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magnifik
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Integrals of the exponential function
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SUMMARY
The discussion focuses on the integral of the exponential function, specifically evaluating the inequality \(\int_0^x e^t dt \ge \int_0^x dt\) for \(x \ge 0\). It is established that since \(e^x \ge 1\) for all \(x \ge 0\), the integral of \(e^t\) from 0 to \(x\) is greater than or equal to the integral of 1 over the same interval. This leads to the conclusion that the area under the curve of the exponential function is always greater than or equal to the area under the constant function 1 for non-negative \(x\).
PREREQUISITES- Understanding of integral calculus
- Familiarity with the properties of the exponential function
- Knowledge of inequalities in calculus
- Basic skills in evaluating definite integrals
- Study the properties of the exponential function, particularly \(e^x\)
- Learn techniques for evaluating definite integrals
- Explore applications of inequalities in calculus
- Investigate the Fundamental Theorem of Calculus
Students of calculus, educators teaching integral calculus, and anyone interested in the properties of exponential functions and their integrals.
- #2
HallsofIvy
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What is the problem? You are told exactly what to do so do it! You are told that [itex]e^x\ge 1[/itex] for all [itex]x\ge 0[/itex] so
[tex]\int_0^x e^t dt\ge \int_0^x dt[/tex]
What does that give?
[tex]\int_0^x e^t dt\ge \int_0^x dt[/tex]
What does that give?
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