Math 106 Lab M. Flashman
Monday September 29th, 2003 11:0011:50 DRAFT

The Anim
Menu: A, B, C,
...

One Function
Menu

Slider...

function
choice

the "slider"
[used for tracing and finding values of f(x).]
 tangent

Zeros... .

Extremes...

Two Function
Menu

Meetings...

Combinations

arithmetic

composition: (f(g(x)) f < g f = top g=bottom
 What functions are needed to create y= (3x+1)^{2} as a composition? Add these to your inventory and see that it works from graphing these functions directly and by the composition combination.
 Compare:

graph of x^{2} with graph
of (3x+1)^{2}
 Find the slope of the tangents when x = 1 algebraically (using the chain rule). Verify this on the graph.
 graph of x^{2} with graph
of 3(x+1)^{2}
 Find the slope of the tangents when x = 1 algebraically using the chain rule. Verify this on the graph.
 graph of x^{2} with graph
of 3x^{2} +1. [What does this have to do with composition? think of u =x^{2} ]
 Find the slope of the tangents when x = 1 algebraically using the chain rule. Verify this on the graph.
 Equa
Menu....
 Implicit functions:
 (i) Graph xx + yy= 13... A circle.
 (ii) 3x^2 + 4y^2 = 25 ... an ellipse
 Find and graph explicit (continuous)
functions that will correspond with a piece of these curves
("implicitly defined by the original equations"):
 (i) y = sqr(13x^2) or y = sqr(13x^2)
 (ii) y = 1/2sqr(253x^2) or y = 1/2sqr(253x^2)
 Problem: How to find the slope of the tangent line to the graphs of an implicitly defined function.
 Idea: Suppose y is a differentiable function of x.
Use L(x) =x^2 + y^2; R(x) = 13
 Find L'(x) ... use the chain rule.
 Find R'(x)..
 Now L'(x) = R'(x) ... and solve for dy/dx.