3.2.2 Continuous, But Not Smooth

As a warm up to magnifying graphs, think about this question: If you magnify a segment by one million and it appears to be 1 cm long, how long is it? We need a general formula to answer this kind of question so that we can predict when sufficient magnification will make a graph appear linear. We begin with some numerics to help present the idea.

Example CD-3.1 Magnification of at x=3/4

The graph in Figure CD-3.1 at the left has 1 cm for each unit. The graph at the right is magnified by 2 so 1 cm equals 1/2 unit. Points are indicated at (x,y)=(3/4,27/64) or (dx,dy)=(0,0), and above .

Figure CD-3.1: y=x³ and at ,

Magnification of at x=3/4

The graph in Figure CD-3.2 at the left has 1 cm for each unit. The graph at the right is magnified by 4 so 1 cm equals 1/4 unit. Points are indicated at (x,y)=(3/4,27/64) or (dx,dy)=(0,0), and above .

Figure CD-3.2: y=x³ and at ,

Magnification of at x=3/4

The graph Figure CD-3.3 at the left has 1 cm for each unit. The graph at the right is magnified by 8 so 1 cm equals 1/8 unit. Points are indicated at (x,y)=(3/4,27/64) or (dx,dy)=(0,0), and above .

Figure CD-3.3: y=x³ and at ,

We are interested in the gap between the curve and the straight line in the last microscopic view (where the change in x was dx=1/8.) We will call the amount we measure (in cm) in the microscopic view "epsilon," , greek "E" (for error). In this example, we want to know (1) How big is (in cm) in the microscopic view? (2) How big is the gap in original unmagnified coordinates?Remember that if we magnify by one million and see 1 cm on the microscopic image, we actually have an error of 10^-6 cm, one one millionth of the apparent error.

The actual distance from the dx-axis, where y=f[x]=f[3/4]=27/64, to the point on the curve above is given by the difference,

Figure CD-3.4: y=x³ and at , , magnification 8

Finally, we want to compare the difference between these vertical distances. Figure CD-3.5 shows the nonlinear and linear graphs at magnification 8 and shows a small segment connecting the linear graph to the nonlinear one above . The actual length of the vertical segment connecting the (dx,dy)-point on the tangent with the (x,y)-point on the curve is the difference between these two values

Figure CD-3.5: The magnified gap between y=f[x] and dy=mdx

The error magnified by 8 (as shown) measures

In the following exercises, draw the sketches accurately on good graph paper. Be careful about the method you use to calculate your results so you are connecting numerics, symbolics, and graphics. You can use the computers if you wish, but you will still need to do algebra.

If you magnify by , so that a segment of length appears to be unit size, and you observe another segment of apparent length , the actual length of the unmagnified segment is . The first exercise tries to help you understand this formula.

1. Let and . Draw a line with a scale of 1 unit = 1cm on the left half of your paper. Put dots at x=0, , and . (What is the difficulty when you draw in centimeter units?) On the right half of your paper, draw a line starting at 0 magnified by 100. How far in centimeters on the magnified picture is the dot for and ?

2. Your First Magnification
   
   • 1. Sketch the graph of y=x³ for using equal x and y scales with 1 cm = 1 unit.
   • 2. We want to focus our microscope over the x-value x=2/3. What is the corresponding y-value?
   • 3. Draw the local (dx,dy)-axes with its origin at the (x,y)-point, (x,y)=(2/3,8/27) on the same graph. The (dx,dy) origin lies at the center of your microscope. We will think of x as fixed and vary dx and dy.
   • 4. The straight line in local coordinates dy=(4/3)dx is tangent to y=x³ at x=2/3. Sketch this as a dotted line on the same graph.
   • 5. Magnify a portion of your graph by 10 and draw the microscopic view with the same scale you used for your original (x,y)-plot. This way (1/10) will appear unit size, that is, measure 1 cm on your graph paper.

     Your result from Exercise CD-3.3.2 should look something like the Figure CD-3.6. Compare your work to the computer animation in the program SecantGapZ.

     
     Figure CD-3.6: y=x³ and at ,
     
   • Measuring the Specific Gap
     Carefully draw the curve y=x³ between the values and on a scale where 1/10 of a unit is 1cm, that is, magnified by 10 if 1cm is the original unit. The curve y=x³ lies above your tangent line , where the local dx-dy axes lie at the center of the "microscope" magnifying the figure.
     • 1. What are the (x,y)-coordinates of the point where (dx,dy)=(0,0)?
     • 2. The x-coordinate of the point dx=1/10=0.1 is , because dx=0 corresponds to What is the y-coordinate on y=x³ when ? What is the change in y from the dx axis to the point on the curve above x=2/3+1/10?
     • 3. What is the change in dy from dy=0 to the point on dy=(4/3)dx above ?
     • 4. What we see in the microscope as on our centimeter scale is the magnified difference between the answer to Part 2 and the answer to Part 3, so its actual size is given by applying the formula from Exercise 1 to this difference. How much is it?

