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(a) Show that $e^{x} \geqslant 1+x$ for $x \geqslant 0$ .(b) Deduce that $e^{x} \geqslant 1+x+\frac{1}{2} x^{2}$ for $x \geqslant 0$ .(c) Use mathematical induction to prove that for $x \geqslant 0$ and any positive integer $n$$$e^{x} \geqslant 1+x+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}$$

A. since $x \geq 0,$ all terms of LHS are positive and the Inequality holdsB. $e^{x} \geq 1+x+\frac{1}{2} x^{2}$C. the given Inequality has been proved by Induction

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Harvey Mudd College

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

06:57

(a) Show that $ e^x \geqsl…

01:12

a. Prove that $e^{x} \geq …

03:52

Proving a Statement Use ma…

05:57

Use mathematical induction…

03:42

use mathematical induction…

Okay, so in part of a we want to show that you to the exits. Greater were you close to one process X when X is greater and greater cost zero. So we defy Oh function Eva to be you to the X minus one process X. So we notice that F zero equals is young. Okay, so we take the purity of life. You would be equally X minus one so you would be positive or no negative when X is greater. Because zero, that means f yes. Increasing on zero to infinity and the dumb use FX is greater recourse to have zero on 0 to 70 and then Amiens FX because of zero recourses zero. So we have this conclusion and then this is exactly to the X minus one pross X territory causes your riches the CMS the statement we want approved here. So a party we have one more term but the ideas lets him. So we define TX because to eat with the ex miners on one process X plus x square over to again G zero with course of the arm so it takes a charity for G. It's because I do this but this is greater because 0102 infinity place on our previous results. So G X is greater because 20 and the zero is equal to zero in the name Eunice You Judy X is greater equals to one process express excrement you circus inductive t If eatery X is greater equals to one plus X plus x square off to pass um X to the power divided by n factorial. Then we can show that into the exits quickly because one plus X plus x square by two in the process one more turn here so you should be in factorial cross extra am Pass one divide and pass one factorial. So to see this, we just defined another function hx because to eat really ex miners One process express up to the term O m plus one term. No, I take it so with notice that hdr because zero So notice that edge crime because to be to the x minus one process X cross X to the end over and factorial which is exactly our assumption. It's that age Crime X is greater because zero on theater infinity That means age crime X Sorry it. That means HX is greater equals to make your own all zero to infinity. Which means HX desecrated decoy causes here and there is proof our statement

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