The population $f\left(x\right)$`f`(`x`) of bacteria present in some food $x$`x` minutes after it was left out of the fridge is given by $f\left(x\right)=x^2+3$`f`(`x`)=`x`2+3.

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a

What is the population of bacteria $1$1 minute after leaving the food out of the fridge?

b

What is the population of bacteria $2$2 minutes after leaving the food out of the fridge?

c

What is the average rate of increase of bacteria between $1$1 and $2$2 minutes?

d

We now want to generalise the average rate of change between any two times.

If the number of bacteria at $x=a$`x`=`a` minutes is $f\left(a\right)$`f`(`a`) and the number of bacteria at $x=a+h$`x`=`a`+`h` minutes is $f\left(a+h\right)$`f`(`a`+`h`), fill in the gaps to form an expression for the average rate of change over this interval of time.

Total change in the quantity | $=$= | $\editable{}-f\left(a\right)$−f(a) |

Total change in time | $=$= | $\editable{}-a$−a |

Average rate of change | $=$= | $\frac{\editable{}-\editable{}}{\editable{}}$− |

Easy

Approx 4 minutes

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