Finding Solutions Using the Intermediate Value Theorem

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SUMMARY

The equation \( e^x = x + 2 \) has exactly two solutions, which can be demonstrated using the Intermediate Value Theorem (IVT). By defining the function \( f(x) = e^x - x - 2 \), one can apply the IVT to show that if \( f(a)f(b) < 0 \), then there exist values \( c \) and \( d \) such that \( f(c) = 0 \) and \( f(d) = 0 \). The key to finding these solutions lies in selecting appropriate intervals where a sign change occurs in the function values.

PREREQUISITES
  • Understanding of the Intermediate Value Theorem (IVT)
  • Basic knowledge of exponential functions
  • Ability to analyze function behavior and sign changes
  • Familiarity with function notation and evaluation
NEXT STEPS
  • Study the Intermediate Value Theorem in detail
  • Learn how to graph exponential functions for better visualization
  • Explore numerical methods for finding roots of equations
  • Practice selecting intervals for applying the IVT with various functions
USEFUL FOR

Students studying calculus, particularly those learning about the Intermediate Value Theorem and root-finding techniques, as well as educators looking for examples to illustrate these concepts.

Vashti
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Homework Statement



Show that the equation e^x = x + 2 had exactly two solutions. Explain your reasoning.

Homework Equations



Okay so I guess I'm using the intermediate value theorum and proving that f(a)f(b) < 0 we can find a c and d such that f(c) = 0 and f(d) = 0 so that by collorary it's true. This is an issue because I don't have boundaries for my function...

The Attempt at a Solution



create a function: f(x) = e^x - x - 2
and that's about as far as I got. I'm assuming I arbitarily select an interval and use that? Are there rule that I should follow though?
 
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I would believe you should select some values and check for a sign change...
 

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