AP Calculus AB: Higher Derivatives, Definite and Indefinite Integrals

In calculus, we call the derivative of y' the second derivative of y and denote it by y" or f" (x). It is also essential to know that other higher-order derivatives, such as the third derivative, y'" can be applied in differential calculus. One obtains y'" by differentiating y" in the same manner in which one differentiates y' to obtain y". In this free online course, you will learn how to calculate the derivatives of inverse trigonometric functions, including arcsine, arctangent and arcsecant. Do you know that the derivatives of trigonometric functions can also help you determine the derivatives for arccosine, arccotangent and arcsecant? You will be able to identify the restrictions that need to be placed on an inverse derivative for it to become functions. The arcsine is given to the inverse of sine. You will study how this function is written and the rationale behind the name. The course uses a basic method that will enable you to derive the formulas for the derivatives of the arccotangent and arccosecant functions. Another interesting concept discussed is logarithms. 

Do you know that the study of logarithms is an essential part of calculus? There are important aspects of logarithms that are relevant to everyday life. One is the ‘base e’. Base e helps in simplifying, differentiating and integrating in calculus. Another important one is the ‘Base ten’. Logarithms using base ten have many applications, including decibels, pH, and the Richter scale. Logarithms can be very useful in simplifying an algebraic expression, thereby making it easier to differentiate. Have you done calculations in calculus involving maxima and minima values? This course explains the underlying principles in maxima and minima values known as the first and second derivatives. You will be introduced to the local extreme value theorem and the technical term used for 'turning point'. The idea behind absolute extrema is also explored. You will learn about the smooth curve, which is also known as the differentiable curve. Several scientists have done some work around finding the absolute extrema of a curve. You will get to know the system used for finding absolute extrema.

Finally, you will study other methods used to calculate a curve's absolute extrema. Have you ever heard about the importance of the mean value theorem? This is a simple theorem in differential calculus. You will learn how to apply the second derivative test, and be introduced to the concepts of concavity and points of inflection. Optimization is a crucial aspect of calculus discussed extensively in this course. An optimization problem seeks a "best" or "optimal" value for a variable quantity, which depends on the value of another variable quantity, over which we have control. Did you know that derivatives have essential applications in business and economics? Generally speaking, derivatives are useful in solving many types of mathematical and non-mathematical problems. This course does a great job of describing the applications of derivatives in several areas. Start today and get real insights into the basics of higher derivatives in calculus and its many applications. The material is useful for those wanting to take the AP (Advanced Placement) calculus exams or anyone who is looking for a comprehensive refresher course.