New Zealand
Level 8 - NCEA Level 3

# Applications of Differentiation

## Interactive practice questions

The position (in metres) of an object along a straight line after $t$t seconds is modelled by $x\left(t\right)=6t^2$x(t)=6t2.

a

State the velocity $v\left(t\right)$v(t) of the particle at time $t$t.

b

Which of the following represent the velocity of the particle after $4$4 seconds? Select all that apply.

$x'\left(4\right)$x(4)

A

$v'\left(4\right)$v(4)

B

$x\left(4\right)$x(4)

C

$v\left(4\right)$v(4)

D

$x'\left(4\right)$x(4)

A

$v'\left(4\right)$v(4)

B

$x\left(4\right)$x(4)

C

$v\left(4\right)$v(4)

D
c

Hence find the velocity of the particle after $4$4 seconds.

Easy
Approx 2 minutes

The position (in metres) of an object along a straight line after $t$t seconds is modelled by $x\left(t\right)=3t^2+5t+2$x(t)=3t2+5t+2.

We want to find the velocity of the object after $4$4 seconds.

The position (in metres) of an object along a straight line after $t$t seconds is modelled by $x\left(t\right)=18\sqrt{t}$x(t)=18t.

In a new mining town, coal needs to be transported from the mine at point $M$M to port at point $P$P. There is an existing train line running $100$100 km from $Q$Q to $P$P such that $M$M is $25$25 km from $Q$Q. A direct road is to be built that connects point $M$M to some part of the train track at point $R$R, a distance of $x$x km from $Q$Q.

### 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