Objective
By the end of this lesson, you will understand exponential functions, their properties and graphs, arithmetic and geometric sequences, their applications, recursive definitions, rates and average rates of change, key concepts of derivatives as an instantaneous rate of change, and derivatives of polynomial functions.
Materials and Prep
- Pen, paper, and calculator
- Basic understanding of algebra and functions
Activities
- Exploring Exponential Functions: Create graphs of various exponential functions and analyze their properties.
- Sequence Investigation: Explore arithmetic and geometric sequences, identify their recursive definitions, and discuss real-world applications.
- Derivative Discovery: Understand the concept of derivative as an instantaneous rate of change, and practice finding derivatives of polynomial functions.
Talking Points
- Understanding Exponential Functions: "Exponential functions grow rapidly as the input increases, and they have a constant ratio between successive outputs."
- Exploring Sequences: "Arithmetic sequences have a common difference between terms, while geometric sequences have a common ratio. Both have interesting real-world applications."
- Recursive Definitions: "A sequence can be defined recursively by specifying the first term and a rule for obtaining subsequent terms."
- Derivatives and Rates of Change: "The derivative of a function at a point gives the slope of the tangent line at that point, representing the instantaneous rate of change."
- Derivatives of Polynomial Functions: "To find the derivative of a polynomial function, apply the power rule by multiplying the coefficient by the exponent and decreasing the exponent by one."