Consider the applet above. The main function f(x) is in blue, the tangent line is in green, and the derivative function f '(x) is graphed in red.
1. Identify the similarities between functions of form xn, such as appear in the applet XToN, and of form ax, such as appear in the applet below. It might help to consider specific examples such as x3 & 3x or x(1/2), & (1/2)x.
2. Identify at least 3 differences between functions of form xn and of form ax. Again, it might help to consider specific examples.
3. Set f(x)=2x using the slider on the right side of the applet. Assume for the moment that the derivative of f(x)=2x is f '(x)=x 2x-1
• a. Given the assumption, find the derivative of f(x)=2x at x=-1, 0, and 1.
• b. Using the applet, estimate the actual derivative of f(x)=2x at x=-1, 0, and 1.
• c. What do (a & b) tell us about our assumption that the derivative of f(x)=2x is f '(x)=x 2x-1? Why?
• d. Since we cannot use the Power Rule to take the derivative of f(x)=ax, how can we find the derivative?
4. Set f(x)=2x using the slider on the right side of the applet. It appears that the derivative of f(x)=2x is just 2x times some constant. Assuming this, estimate the constant. (Hint: your answers to 3b may be helpful.)
5. Set f(x)=3.7x using the slider on the right side of the applet. It appears that the derivative of f(x)=3.7x is just 3.7x times some constant. Assuming this, estimate the constant.
6. Is there any number a so that the derivative of f(x)=ax is just ax (that is, the constant in a question similar to (4 or 5) is just 1)? If so, estimate a. If not, explain why not.