Math 106 Lab M. Flashman
Monday September 29th, 2003 11:00-11:50 DRAFT
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The Anim
Menu: A, B, C,
...
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One Function
Menu
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Slider...
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function
choice
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the "slider"
[used for tracing and finding values of f(x).]
- tangent
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Zeros... .
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Extremes...
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Two Function
Menu
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Meetings...
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Combinations
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arithmetic
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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:
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graph of x2 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 x2 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 x2 with graph
of 3x2 +1. [What does this have to do with composition? think of u =x2 ]
- 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(13-x^2) or y = -sqr(13-x^2)
- (ii) y = 1/2sqr(25-3x^2) or y = -1/2sqr(25-3x^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.