NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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.

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

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)=65x.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems