Problem:
Graphically, find the tangent line to the graph of the function

f(x) = x4

at the point (0.5, 0.0625) as the limit of secant lines.


Visualization:
Using Microcalc:

  1. Choose Beginning Calculus from the initial menu and press .
  2. Use the arrow keys to move to the menu item Secants and Tangents, press and .
  3. At the prompt F(x) =, enter the formula
    x^4
  4. and use the bounds
    x0 = -1.5 x1 = 1.5
    y0 = -1 y1 = 4
  5. Use the arrow keys to move to Key in Base Point and press .
  6. At the prompt a =, type in 0.5 .
  7. You will then see the point (0.5, 0.0625) on the graph of this function.
  8. Press and use the arrow keys to choose Choose Positive h and press and .
  9. You will then see the following:

  10. (*) Press and and will see:
         Point-slope equation of the secant:
    
           y = c(x - a) + b
    
         c = 1.500000
         a = 0.500000
         b = 0.062500, (h = 1.000000)
    
  11. Press and use the arrow keys to move to h <<-- h/2 and press twice.
  12. You will now see, superimposed on the previous graph, the secant line from (0.5, 0.0625) to (0.75, 0.3164). As you repeat these steps from (*) above, you will see the secant lines approaching the tangent line.