Intermediate Value Theorem

Reading

• Sections 2.8

Practice problems

• Section 2.8: 1, 5, 7, 19
• In class: 2.8 14: $\tan x = x$ has infinitely many solutions
• To turn in (due Wednesday, together with 2.7): 2.8 2, 6

Notes

Intermediate Value Theorem

Intermediate Value Theorem

If $f(x)$ is a continuous function on $[a,b]$ and $L$ is a number between $f(a)$ and $f(b)$, then there must be a $c\in[a,b]$ such that $f(c) = L$.

Simply put: A continuous function takes all values between $f(a)$ and $f(b)$.

Important special case: If $f(a)$ and $f(b)$ have opposite signs, then the function $f$ must have a zero somewhere between $a$ and $b$.

• Make sure to have a good graphical representation of this result.
• The importance of the theorem is that it allows us to guaranteed the existence of solutions to equations.
• Example: We will show that $\sqrt{2}$ exists:
  • The square root is a number $c$ such that $c^2 = 2$.
  • Consider the function $f(x) = x^2$ on the interval $[1,2]$.
  • Then $f(1) < 2 < f(2)$.
  • The IVT tells us there must be a $c\in[1,2]$ with $f(c) = 2$.
• Example 2: Show that there is a positive $x$ with $x = \sin x$.
  • Idea: Bring everything to one side: $f(x) = x - \sin x$.
  • Look for a zero.
• Bisection method: Allows us to zero in on a solution:
  • In the previous example, consider $f(1.5)$. Since it is bigger than $2$, the $c$ is actually in $[1,1.5]$.
  • Try $f(1.25)$ next.
  • Keep bisecting the interval and trying the middle value.
  • Practice: Find the first 3 decimals points of $\sqrt{2}$.