Understanding Rolle's Theorem, Mean Value Theorem, and Calculus Applications
Explore Rolle's Theorem and the Mean Value Theorem in calculus, including their applications and examples. Learn how to find x-intercepts, determine values at specific points, and understand the concepts behind these theorems. Dive into the mathematical principles that govern these fundamental calculus concepts.
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Presentation Transcript
Chapter Chapter 3 3.2: Rolle s Theorem and Rolle s Theorem and The Mean Value Theorem The Mean Value Theorem .2: HONORS CALCULUS/CALCULUS HONORS CALCULUS/CALCULUS
Rolles Theorem Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If ? ? = ?(?) then there is at least one number c in (a, b) such that ? ? = ?.
Ex. 1) Find the two x-intercepts of ? ? = ?? ?? + ? and show that ? ? = ? at some point between two intercepts.
Ex. 2) Let ? ? = ?? ???. Find all c in the interval (?,?) such that ? ? = ?.
The Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then a number c in (a, b) such that ? ? =? ? ?(?) ? ?
Ex. 3) Given ? ? = ? ? such that the slope of the secant line is equal to the slope of the tangent line. ?, find all c in the interval (1, 4)
