Difference Quotients
Interactive practice questions
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)=x2+3.
The graph of the function has been provided.
What is the population of bacteria $1$1 minute after leaving the food out of the fridge?
What is the population of bacteria $2$2 minutes after leaving the food out of the fridge?
What is the average rate of increase of bacteria between $1$1 and $2$2 minutes?
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 | $=$= | $\frac{\editable{}-\editable{}}{\editable{}}$− |
Consider the function $f\left(x\right)$f(x) graphed below. The points $A\left(a,f\left(a\right)\right)$A(a,f(a)) and $B\left(a+h,f\left(a+h\right)\right)$B(a+h,f(a+h)) lie on the graph.
Consider the function $f\left(x\right)=3x+2$f(x)=3x+2.
Consider the function $f\left(x\right)=6-5x$f(x)=6−5x.