Chapter 3 - Derivatives

u3.1 Derivatives:

• Derivatives presented geometrically, numerically, and analytically.
• Derivatives interpreted as an instantaneous rate of change, and defined as the limit of a difference quotient.
• Slope of a curve at a point, vertical tangents, no tangents.
• Tangent lines to curves and local linear approximations.

We will extend the concept of tangent lines into the calculus concept of a derivative.

The derivative as the slope of a tangent

The derivative as a rate of change

u3.2 Derivatives as Functions

• Relationship between differentiability and continuity.
• Corresponding characteristics of f and f '.

Armed with the definition of the derivative, we can now discuss what the derivative of a function can tell us about the function, using graphs, values, and equations.

The Derivative as a Function

u3.3 Differentiation Formulas:

• Derivatives of power functions, basic rules for the derivative of sums, products, and quotients of functions.

We will discover mathematical patterns that help us find derivatives quickly.

Constant rule, constant multiple rule

Power rule

Sum rule, difference rule

Product rule

Quotient rule

u3.4 Rates of Change in the Natural and Social Sciences:

• Instantaneous rate of change as the limit of average rate of change.
• Equations involving derivatives, and verbal descriptions translated into equations involving derivatives (and vice versa).
• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

We can now apply the concept of derivatives to solving real-world problems, given written descriptions, data, and graphs. We will explain and interpret our answers in the context of our real-world problems.

u3.5 Derivatives of Trigonometric Functions

·    Derivatives of basic trig functions, such as sin(x), cos(x), and tan(x).

We will learn how to find the derivatives of cyclical or periodic functions. We will analyze the functions and their derivatives graphically and analytically.

Trigonometric Rules

u3.6 The Chain Rule

·    Derivatives of compound functions.

Functions in nature are often complex. We will complete an in-depth exploration using Excel’s numerical and graphical features to derive techniques for finding derivatives of compound functions.

The Chain Rule

The Chain Rule With Other Rules

u3.7 Implicit Differentiation

·    The concept of implicit functions and implicit curves.

·    The technique of implicit differentiation.

Many functions and models cannot be express in y=f(x) form. Some models may not even be functions (e.g. x²+y²=8). We will learn how to find derivatives of implicit functions.

 Derivatives of Implicitly Defined Functions

u3.8 Higher Derivatives

·    The derivative of a derivative.

·    Corresponding characteristics of the graphs of f, f', and, f".

·    Interpretation of the derivative as a rate of change in varied applied contexts.

What is the derivative of a derivative? What can it tell us about the function?

Higher Order Derivatives

u3.9 Related Rates

·    Modeling rates of change, including related rates problems, numerically, graphically, and analytically.

·    Verbal descriptions are translated into equations involving derivatives and vice versa.

·    The value of careful diagrams and good notation.

Nature is dynamic. Inputs to functions can change with time. How do changing inputs affect the output of a function and how does the derivative help us to analyze rates of change in the natural world?