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Calculus 131 A Donut-hole Activity on Estimating Derivatives
Tom Linton, February 25, 2000
The goal today will be to draw a graph of the derivative, or rate function, f '(x) for the function f(x) plotted below. However, we will exploit several techniques (graphical, numerical and algebraic) for creating this plot, which is NOT a typical manner in which to draw a graph of the derivative from a graph of the function, but is good practice. Here is a graph of the function f(x) and a set of axis (labeled with appropriate units) for creating our graph of f '(x).

  1. Start by plotting the easy points (where f '(x) = 0) on the graph of the derivative.












  2. Draw in tangent lines at x = 3 and x = 5 (on the graph of f(x) and make them long) and use the slopes of these tangent lines (pick 2 nice points on them to calculate the slopes) to plot derivative points for x = 3 and 5 on the second set of axes. Use the space below to calculate your slopes and indicate the points you use for these slopes on the graph of f(x).

    3.  

     
     
     
     
     
     
     
     
     
     
     

  3. There's an inflection point of f(x) at x = 1.55 (roughly). Since f(x) switches from concave up to concave down at x = 1.55, this is a hilltop on the graph of f '(x) (the function f(x) is increasing fastest at x = 1.55). The table below gives (rounded, but fairly accurate) values of the function f(x) near x = 1.55.

    4.  

     

    x f(x)
    1.53 -4.494
    1.54 -4.412
    1.55 -4.329
    1.56 -4.246
    1.57 -4.163
    1. By looking at the graph of f(x) above, is f(x) increasing or decreasing near x = 1.55?

      2.  

       
       
       
       
       

    2. Now look at the table of values. How does the table show that f(x) is increasing near x = 1.55?

      3.  

       
       
       
       
       
       
       

    3. Use the table of values and centered difference quotients to estimate f '(1.54), f '(1.55) and f '(1.56) (do not plot these values yet however). If x = 1.55 is a hilltop on the graph of f '(x), then f '(1.55) should be larger than the derivatives at the other two neighboring points. Is this the case?

      4.  

       
       
       
       
       
       
       
       
       
       
       
       
       
       
       

    4. Plot the point (1.55, f '(1.55) ) on the graph of f '(x). This should be a hilltop on the graph we are creating.

      5.  
  4. Normally, you wouldn't have a formula for f(x), but since the calculator does a nice job of estimating f '(a) when we have a formula, I'll let the cat out of the bag. The graph above shows the function f(x) = (4 - 42x + 19x2)*0.44x . Use your calculator's numeric derivative function (nDeriv or der1) to estimate f '(x) at x = 0, 0.5 and 7. The value of f '(0) is too negative to fit on the graph, however the true value (to check that you are calculating your estimates correctly) is f '(0) = -45.2839. Plot the other two points on the graph of f '(x). You now have enough points on the graph of f '(x) to connect the dots, so long as you look at the graph of f(x) and consider how f(x) increases or decreases between the points. Complete your graph of f '(x) and then see Tom to check your answer.