NCEAce - NCEA Calculus Question Bank

2016 ONE(a)

Achievement

Differentiate

y = 1 + x - \frac{1}{x} + \frac{1}{x^2}

.

Quotient Rule Mixed Find Derivative

2016 ONE(b)

Achievement

The height of the tide at a particular beach today is given by the function

h(t) = 0.8\sin\left(\frac{4\pi}{25}t+\frac{\pi}{2}\right)

where

h

is the height of water, in metres, relative to the mean sea level
and

t

is the time in hours after midnight.

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At what rate was the height of the tide changing at that beach at 9.00 a.m. today?

Chain Rule Trigonometric Find Rate

2016 ONE(c)

Merit

A curve is defined by the parametric equations

x = 2\cos 2t

and

y = \tan^2 t

.

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

t = \dfrac{\pi}{4}

.

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

Parametric Chain Rule Tangent Normal Trigonometric Find Gradient

2016 ONE(d)

Merit

The tangents to the curve

y = \frac{1}{4}(x - 2)^2

at points P and Q are perpendicular.

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Q is the point

(6, 4)

.

What is the

x

-coordinate of point P?
*You must use calculus and show any derivatives that you need to find when solving this problem.*

Tangent Normal Chain Rule Polynomial Find Gradient

2016 ONE(e)

Excellence

A curve is defined by the function

f(x)=e^{-(x-k)^2}

.

Find, in terms of

k

, the

x

-coordinate(s) for which

f''(x)=0

.

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

Inflection Chain Rule Product Rule Exponential Find Inflection

2016 TWO(a)

Achievement

Differentiate

f(x)=x\ln(3x-1)

.

Product Rule Chain Rule Mixed Logarithmic Find Derivative

2016 TWO(b)

Achievement

Find the gradient of the tangent to the function

y = \sqrt{2x - 1}

at the point

(5, 3)

.

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

Chain Rule Tangent Normal Radical Find Gradient Find Derivative

2016 TWO(c)

Merit

The graph below shows the function

y = f(x)

.

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For the function

y = f(x)

above:

(i) Find the value(s) of

x

that meet the following conditions:

1.

f

is not continuous:

2.

f

is not differentiable:

3.

f'(x) = 0

:

4.

f''(x) < 0

:

(ii) What is the value of

\lim_{x\to-1} f(x)

?
State clearly if the value of the limit does not exist.

Graph Properties Maxima Minima Inflection Mixed Find Stationary Solve Inequality

2016 TWO(d)

Merit

A large spherical helium balloon is being inflated at a constant rate of

4800

cm

^3

s

^{-1}

.

At what rate is the radius of the balloon increasing when the volume of the balloon is

288\,000\pi

cm

^3

?

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

2016 TWO(e)

Excellence

A cone of height

h

and radius

r

is inscribed, as shown, inside a sphere of radius

6

cm.

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The base of the cone is

s

cm below the

x

-axis.

Find the value of

s

which maximises the volume of the cone.
*You must use calculus and show any derivatives that you need to find when solving this problem.*

*You do not need to prove that the volume you have found is a maximum.*

Optimisation Maxima Minima Polynomial Optimisation Find Stationary

2016 THREE(a)

Achievement

Differentiate

f(x)=\sqrt[4]{3x+2}

.

Chain Rule Radical Find Derivative

2016 THREE(b)

Achievement

Find the

x

-value at which a tangent to the curve

y = 6x - e^{3x}

is parallel to the

x

-axis.

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

Chain Rule Tangent Normal Mixed Find Stationary