In this module, we define the derivative and explore methods to differentiate various functions. We begin by learning the power rule to differentiate power functions, followed by learning the sum, product, quotient, and chain rules. We then learn how to differentiate exponential functions, natural logarithms, trigonometric functions, and finally, inverse trigonometric functions.

In this module, we define the integral and explore methods to integrate various functions. We begin by learning how the definite integral is used to calculate areas. We then find a connection between integration and differentiation by proving the first and second fundamental theorems of calculus. These theorems motivate us to define an indefinite integral as an anti-derivative. Throughout the module, we will examine various integration techniques, including integration by substitution, integration by parts, integration of trigonometric functions, trigonometric substitution, and integration by partial fractions.

In this module, we explore sequences and series. We learn how an infinite power series can converge to a function. These convergent series are known as Taylor series, and we will determine the Taylor series for the most important functions of calculus, including the exponential function, sine and cosine functions, the natural logarithm, and the arctangent. We also learn L’Hospital’s rule, a very useful tool for finding indeterminate limits.

In this module, we begin to apply the calculus. Using Taylor series, we define the complex exponential function and use it to prove key trigonometric identities. We employ calculus to derive the circumference and area of a circle, as well as the surface area and volume of a sphere. Finally, we show how calculus can be used in numerical methods to find the roots of equations and to integrate and differentiate functions.

In this module, we continue exploring applications of calculus. We learn how to use derivatives to find local extrema of functions. We prove that among rectangles with a given perimeter, the one that maximizes the area is a square. We find the shortest path between two villages after collecting water from a river. We determine the optimal position on a beach for a lifeguard to enter the sea to rescue a swimmer in distress. We discuss how calculus is used in physics to define velocity and acceleration, and how to determine the position and velocity of an object falling under gravity. Lastly, we explore differential equations related to growth, decay, and oscillation, including equations for compound interest and the oscillating pendulum.