Ep 11. Implicit Differentiation and Related Rates

Understanding Calculus is a show for calculus beginners to aspire to take their math skills to the next level. The eleventh episode of the show's first season sheds light on two topics - Implicit Differentiation and Related Rates.

The episode teaches Implicit Differentiation, which is a method of finding the derivative of an implicit function. An implicit function is one where the dependent variable cannot be expressed in terms of the independent one. The episode guides the viewers with step-by-step examples of finding derivatives of functions like x² - y² = 4 and xy = 6. The calculations might get tricky in such cases, but the show makes sure that the viewer doesn't get lost in the numbers. The hosts explain each step with graphs and diagrams to make the learning more lucid.

Moving on, the episode dives into the concept of Related Rates, which is the application of implicit differentiation. When two variables in a given equation are changing with respect to time, we call them related rates. The episode protagonist Steve takes the example of a ladder sliding down a wall and a balloon inflating to explain this concept. These variables not only form an equation but also link the two rates, so they are called related. The episode highlights how we can find the rate of change of one variable, given the rate of change of the other.

The hosts simplify the equations for the viewers, walking them through each step of the calculations. The team goes out of its way to make sure that the students understand the importance of related rates by giving several real-life examples. It's always fantastic to see a mathematical problem being derived from a given situation and how math can help us calculate various physical quantities in the real world.

Moving forward in the episode, the hosts take specific examples of related rates, like a cone-shaped cup being filled with juice. The given equation is volume of the juice in the cup increasing by 5 cubic centimeters per second, and the height of the liquid column increasing at 6 centimeters per second. The show teaches how we can use the Law of Similarity to derive the equation and calculate the remaining variables.

The episode encapsulates the two concepts of Implicit Differentiation and Related Rates phenomenally with relatable examples. The team ensures that each step gets explained in detail, and the visualization of the graphs and diagrams make the episode more attractive. The presentation style of the team is exceptional, putting the math into context with various scenarios and walking the viewers through the calculations.

In conclusion, Understanding Calculus's 11th episode, Implicit Differentiation and Related Rates, is a powerful teaching tool for calculus students. The episode nuances the two related concepts and teaches them with practical, real-life examples that make the viewing easy to understand. This episode of the show is a must-watch for anyone who wants to grasp these two essential ideas of calculus.