## Answers for Chapter 11 Sample Problems

**4. Compound interest**

What would the amount you have in
the bank after 1 year, putting in $1, at 6%, compounded quarterly
(**4 times per year**)?(1 +
.06/4)^{4} = $1.061363551

What would the amount you have in
the bank after 1 year, putting in $1, at 6%, compounded monthly (**12 times per year**)?(1 +
.06/12)^{12} = $1.061677812

What would the amount you have in
the bank after 1 year, putting in $1, at 6%, compounded daily (**365 times per year**)?(1 +
.06/365)^{365} = $1.061831311

What would the amount you have in
the bank after 1 year, putting in $1, at 6%, compounded **10,0000 times per year**?(1 +
.06/10,000)^{10,000} = $1.061836355

What would the amount you have in
the bank after 1 year, putting in $1, at 6%, compounded **continuously,
(an infinite number of times per year**)?

We end up with an infinite sequence

1.12, 1.061363551, 1.061677812, 1.061831311, 1.061836355, 1.0618363547...
which approaches the limit_{n->inf}(1 + .06/n)^{n} =
1.0618363547...
= **e**^{.06}
WOW!

Kirsten, now 15, started with me at age 5. On Sept. 10, 1996 she was
working on this
problem (using .07 instead of .06) for her calculus class. We graphed (1 + .07/x)^{x} in *Derive*, but
couldn't import it into a paint program and then to a gif file. So we
graphed it in *Mathematica* and got this:

Here are some of the
things we explored in *Mathematica* and *Derive* with some
surprising and exciting results:

(1 + .07/-0.1)^{-0.1} =
1.12794, a real number. Check the graph.

(1 + .07/0)^{0} = 1 This was hard to believe, but
the graph bears it out!

(1 +
.07/0.001)^{0.001} = 1.00427

(1 +
.07/-0.07)^{-0.07} *Mathematica* gives:
"Power::infy: Infinite expression 0.^{-0.07} encountered. ComplexInfinity." as the
answer. *Derive* gives " + or - infinity" as the answer.

(1 + .07/-0.01)^{-0.01} =
0.981757 - 0.030853**i** ; which makes sense when
you look at the graph.

limit_{x->0}(1 + .07/x)^{x} = 1

limit_{x->0,Direction->-0}(1 + .07/x)^{x}
*Mathematica* gives: "Power::infy: Infinite expression 1/0
encountered. Infinity::indet: Indeterminate expression ComplexInfinity"
^{0}
encountered.

Indeterminate"

limit_{x->infinity}(1 + .07/x)^{x} = 1.072508181254216
and e^{.07} = 1.072508181254216

The first 6 terms of the binomial expansion are:

**9.** Ian looked for patterns always, and found
^{} = **e** .

How about that.

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