Exploring Instantaneous Rate Of Change and the calculus of motion

Explore the use of product and quotient rules for derivatives

Time is spent to develop patterns so that we no longer need to take the limit of the difference quotient. The proof is provided as to why if f(x)=x^n the f'(x)=n*x^(n-1)

Examines the idea of locally linear. Verifies that at a specific value a function has a vertical tangent. Examines how the calculator can give a numerical derivative answer--also its limitations regarding the use of a symmetric difference quotient.

Investigates four cases as to why the derivative may fail to exist. Sharp point, cusp, vertical tangent, and discontinuity. A discussion of the limit values at the points motivates why the derivate will fail to exist. A review of the alternate form of a derivate is also presented.

The derivative of a function and the alternate defintion

Explore the relationship between the graph of a function and how the graph of a derivative looks. Attention is paid to the the behavior of the slope of the tangent in helping understand this relationship. Images of a GSP sketch are provided to help clarify and explore the graphs.

Types of discontinuity is discussed. The intermediate value theorem is also mentioned.

Address limits as x->infinity or as x->-infinity The connection to horizontal asymptotes is explored as well as end behavior. Graphing f(1/x) is discussed as a means of better "seeing" limits as x->infinity

Lesson finishes the intro to limits. A specific case of the limit of a polynomial is investigated leading to an easy way to evaluate many limits. The sandwich theorem is explored for a situation where substitution will not work.