HACT Workshop Derivatives, Slope Fields and more Bekki George

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Presentation transcript:

HACT Workshop Derivatives, Slope Fields and more Bekki George

AP Calculus Resource Site

Over 40 Online Quizzes and AP Practices Exams

Some Free Stuff Online resources for Calculus and Finite Math: Sage: Geogebra: Winplot: GraphFunc Online: SpaceTime: Graph:

Use online Can save worksheets

Allows you to create dynamic worksheets for the web (ex: ) Easy to use Free for both PC and Mac 5.0 has built in CAS and 3-D

Winplot Easy to use with 2-D and 3-D PC only

Online only 2-D and 3-D views Need latest JRE

SpaceTime Free for PC but $$ for Mac and iPhone/iPad Great graphics Powerful Tutorial during first download

Graph 2-D only Download free for PC only (won’t run on Mac)

Break!

Derivatives Some problems: Derivative defined as the limit of the difference quotient. Evaluate the limit:

The position of a particle moving along a line is given by for 0 ≤ t ≤ 4. For what value(s) of t is the speed of the particle smallest?

The function g is differentiable, and the tangent line to the graph of y = g(x) at x = -2 is y = 4x - 2. Let f (x) = 2g(x) + 3x + 2. Give f ' (-2).

The function g is differentiable, and the tangent line to the graph of y = g(x) at x = -1 is y = 2x - 1. Let f (x) = 2[g(x)] 2 + 2x + 4. Give f ' (-1).

Suppose that f is a differentiable function, f (-2) = 3, and f ' (-2) = 2. Find g ' (-2) given that

The functions f and g are differentiable and h(x) = f (g(x)). Use the information below to find h ' (4): f(4) = - 3, f ' (4) = 2 f(2) = - 3, f ' (2) = 3 g(4) = 2, g ' (4) = - 2 g(- 3) = 3, g ' (- 3) = - 2

The functions f and g are differentiable and h(x) = f(x)/g(x). Use the information below to find h ' (3): f(3) = - 1, f ' (3) = 6 f(6) = - 4, f ' (6) = 0 g(3) = 6, g ' (3) = - 6 g(- 4) = 4, g ' (- 4) = - 6

The graph of f ' (the derivative of f) is shown below. At what value of x does the graph of f (x) change from increasing to decreasing?

The graph of f is given below. If f is twice differentiable, then which of the following is true? a) f (3) < f '' (3) < f ' (3) b) f ' (3) < f (3) < f '' (3) c) f '' (3) < f ' (3) < f (3) d) f '' (3) < f (3) < f ' (3) e) f (3) < f ' (3) < f '' (3)

The graph of y = f (x) is given below. For which of the five domain values shown is f ‘’ (x) < 0 and f ’ (x) < 0 ?

Slope Fields Intro Activity Give all students a small square cut from an overhead transparency with a dot on it. The dots will each represent a coordinate point. Then give the class a problem such as or and talk about what this means (slope of tangent line at the point). Have them each draw small tangent lines and come place on the overhead. Expand to more difficult differential equations (some they cannot integrate easily).

Creating Slope Fields with Winplot Equation->Differential->dy/dx=

Now choose One->Initial Value Problem Pick your x and y values then select draw:

Ti Activities (some old ones):

Some Problems: Which of the following differential equations has the following slope field? a) b) c) d) e)

Which of the following is a slope field for the given differential equation?

From Sample AP* exam

A few more questions…

The function f is differentiable and Determine the value of f ' ( π ⁄ 3 ).

The area of the region bounded between the graph of f and the x-axis from x = -1 to x = 0 is 2, from x = 0 to x = 2 is 4, and from x = 2 to x = 4 is 36. Evaluate: