Calculus Rhapsody By Phil Kirk & Mike Gospel (Lyrics below) We've had this little school project of ours here on YouTube for over 10 years!! It's been so lovely seeing all of your nice comments over the years. We've updated this description to show you what we're up to nowadays (we're both adults with jobs now!). You can find us online at the links below. Thanks everyone! ******************* Follow Mike: Mike's podcast "How Star Wars Is It?": https://podcasts.apple.com/us/podcast/how-star-wars-is-it/id1454705203 HSWII Instagram: https://www.instagram.com/howstarwarsisit/ HSWII Twitter: https://twitter.com/howstarwarsisit Mike's art and face: https://www.instagram.com/wordgospel09/?hl=en Mike's Twitter: https://twitter.com/mikegospel?lang=en Follow Phil: Indy Rustic: https://www.indyrustic.com/ Plat Collective Instagram: https://www.instagram.com/phil_plat_collective/ ******************* Lyrics: Is this x defined? Is f continuous? How do you find out? You can use the limit process. Approach from both sides, The left and the right and meet. Im a just a limit, defined analytically Functions continuous, Theres no holes, No sharp points, Or asymptotes. Any way this graph goes It is differentiable for me for me. All year, in Calculus Weve learned so many things About which we are going to sing We can find derivatives And integrals And the area enclosed between two curves. Y prime oooh Is the derivative of y Y equals x to the n, dy/dx Equals n times x To the n-1. Other applications Of derivatives apply If y is divided or multiplied You use the quotient And product rules And dont you forget To do the dance Also oooh (dont forget the chain rule) Before you are done, You gotta remember to multiply by the chain (Instrumental solo) I need to find the area under a curve Integrate! Integrate! You can use the integration Raise exponent by one multiply the reciprocal Plus a constant Plus a constant Add a constant Add a constant Add a constant labeled C (Labeled C-ee-ee-ee-ee) Im just a constant Nobody loves me. Hes just a constant Might as well just call it C Never forget to add the constant C Can you find the area between f and g In-te-grate f and then integrate g (then subtract) To revolve around the y-axis (integrate) outer radius squared minus inner radius squared (multiplied) multiplied by pi (multiply) Multiply the integral by pi! Pi tastes real good with whipped cream! Mama-Mia, Mama-Mia Mama-Mia let me go. Pre-calculus did not help me to prepare for Calculus, for Calculus, help me! So you think you can find out the limit of y? So you think youll find zero and have it defined Oh baby cant define that point baby Its undefined Goes to positive and negative infinity Oooh. Oooh yeah, oooh yeah. Differentiation Anyone can see Any mere equation It is differentiable for me. (Any way this graph goes)