NCEAce - NCEA Calculus Question Bank

A car is being pulled along by a rope attached to the tow-bar at the back of the car.

The rope passes through a pulley, the top of which is 3 m further from the ground than the tow-bar.

The pulley is

x

m horizontally from the tow-bar, as shown in the diagram above.

The rope is being winched in at a speed of

0.6

m s

^{-1}

.

The wheels of the car remain in contact with the ground.

At what speed is the car moving when the length of the rope,

L

, between the tow-bar and the pulley is

5.4

m?

*You must use calculus and show any derivatives that you need to find when solving this problem.*

2018 ONE(e)

Excellence

A curve is defined by the parametric equations

x = t^3 + 1

y = t^2 + 1

Show that

\dfrac{\dfrac{d^2y}{dx^2}}{\left(\dfrac{dy}{dx}\right)^4}

is a constant.

Parametric Chain Rule Proof Polynomial Proof Find Derivative

2018 TWO(a)

Achievement

Differentiate

y = 3\sqrt{x} + \mathrm{cosec}\,5x.

Chain Rule Radical Reciprocal Trig Find Derivative

2018 TWO(b)

Achievement

A particle is travelling in a straight line. The distance, in metres, travelled by the particle may
be modelled by the function

s(t)=\ln(3t^2+3t+1)

t\geq 0

where

t

is time measured in seconds.

Find the velocity of this particle after 2 seconds.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Chain Rule Logarithmic Polynomial Find Rate

2018 TWO(c)

Merit

The diagram below shows the graph of the function

y = f(x)

.

Loading diagram...

For the function above:

(i) What is the value of

f(1)

?
State clearly if the value does not exist.

(ii) For what value(s) of

x

does the function

f(x)

not have a limit?

(iii) Find all the value(s) of

x

that meet the following conditions:

(1)

f'(x) > 0

:

(2)

f'(x) = 0

and

f''(x) < 0

:

(3)

f(x)

is continuous but not differentiable:

Graph Properties Maxima Minima Mixed Solve Inequality Classify Stationary

2018 TWO(d)

Merit

If

y = e^x(2x^2 - x - 1)

, find the value(s) of

x

for which

\dfrac{dy}{dx} = 0

.

You must use calculus and show any derivatives that you need to find when solving this
problem.

Product Rule Exponential Polynomial Find Stationary

2018 TWO(e)

Excellence

A water tank is in the shape of an inverted right-circular cone.

The height of the cone is 200 cm and the radius of the cone is 80 cm.

Loading diagram...

The tank is being filled with water at a rate of

150\ \mathrm{cm}^3

per second.

At what rate will the surface area of the water in the tank be increasing when the depth of
water in the tank is 125 cm?

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

2018 THREE(a)

Achievement

Differentiate

y = \dfrac{e^{2x}}{x^2 + 1}

.

You do not need to simplify your answer.

Quotient Rule Chain Rule Exponential Rational Find Derivative

2018 THREE(b)

Achievement

A curve is defined parametrically by the parametric equations

x = 5e^{2t}

y = 2e^{5t}

Find the gradient of the tangent to this curve at the point where

t = 0

.
*You must use calculus and show any derivatives that you need to find when solving this problem.*

Parametric Chain Rule Tangent Normal Exponential Find Gradient