Tuesday – AB. Morning (Part 1) Developing Understanding of the Derivative Upload TI 84 Programs Break Morning (Part 2) Ideas That Can Be Explored Before Working with Formulas Connecting Graphs of f, f’, and f” Connecting Differentiability with Continuity Local Linearity. Lunch
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Topics listed in the course description relating to the introduction of the derivative and the definition of the derivative are:
Forward Difference analyticallyQuotient
Backward Difference Quotient analytically
Symmetric Difference Quotient analytically
All lead to the derivative of a function at a point analyticallyx=a.
Activities with the graphing calculator can numerically and graphically develop understanding for the algebraic approach.
Do you see how this relates to the activity we did on the difference quotients?
When y1 is increasing, what do you notice about the values of y2?
When y1 is decreasing, what do you notice about the values of y2?
When y1 reaches a maximum, what do you notice about the value of y2?
Making Observations about the Function and Its Derivative derivative and uses of the derivative before you actually derive the formulas for the derivatives.
When y2 is equal to zero, what do you notice about the behavior of y1?
Would you describe y1 as concave down or concave up? How would you describe the slope of y2?
Using any of the difference quotients derivative and uses of the derivative before you actually derive the formulas for the derivatives. (with small h values)obtain graphical (and sometimes numerical)information that can be generalized.Ideas That Can Be Explored Without the Knowing Derivative Formulas
The graph of f ’
using the difference
quotient with f
The graph of f “
using the difference
quotient with f ‘
The graph of f
N derivative and uses of the derivative before you actually derive the formulas for the derivatives. otice that
Extreme Value Theorem derivative and uses of the derivative before you actually derive the formulas for the derivatives.
Match graphs of f, f ‘ and descriptions of f and f’
Write a rule for each of the three lines. Give justification for why you
wrote each equation.
Zoom in to a small window and write the equation of the line that can be used as the linear approximation for this function atx = 0.
Create a slope field from a differential equation but since a differential equation is nothing more than an equation that involves a derivative, differential equations occur throughout the course. A solution to a differential equation is simply a function that satisfies the equation.
Practice Reading Information from a Slope Field
Viewing Slope Fields on a TI-84
Build activities so that student
2008 Curriculum Module
Pages 1 and 2
Approximate y’(12) and explain the meaning of y’(12) in terms of the population of the town.
Pages 1 and 2
Approximate, with a trapezoidal rule, the average population of the town over the 20 years.
Pages 3 and 4
Use a midpoint Riemann sum with three subintervals to approximate
Explain the meaning of this definite integral in terms of the water flow, using correct units.
Pages 4 and 5
Use P(t) to find the average rate of water flow during the 12-hour period. Indicate units of measure.
Pages 5 and 6
Approximate the distance traveled over
Using a right Riemann sum with four intervals.
Use P(t) to find the average rate of water flow during the 12-hour time period. Indicate units of measure.
Other AP Free Response questions that reference tabular data 12-hour time period. Indicate units of measure.