In this unit, you will examine values of the average rate of change over an interval to approximate the instantaneous rate of change at a point. The concept of a limit will be formally defined, and you will use a graphs and the properties of limits to evaluate limits of a variety of functions.
This unit will introduce you to the formal definition of the derivative. You will examine graphs and use the definition of the derivative to verify the rules for determining derivatives: constant function rule, power rule, constant multiple rule, sum and difference rules, product rule, chain rule, and quotient rule. You will apply these rules to differentiate polynomial, rational, radical, and composite functions. You will be able to connect the value of the derivative at a particular value of x with the slope of the tangent line at a point on a curve, and you will use this slope and point to determine the equation of the tangent line.
In this unit, you develop an algorithm for sketching a curve given the algebraic equation of the curve. You find out about the extreme value and mean value theorems, and you examine the notion of a turning point, an absolute extreme, an interval of increase or decrease, concavity, and a point of inflection.
Now that you are familiar with how to calculate derivatives, we will use them in this unit to solve real-world problems in optimization. Optionally you can use them as a way to determine related rates. You will also introduce Newton’s method as a way to approximate roots of equations.
This unit begins with an introduction to Euler’s number, e. In addition to developing the derivatives of the exponential, logarithmic, and trigonometric functions, you will also extend your algebraic and equation solving skills with these three function types.
This unit introduces the concept of a vector as being a mathematical object having both magnitude and direction. The mathematical operations on geometric vectors developed will culminate in the modeling and solving of problems involving the physical quantities of force and velocity.
This unit introduces vectors in a Cartesian coordinate system. The new model allows us to perform operations on vectors and to investigate interesting geometrical and physical applications.