Question Bank

Official NZQA past exam questions • AS91578 Differentiation

With video explanations by infinityplusone

15 questions

Achievement

Differentiate

y = 5\cos(3x)

.

Chain Rule Trigonometric Find Derivative

Achievement

Find the gradient of the normal to the function

y = (3x^2 - 5x)^2

at the point

(1,4)

.

*Show any derivatives that you need to find when solving this problem.*

Chain Rule Tangent Normal Polynomial Find Gradient Find Derivative

Merit

If

x = 2\sin t

and

y = \cos 2t

show that

\dfrac{dy}{dx} = -2\sin t

.

Parametric Chain Rule Trigonometric Find Derivative

Merit

Find the

x

-value at which the tangent to the function

y = \dfrac{4}{e^{2x-2}} + 8x

is parallel to the

x

-axis.

*Show any derivatives that you need to find when solving this problem.*

Chain Rule Tangent Normal Mixed Find Stationary

Excellence

What is the maximum volume of a cone if the slant length of the cone is 20 cm?

Loading diagram...

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

*Show any derivatives that you need to find when solving this problem.*

Optimisation Maxima Minima Chain Rule Mixed Optimisation Find Stationary

Achievement

Differentiate

f(x)=\dfrac{e^{4x}}{2x-1}

.

You do not need to simplify your answer.

Quotient Rule Chain Rule Exponential Rational Find Derivative

Achievement

Find the gradient of the curve defined by

y = 8\ln(3x - 2)

at the point where

x = 2

.

Show any derivatives that you need to find when solving this problem.

Chain Rule Logarithmic Find Gradient Find Derivative

Merit

The graph below shows the function

y = f(x)

.

Loading diagram...

For the function

f(x)

above:

(i) Find the value(s) for

x

that meet the following conditions:

1.

f(x)

is not differentiable: ______________________________

2.

f''(x) < 0

: ______________________________

3.

f(x)

is not defined: ______________________________

(ii) What is the value of

f(2)

? ______________________________

*State clearly if the value does not exist.*

(iii) What is the value of

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

? ______________________________

*State clearly if the value does not exist.*

Graph Properties Inflection Solve Inequality

Merit

The hourly cost of running an aeroplane depends on the speed at which it flies.

For a particular aeroplane this is given by the equation

C = 4v + \dfrac{1\ 000\ 000}{v},\ 200 \le v \le 800

where

C

is the hourly cost of running the aeroplane, in dollars per hour
and

v

is the airspeed of the aeroplane, in kilometres per hour.

Find the minimum hourly cost at which this aeroplane can be flown.

*Show any derivatives that you need to find when solving this problem.*

Optimisation Maxima Minima Mixed Optimisation Find Stationary

Excellence

A rectangle is drawn inside a right angled triangle, as shown in the diagram below.

Loading diagram...

Point B moves along the base of the triangle AC, beginning at point A, at a constant speed of 3 cm s

^{-1}

.

At what rate is the area of the rectangle changing when point B is 20 cm from point A?

*Show any derivatives that you need to find when solving this problem.*

Related Rates Chain Rule Mixed Related Rates

Achievement

Differentiate

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

.

Chain Rule Radical Find Derivative

Achievement

Find the value(s) of

x

for which the graph of the function

y = x + \dfrac{32}{x^2}

has stationary points.

*Show any derivatives that you need to find when solving this problem.*

Maxima Minima Mixed Find Stationary Find Derivative