10.11 Comparing exponential and quadratic functions
Interactive practice questions
Consider the functions $f\left(x\right)=2^x$f(x)=2x and $g\left(x\right)=2x^2$g(x)=2x2, for $x\ge0$x≥0.
Complete the table of values for each function.
| $x$x | $f\left(x\right)$f(x) | Increase in $f\left(x\right)$f(x) | $g\left(x\right)$g(x) | Increase in $g\left(x\right)$g(x) |
|---|---|---|---|---|
| $0$0 | $\editable{}$ | $\editable{}$ | ||
| $1$1 | $\editable{}$ | $1$1 | $\editable{}$ | $2$2 |
| $2$2 | $\editable{}$ | $2$2 | $\editable{}$ | $6$6 |
| $3$3 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
| $4$4 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
| $5$5 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
| $6$6 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
| $7$7 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Which of the following statements is true?
The exponential function increases more rapidly at first, but then the quadratic function starts to increase more rapidly.
For $x\ge0$x≥0, the exponential function increases more rapidly than the quadratic function.
The quadratic function increases more rapidly at first, but then the exponential function starts to increase more rapidly.
For $x\ge0$x≥0, the quadratic function increases more rapidly than the exponential function.
The graph of the exponential function $P$P, given by $y=-4^x$y=−4x is shown below.
Mint Corporation’s operations are such that the total amount mined by the $n$nth week of operations is given by $M=100\left(1.15\right)^{n-1}$M=100(1.15)n−1.
Crest Corporation’s operations are such that the total amount mined by the $n$nth week is given by the equation $C=100n^2$C=100n2.
To accommodate for its distributing population, a country creates a new city and immediately relocates $400000$400000 of its citizens there. The city’s land is allocated such that it can immediately produce enough food for $600000$600000 people in the first year. The table shows the functions that can be used to predict the city's population ($P$P) and the number of people who can be fed ($Q$Q), after $t$t years.