Rolle's Theorem Explained (with proof) | Calculus 1

We introduce Rolle's theorem which we will later use in a proof of the Mean Value theorem. Rolle's theorem states that if a function is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b), then there exists a point c in (a,b) so that f'(c)=0, that is - a place where the derivative is 0. We'll prove Rolle's theorem and go through two examples of using Rolle's theorem. In the first example we will find two x-intercepts of our function then use Rolle's theorem, and in the other example we will use Rolle's theorem to prove a cubic has only one root. #calculus1 #apcalculus Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join Calculus 1 Exercises playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxUEqGGgvL3EuIQUNcAdmVhx Calculus 1 playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxWVDpl8oaz_Co6CW50KtGJy Get the textbook for this course! https://amzn.to/3PieD1M 0:00 Intro 0:15 Rolle's Theorem 1:00 Rolle's Theorem Visualized 2:27 Proof of Rolle's Theorem 5:33 Example 1 of using Rolle's Theorem 7:30 Example 2 of using Rolle's Theorem 10:25 Conclusion ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Outro music by Ben Watts and is available for channel members. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath