Let f be defined as follows: f(x) = x/3+x^2*sin(1/x) if x is not 0, f(x)=0 if x=0.

1. Is f continuous at x=0? Explain.
2. Graph f on your calculator, computer, or other suitable device making sure that the scale on the horizontal and vertical axis are the same. The default setting on the HP49 calculator is such that the scale on the x and y axis are the same. Center the graph at the origin and zoom in until you can estimate the slope of the tangent line at x=0 . When you zoom in you must also be careful to make sure that the scale on the x and y axis are the same. If the H-Factor and the V-Factor in the ZFACT menu of the HP49 are both the same then the ZIN function will keep the scales on the two axis the same. Make a rough sketch of a couple of the pictures you get. (If you have access to a printing device and know how to use it you may just print out a couple of pictures.) What is your estimate of the slope of the tangent line to the graph of f at (0,0)?
3. Find f'(0). Note that the differentiation rules do not apply at this point since the function is defined differently at 0 than at any nearby point.
4. Find f'(x) when x is not 0. Note the differentiation rules would apply here.
5. Is f'(x) continuous at x=0? Explain.
6. Is f'(x) differentiable at x=0? Explain.