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 \(f(t)=b^t\) (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

\(\frac{f(0+\Delta t)-f(0)}{\Delta t}\)

as \(\Delta t\) shrinks to zero. If we replace \(f(t)\) by \(b^t\), we have

\(\frac{\Delta f}{\Delta t}=\frac{b^{\Delta t}-b^0}{\Delta t}=\frac{b^{\Delta t}-1}{\Delta t}\)

so \(L(b)\)  is the limiting value as  \(\Delta t \rightarrow 0\)  of  \((b^{\Delta t}-1)/(\Delta t)\).

Table P1   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 P1.


2. From this calculation, how many digits in \(L(2)\) 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 \(L(2)\)? Do you get evidence for a different value of \(L(2)\)?
6. Repeat Problem 4 for \(b=3\). What do you conclude about \(L(3)\)?
7. Repeat Problem 4 for \(b=4\). What do you conclude about \(L(4)\)?
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 \(L(2)\)? Why or why not?