NCEAce

Question Bank

Official NZQA past exam questions • AS91578 Differentiation

With video explanations by infinityplusone

15 questions
2020 ONE(a)
Achievement
Differentiate y=(3x−x2)5y = (3x - x^2)^5.

*You do not need to simplify your answer.*
Chain RulePolynomialFind Derivative
2020 ONE(b)
Achievement
Find the gradient of the tangent to the curve y=3sin⁡2x+cos⁡2xy = 3\sin 2x + \cos 2x at the point where x=π4x = \dfrac{\pi}{4}.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Chain RuleTangent NormalTrigonometricFind GradientFind Derivative
2020 ONE(c)
Merit
Find the value of xx for which the graph of the function y=x1+ln⁡xy = \dfrac{x}{1+\ln x} has a stationary point.

*You must use calculus and show any derivatives that you need to find when solving this problem.*
Quotient RuleMaxima MinimaRationalLogarithmicFind Stationary
2020 ONE(d)
Merit
A curve has the equation y=x2cos⁡xy = x^2 \cos x.

Show that the equation of the tangent to the curve at the point (π,−π2)(\pi, -\pi^2) is

y+2πx=π2y + 2\pi x = \pi^2

*You must use calculus and show any derivatives that you need to find when solving this problem.*
Product RuleTangent NormalProofMixedEquation TangentFind Gradient
2020 ONE(e)
Excellence
A cylinder of height hh and radius rr is inscribed, as shown to the
right, inside a sphere of radius 20 cm.
Loading diagram...
Find the maximum possible volume of the cylinder.

*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.*
OptimisationMaxima MinimaPolynomialOptimisation
2020 TWO(a)
Achievement
Differentiate y=tan⁡xx3y = \dfrac{\tan x}{x^3}.

*You do not need to simplify your answer.*
Quotient RuleMixedFind Derivative
2020 TWO(b)
Achievement
The value of a car is modelled by the formula

V=17 000e−0.25t+2000e−0.5t+500V = 17\,000e^{-0.25t} + 2000e^{-0.5t} + 500 for 0≤t≤200 \le t \le 20

where VV is the value of the car in dollars (\),and), and t$ is the age of the car in years.

Calculate the rate at which the value of the car is changing when it is 8 years old.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Chain RuleExponentialFind RateFind Derivative
2020 TWO(c)
Merit
Find the xx-coordinates of any stationary points on the graph of the function

f(x)=(2x−3)ex2+kf(x) = (2x - 3)e^{x^2 + k}

You must use calculus and show any derivatives that you need to find when solving this
problem.
Product RuleChain RuleMixedExponentialFind Stationary
2020 TWO(d)
Excellence
A rocket is fired vertically upwards. Its height above the launch point is given by the formula
h(t)=4.8t2h(t) = 4.8t^2, where hh is the height in metres, and tt is the time in seconds from firing.
Loading diagram...
An observer at point A is watching the rocket. She is at the same level as the launch point of
the rocket, and 500 m from the launch point.

Find the rate at which the angle of elevation at A of the rocket is increasing when the rocket is
480 m above the launch point.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Related RatesImplicitChain RuleMixedRelated RatesFind Rate
2020 TWO(e)
Excellence
A curve is defined by the parametric equations x=ln⁡(t)x = \ln(t) and y=6t3y = 6t^3 where t>0t > 0.

The point P lies on the curve, and at point P, d2ydx2=2\dfrac{d^2 y}{dx^2} = 2.

Find the exact coordinates of point P.

*You must use calculus and show any derivatives that you need to find when solving this problem.*
ParametricChain RuleLogarithmicPolynomialFind Derivative
2020 THREE(a)
Achievement
Differentiate y=3ln⁡(x2−1)y = 3\ln (x^2 - 1).

*You do not need to simplify your answer.*
Chain RuleLogarithmicPolynomialFind Derivative
2020 THREE(b)
Achievement
For what value(s) of xx does the tangent to the graph of the function

f(x)=2x−2x, x>0f(x)=2x-2\sqrt{x},\ x>0, have a gradient of 1?

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