Explorations With Differentiability

1. Case 1:
   
   1. Graph f(x) = x^^(2/3) on your calculator.
   2. Zoom-in several times.
   3. Can you draw a unique tangent line to the graph of f at x = 0? If so, what is its slope?
   4. Is f(x) differentiable at x = 0? Support your answer.
   5. Try a numerical approach. WRITE A PROGRAM to calculate the difference quotient given values for x and h. Use the program to construct a chart calculating the difference quotient using x = 0, and h = 0.01, 0.0001, 0.0000001. What do you observe about f '(0)?
   6. Try the dy/dx key (under 2nd CALC) on your calculator if you have a TI-82. Is this result reliable?
   7. How would you summarize your findings about differentiability based on the example above?

   
   

2. Case 2:
   
   1. Graph f(x) = 2^x on your calculator.
   2. Zoom-in several times.
   3. Can you draw a unique tangent line to the graph of f at x = 0? If so, what is its slope?
   4. Is f(x) differentiable at x = 0? Support your answer.
   5. Try a numerical approach. WRITE A PROGRAM to calculate the difference quotient given values for x and h. Use the program to construct a chart calculating the difference quotient using x = 0, and h = 0.01, 0.0001, 0.0000001. What do you observe about f '(0)?
   6. Try the dy/dx key (under 2nd CALC) on your calculator if you have a TI-82. Is this result reliable?
   7. How would you summarize your findings about differentiability based on the example above?

   
   

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