Solutions
Solution:
Table
(a) The table can be extended for whole number values of $x$ up to $x = 10$
and the values of $2x^3 + 1$ remain larger than those for $2^x$:
| $x$ |
$2^x$ |
$2x^3+1$ |
| 6 |
64 |
433 |
| 7 |
128 |
687 |
| 8 |
256 |
1025 |
| 9 |
512 |
1459 |
| 10 |
1024 |
2001 |
(b) If the table is continued,
for all values of $x$ up to and including 11 the polynomial $2x^3 + 1$ takes
a larger value than the exponential $2^x$. But
$$
2^{12} > 2(12)^3 + 1.
$$
| $x$ |
$2^x$ |
$2x^3+1$ |
| 11 |
2048 |
2663 |
| 12 |
4096 |
3457 |
We know that the exponential $2^x$ will eventually exceed in value
the polynomial $2x^3 + 1$ because its base, 2, is larger than one and
an exponential functions grow faster, as the size of $x$ increases, than
any particular polynomial function. This is explained in greater
detail in the second solution below by examining quotients
of $2^x$ and $2x^3+1$ when evaluated at successive whole numbers.
Solution:
2. Abstract argument
The argument presented here does not find the smallest whole number (12) where
the value of $2^x$ first exceeds the value of $2x^3 + 1$ but rather explains
why there must be such a whole number. The argument would apply not only
to $2x^3+1$ but also to any other polynomial.
Each time the variable $x$ is increased by one unit, the exponential function
$2^x$ doubles:
$$
\frac{2^{x+1}}{2^x} = 2.
$$
For the polynomial function $2x^3 + 1$, an increase in $x$ by one unit
increases the value of the function by a factor of
$$
\frac{2(x+1)^3+1}{2x^3+1} = \frac{2x^3+6x^2+6x+7}{2x^3+1}.
$$
Unlike the exponential function, these growth factors for the polynomial
function depend on the value of $x$. Notice that as $x$ increases, the
expression
$$
\frac{2x^3+6x^2+6x+7}{2x^3+1}
$$
gets closer and closer to one (because for large positive values of $x$,
the terms $6x^2, 6x, 7,$ and $1$ influence the value of the quotient by
a small quantity). Thus, as $x$ is continually incremented by one unit,
the value of $2^x$ always doubles while value of $2x^3+1$ only increases
by a factor closer and closer to one, thereby allowing the exponential values to eventually
surpass the polynomial values.