     • A Gap in Function Notation
       Let f[x]=x³ in the following computations, and verify that the function procedures produce the answers to the measurement of as in the examples and exercises above, especially Exercise CD-3.3.3. Basic units are measured in centimeter, whereas magnified units appear 10 times larger.
       • 1. When x=2/3 and the (dx,dy)-coordinates are centered on y=f[x] above this value of x, then the (x,y)-value of (dx,dy)=(0,0) is (x,f[x])=(2/3,f[2/3])=(2/3,??).
       • 2. On the sketch of the portion of y=f[x] between and with a scale of 1/10 unit = 1cm, the length of the vertical segment from the dx-axis to the (x,y)-point is 10 times the unit value of
         
         or simply,
         
         
       • 3. If we sketch the line dy=mdx, with m=4/3, the length of the vertical segment (in original units) between the dx-axis and the point above is
         
         If we magnify by so that appears unit size or specifically, 1/10 unit equals 1cm, then this distance measures
         
         
       • 4. The magnified gap measures
         
         
       • 5. The actual unmagnified size of the gap between the curve and its tangent approximation above is
         
         

Figure CD-3.7: Slight magnification of y=f[x]

Figure CD-3.7 is a sketch of a general function y=f[x] with 1unit=1cm. A pair of local coordinate (dx,dy)-axes is centered on the graph over a fixed point x. A line in local coordinates dy=mdx is shown in grey.

On the right, an image of these graphs is shown magnified by , so that the small number appears unit size. Since we magnify by an amount that makes appear 1 cm in size, if we measure a distance in the microscopic image, the actual size is really . (Check this formula intuitively when . We magnify by one million and see a gap of 0.3, for example, but it is really only a gap of 0.3/1000000.)

Problem CD-3.1 Symbolic Magnification for an Unknown y=f[x]

Explain the following statements about Figure CD-3.7:

• 1. The vertical distance from the dx-axis up to the curve is
  
  but the magnified view of this vertical segment measures
  
  
• 2. The vertical distance from the dx-axis up to the line dy=mdx above the point is
  
  but the magnified view of this vertical segment measures
  
  
• 3. The magnified gap measures
  
  but the actual change in the function as x moves to is
  
  

The condition for local linearity of a graph y=f[x] is that the magnified error gap between the curve and line is small, , when the magnification is large. In other words, if the local change in x, , then the MAGNIFIED change along the curve is -close to the change along the line. (The lowercase [small] Greek delta ,, indicates intuitively that the difference in x is a very small amount.)

When the gap is small for , the slope of the local linear approximation, f'[x], is called the "derivative." This is what we saw in the examples and exercises of the last section:

The condition of (uniform) "tangency" is expressed by the microscopic error formula

The approximation means that a microscopic view of a tiny piece of the graph y=f[x] looks the same as the linear graph on the scale of . This looks like

Figure CD-3.8: A Symbolic Microscope

When we say f'[x] is the derivative of f[x], we mean that this local approximation is valid, when , or as .

Chapter 5 is devoted to symbolic computations of the gap and symbolic ways to show that it becomes small when is small. This is easy to verify in the case of y=x³.

We expand

1. Check Your General Formula with y=x³
   When f[x]=x³, the microscopic gap is .
   • 1. Use this formula for to check the errors you measured in Exercise CD-3.3.3 and Problem 3.3 .
   • 2. We are interested in the size of the gap at x+1/100. How much is this gap when in a microscope of power 100 focused at the point x=2/3? How much is it really? Use your general formula to show that and (see Figure CD-3.9).
     
     Figure CD-3.9: y=x³ and at ,
     
   • Your General Formula with y=x⁴
     Let f[x]=x⁴ and show that the microscope equation
     
     becomes
     
     with .

   • There is only one possible line that can `fit' a graph y=f[x] when magnified. To see this, let near x=0, but suppose the approximation we try is dy=dx, or m=1, so the microscope equation is
     
     (see Figure CD-3.10).
     
     Figure CD-3.10: y=x/2 and dy=dx near (x,y)=(0,0), magnified 5

     Show that no matter what magnification, we always have above the point that appears to be one unit to away from the intersection. (Do this by writing f[x] and f'[x] explicitly in the microscope equation and solving for .)

Problem CD-3.2

The graph of the function

is actually two half lines meeting at the point (1,1).

Figure CD-3.11: near (x,y)=(1,1)

To the right, the line has slope , and to the left it has slope . There is no tangent line at this point because the "gap" does not go to zero. (Rules of differentiation from Chapter 6 applied to this formula give a formula that is not defined at x=1.)

We want to magnify this graph at (1,1) anyway and compare it with the "best" local linear approximation we can make. Since the slopes average out to slope 1, compute the gap between this function and the line dy=dx.

Begin with several numerical cases. Let x=1 and calculate in the equation

for . (ANS: )

Graph the function and the line dy=dx on the same axes for the scales in the previous part of the problem and show that the gap remains the same size at all these scales. Magnifying the graph never makes y=f[x] appear linear and never makes the gap between y=f[x] and dy=dx get smaller.

Figure CD-3.12: near (x,y)=(1,1), magnified 5