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

## Step by step solution

## Integrate the left side of the inequality

## Integrate the right side of the inequality

## Combine the results and derive the inequality

## Key Concepts

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

###### Exponential Function

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.

###### Definite Integral

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.

###### Mathematical Proof

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.