyze the motion of the object during its upward flight. (a) If air resistance is neglected, find the velocity of the object as a function of time. Use a graphing utility to graph this function. (b) Use the result in part (a) to find the position function and determine the maximum height attained by the object. (c) If the air resistance is proportional to the square of the velocity, you obtain the equation $$ \frac{d v}{d t}=-\left(32+k v^{2}\right) $$ where \(-32\) feet per second per second is the acceleration due to gravity and \(k\) is a constant. Find the velocity as a function of time by solving the equation $$ \int \frac{d v}{32+k v^{2}}=-\int d t $$ (d) Use a graphing utility to graph the velocity function \(v(t)\) in part (c) if \(k=0.001\). Use the graph to approximate the time \(t_{0}\) at which the object reaches its maximum height. (e) Use the integration capabilities of a graphing utility to approximate the integral \(\int_{0}^{t_{0}} v(t) d t\) where \(v(t)\) and \(t_{0}\) are those found in part (d). This is the approximation of the maximum height of the object. (f) Explain the difference between the results in parts (b) and (e).

Let \(f\) and \(g\) be one-to-one functions. Prove that (a) \(f \circ g\) is one-to- one and (b) \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\).

Prove that \(\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.