Problems

1. 1. Sketch the graph of the natural exponential function `y=e^t`.
   
   
   2. Use properties of exponents to show that, for every number `t`,

   `e^-t=1/e^t`.

   3. On the same axes as you used in part (a), sketch the graph of `y=e^-t`.
   
   
   4. Explain the symmetry the two graphs exhibit.
   
   
   5. Now sketch the graph of `y=e^t+e^-t`. Is this function even, odd, or neither? Confirm your answer algebraically.
2. 1. Find the derivative of `y=e^t+e^-t`.
   
   
   2. Graph the derivative of `y`.
   
   
   3. Is the derivative even, odd, or neither? Confirm your answer algebraically.
3. 1. Sketch the graph of `y=e^t-e^-t`.
   
   
   2. Find the derivative of `y=e^t-e^-t`.
   
   
   3. Graph the derivative of `y`.
   
   
   4. Is the derivative even, odd, or neither? Confirm your answer algebraically.
4. You can use your calculator to determine the slope at `t=0` on the graph of `ftext[(]t text[)]=b^t` (for any particular value of `b`) by a procedure that does not depend on graphing. This slope is just the limiting value of the difference quotients

`(ftext[(]0+Delta t text[)]-ftext[(]0text[)])/(Delta t)`

as `Delta t` shrinks to zero. If we replace `ftext[(]t text[)]` by `b^t`, we have

`(Delta f)/(Delta t)=(b^(Delta t)-b^0)/(Delta t)=(b^(Delta t)-1)/(Delta t)`

so

`Ltext[(]btext[)]`  is the limiting value as  `Delta t ->0`  of  `(b^(Delta t)-1)/(Delta t)`.

Table E1   Experimental
determination of the
limiting value for L(2)

`Delta t`	`2^(Delta t)`	`(2^(Delta t)-1)/(Delta t)`
0.01	1.00695555	0.695555
0.001
0.0001
0.00001

1. Estimate this limiting value for `b=2` by filling in the blanks in a table like Table E1.


2. From this calculation, how many digits in `Ltext[(]2text[)]` are you sure about?


3. How does this fit with your calculations in Activities 2 and 3?


5. Repeat Problem 4 with `Delta t=-0.01,`` -0.001, -0.0001, -0.00001`. Do you get more evidence, less evidence, or about the same evidence about the value of `Ltext[(]2text[)]`? Do you get evidence for a different value of `Ltext[(]2text[)]`?
6. Repeat Problem 4 for `b=3`. What do you conclude about `Ltext[(]3text[)]`?
7. Repeat Problem 4 for `b=4`. What do you conclude about `Ltext[(]4text[)]`?
8. What happens if you extend Table E1 several lines further? Try it and see. Does this alter your opinion about the correct value for `Ltext[(]2text[)]`? Why or why not?