Integral Calculus - Example on Intermediate Value Theorem

  • The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval, and takes on two values, then it must also take on every value in between those two values.
  • Let’s consider a simple example to understand the application of the IVT.

Example Problem

  • Given the function f(x) = x^2 - 4x + 3 on the interval [0, 3].
  • Determine if the function takes on the value of 2 on this interval.
  • We will use the IVT to solve this problem.

Steps to Solve

  1. Determine the function values at the endpoints of the interval.
    • f(0) = (0)^2 - 4(0) + 3 = 3
    • f(3) = (3)^2 - 4(3) + 3 = -3
  1. Check if the desired value (2 in this case) lies between the function values at the endpoints.
    • 3 > 2 > -3
  1. Since the desired value lies between the function values at the endpoints, the function must indeed take on the value of 2 on the interval [0, 3].

Visualization

  • We can visualize this by plotting the graph of the function f(x) = x^2 - 4x + 3.
  • The graph will intersect the horizontal line at y = 2 at some point between x = 0 and x = 3.
  • Hence, we can conclude that the function takes on the value of 2 on the interval [0, 3].

Solution Summary

  • The function f(x) = x^2 - 4x + 3 takes on the value of 2 on the interval [0, 3].
  • This is determined using the Intermediate Value Theorem (IVT), which states that if a function is continuous on a closed interval and takes on two values, then it must also take on every value in between those two values.

Importance in Calculus

  • The Intermediate Value Theorem is an important concept in calculus.
  • It allows us to make conclusions about the existence of roots or solutions of equations.
  • It also helps in analyzing the behavior of continuous functions on closed intervals.

Key Takeaways

  • The Intermediate Value Theorem states that if a function is continuous on a closed interval and takes on two values, then it must also take on every value in between those two values.
  • By evaluating the function at the endpoints of the interval and comparing with the desired value, we can determine if the function takes on that value on the given interval.
  • The IVT is a powerful tool in calculus for analyzing the behavior of functions and determining the existence of solutions.

Example Application

  • Suppose we have a function representing the temperature of a body over time.
  • If we measure the temperature at the beginning and end of a certain time interval, and find that the temperature crosses a certain threshold value within that interval, then we can infer that the temperature must have reached that threshold at some point in between.
  • This application of the Intermediate Value Theorem helps us in understanding the behavior of physical systems.

Summary

  • The Intermediate Value Theorem is a fundamental theorem in calculus.
  • It helps in determining the existence of values of functions on intervals.
  • By checking the function values at the endpoints of an interval and comparing them with the desired value, we can apply the IVT to make conclusions about the function’s behavior.
  • The IVT has various applications in both mathematics and other fields to understand continuous phenomena.