Rate of change from explicit formula

A question is this type if and only if it gives an explicit formula for a quantity (depth of water, height of bird, volume in tank) as a function of time and requires finding the rate of change (derivative) at a specific time or finding when a stationary value occurs.

2 questions · Moderate -0.8

1.07b Gradient as rate of change: dy/dx notation
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AQA C1 2013 January Q2
8 marks Moderate -0.8
2 A bird flies from a tree. At time \(t\) seconds, the bird's height, \(y\) metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
    1. Find the rate of change of height of the bird in metres per second when \(t = 1\).
    2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when \(t = 1\).
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) when \(t = 2\).
    2. Given that \(y\) has a stationary value when \(t = 2\), state whether this is a maximum value or a minimum value.
AQA C1 2011 June Q3
11 marks Moderate -0.8
3 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank after time \(t\) seconds is given by $$V = \frac { t ^ { 3 } } { 4 } - 3 t + 5$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
    1. Find the rate of change of volume, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 1\).
    2. Hence determine, with a reason, whether the volume is increasing or decreasing when \(t = 1\).
    1. Find the positive value of \(t\) for which \(V\) has a stationary value.
    2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\), and hence determine whether this stationary value is a maximum value or a minimum value.
      (3 marks)