Fun With Tangent Lines

Jeff MorganUniversity of Houston

Before We Start…

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Three Challenge Questions

Coming Events at UH

• AP Calculus Workshop II – 10/21/2006 http://www.HoustonACT.org

• Algebra I Workshop II – 10/28/2006 http://www.EatMath.org

• High School Math Contest – 2/17/2007 http://www.mathcontest.uh.edu

1. Single Point Identification

There are two rectangles on the right. The original one is depicted in blue. The yellow rectangle is the result of shrinking the blue rectangle in both the vertical and horizontal directions, rotating it, and repositioning it on top of the blue rectangle.

Question: Can you show that the rectangles have exactly one common point?

The solution requires trigonometry.

2. A Geometric Puzzle

The solution requires thought and geometry.

This problem was given to a large group of students who had never seen geometry. Many of them solved the problem (although not immediately!!).

The Problem: Divide the circle into at least three pieces so that all pieces are the same size and shape, and at least one of the pieces does not touch the center of the circle.

3. Radio PlayJoe Smith tunes into the same radio programming for an average length of 30 minutes the same time each day, seven days each week. What he listens to is a pre-recorded program that loops continuously through the 7-day week (meaning it repeats over and over again.) It is easy to see that 3.5 hours is the minimum amount of recording time. Suppose the station decides this just isn’t enough recording time and they want to know if there are other options. What are the other possible recording times which allow Joe to hear a different show every day, while remaining under 24 hours of recording?

The solution requires thought and arithmetic.

Fun With Tangent Lines

(Using a function and its tangent lines to create a new function.)

A function is plotted below. Use this graph to create a new function by

taking each value of to the value given by the -intercept of the tangent

line to the graph at the point , ( ) .

x y y

x f x

A function is plotted below. Use this graph to create a new function by

taking each value of to the value given by the -intercept of the tangent

line to the graph at the point , ( ) .

x y y

x f x

2

Let be a real number. Determine the -intercept of

the tangent line to the graph of ( ) at the point

where . What new function is created through this

process?

a y

f x x

x a

Suppose you start with a differentiable function and you create a new function by

taking each value of to the value given by the -intercept of the tangent line to the

graph at the point , ( ) .

f

x y y

x f x What function will be created by each of the following

choices of ?f

2x3x4x5x6x

lnx x

f x New Function

Suppose you start with a differentiable function and you create a new function by

taking each value of to the value given by the -intercept of the tangent line to the

graph at the point , ( ) .

f

x y y

x f x What function will be created by each of the following

choices of ?f

2 3x x3 22x x4 23x x

f x New Function

( )g x ( )G x

( )h x ( )H x

2 ( ) 3 ( )g x h x

Can you make a conjecture?

Can you prove your conjecture?

Suppose you start with a differentiable function and you

create a new function by taking each value of to the value

given by the -intercept of the tangent line to the graph at the

point , ( ) .

f

x y

y

x f x Give a formula for the new function.

Define a function that takes a differentiable function

to a new function using the process above.

Find the solutions to 0.

T f

g

T f

A function is plotted below. Use this graph to create a new function by

taking each value of to the value given by the -intercept of the tangent

line to the graph at the point , ( ) . What are your

x y y

x f x observations?

Can you make a conjecture concerning the relationship of critical points and points of

inflection from the original function, and the behavior of the resulting function?

Can you prove your conjecture?

Make a slight change in this process – Part I Fix a real number . Suppose you start with a differentiable

function and you create a new function by taking each value

of to the coordinate given by the intersection of the tangent

line to th

a

f

x y

e graph at the point , ( ) with the line .

How will the process change?

x f x x a

Make a slight change in this process – Part II

Suppose you start with a differentiable function and you

create a new function by taking each value of to the value

given by the -intercept of the tangent line to the graph at

the point , ( )

f

x

c x

x f x (if it exists). How will the process change?

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