Can you explain the inequalities in exponential functions?

AI Thread Summary
The discussion focuses on understanding the inequalities involving exponential functions, specifically 1-exp(-μt) ≤ μt and (1-exp(-μt))exp(-λt) ≥ μt - (μ^2t^2/2)(1-λt). Participants seek clarification on why these inequalities are true, as they appear in lecture notes. The request emphasizes the need for a mathematical explanation rather than a general overview. A clear understanding of the properties of exponential functions and their behavior in these contexts is essential for grasping the inequalities. The conversation highlights the importance of foundational knowledge in calculus and exponential growth for interpreting these mathematical statements.
stukbv
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Hello, could somebody please explain to me how

1-exp(-μt) ≤ μt

and similarly

(1-exp(-μt))exp(-λt) ≥ μt-μ2t2\2)(1-λt)

Thanks a lot
 
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You haven't said what you want explained...
 
Sorry, I just want to know why they're true, they are in my lecture notes and I can't work out why we know these inequalities hold.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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