1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),

1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
3. find the rate at which the area of the stain is increasing when \(t = 15\).

6

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
   2. \(y = x ^ { 2 } \tan x\).
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
   2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 2 x ^ { 2 } - 5 x + 1 \right) ^ { 20 }\);
   2. \(y = x \cos x\).
2. Given that $$y = \frac { x ^ { 3 } } { x - 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k x ^ { 2 } ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$$ where \(k\) is a positive integer.

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } - 1 \right) ^ { 6 }\).
2. A curve has equation \(y = x \ln x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln x\) at the point on the curve where \(x = \mathrm { e }\).

3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when $$y = \mathrm { e } ^ { 3 x } + \ln x$$
2. 1. Given that \(u = \frac { \sin x } { 1 + \cos x }\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 1 } { 1 + \cos x }\).
   2. Hence show that if \(y = \ln \left( \frac { \sin x } { 1 + \cos x } \right)\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } x\).

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \sin 2 x\).
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 2 } - 6 \right) ^ { 4 }\).
   2. Hence, or otherwise, find \(\int x \left( x ^ { 2 } - 6 \right) ^ { 3 } \mathrm {~d} x\).

2

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = ( 3 x - 1 ) ^ { 10 }\).
2. Use the substitution \(u = 2 x + 1\) to find \(\int x ( 2 x + 1 ) ^ { 8 } \mathrm {~d} x\), giving your answer in terms of \(x\).

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:

1. \(y = ( 3 x + 1 ) ^ { 5 }\);
2. \(y = \ln ( 3 x + 1 )\);
3. \(y = ( 3 x + 1 ) ^ { 5 } \ln ( 3 x + 1 )\).

6 The diagram shows the curve with equation \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-5_483_611_402_717}

1. Find the gradient of the curve \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) at the point where \(x = \ln 2\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 2 } \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The shaded region \(R\) is bounded by the curve, the lines \(x = 0 , x = 2\) and the \(x\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2

1. Given that \(y = x ^ { 4 } \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. Find the gradient of the curve with equation \(y = \frac { x ^ { 2 } } { x - 1 }\) at the point where \(x = 3\).
   (3 marks)

3

1. 1. Differentiate \(\left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }\) with respect to \(x\).
   2. Given that \(y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 0\).
2. A curve has equation \(y = \frac { 4 x - 3 } { x ^ { 2 } + 1 }\). Use the quotient rule to find the \(x\)-coordinates of the stationary points of the curve.
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-06_1855_1709_852_153}

1

1. Given that \(y = ( 4 x + 1 ) ^ { 3 } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that \(y = \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p x } { \left( 3 x ^ { 2 } + 4 \right) ^ { 2 } }\), where \(p\) is a constant.
3. Given that \(y = \ln \left( \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).

7

1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
   [0pt] [6 marks]

4. (a) Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\sqrt { 1 - \cos x }\)
2. \(x ^ { 3 } \ln x\) (b) Given that $$x = \frac { y + 1 } { 3 - 2 y } ,$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

3. Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\quad \ln ( \cos x )\)
2. \(x ^ { 2 } \sin 3 x\)
3. \(\frac { 6 } { \sqrt { 2 x - 7 } }\)

1. (a) Given that

$$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$ (b) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, \(N\), in the population \(t\) weeks after the start of the investigation. Use this model to answer the following questions.

1. 1. Find the size of the population at the start of the investigation.
   2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
   3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form \(t = r \ln s\), where \(r\) and \(s\) are integers.
2. 1. Show that the rate of growth, \(\frac { \mathrm { d } N } { \mathrm {~d} t }\), is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
   2. The maximum rate of growth occurs after \(T\) weeks. Find the value of \(T\).

2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .

6

1. Given that $$x = \ln ( \sec t + \tan t ) - \sin t$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
2. A curve is given parametrically by the equations $$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$ The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\). Show that \(s = \ln p\), where \(p\) is an integer.

5 The function f , where \(\mathrm { f } ( x ) = \sec x\), has domain \(0 \leqslant x < \frac { \pi } { 2 }\) and has inverse function \(\mathrm { f } ^ { - 1 }\), where \(\mathrm { f } ^ { - 1 } ( x ) = \sec ^ { - 1 } x\).

1. Show that $$\sec ^ { - 1 } x = \cos ^ { - 1 } \frac { 1 } { x }$$
2. Hence show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 4 } - x ^ { 2 } } }$$

7

1. Given that \(y = \tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right)\) and \(x \neq 1\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
   [0pt] [4 marks]
2. Hence, given that \(x < 1\), show that \(\tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right) - \tan ^ { - 1 } x = \frac { \pi } { 4 }\).
   [0pt] [3 marks]

1. A particle is moving along the \(x\)-axis. At time \(t\) seconds the particle is \(x\) metres from the origin, \(O\), and its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by

$$v = \frac { 24 } { 4 x + 9 }$$

1. Find, in terms of \(x\), an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
2. At \(t = T\) the acceleration of the particle is \(- \frac { 4 } { 3 } \mathrm {~ms} ^ { - 2 }\).
   1. Determine the value of \(x\) when \(t = T\).
   2. Given that \(x = - 2\) when \(t = 0\), find an expression for \(t\) in terms of \(x\) and hence find the value of \(T\).

1. The curve \(C\) has equation

$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\) There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.

1. Show that the range of possible values for \(p\) is $$2 < p < 4$$
2. Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
   The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres.
   Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
   Given that the pond is initially empty,
3. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
4. State a limitation of the model.

1. (a) Given that

$$y = \arcsin x \quad - 1 \leqslant x \leqslant 1$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$$ (b) $$\mathrm { f } ( x ) = \arcsin \left( \mathrm { e } ^ { x } \right) \quad x \leqslant 0$$ Prove that \(\mathrm { f } ( x )\) has no stationary points.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$ Figure 1 also shows the tangent to \(C\) at the point \(A\).
This tangent is perpendicular to the initial line.

1. Use differentiation to prove that the polar coordinates of \(A\) are \(\left( 2 , \frac { \pi } { 4 } \right)\) The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the tangent at \(A\) and the initial line.
2. Use calculus to show that the exact area of \(R\) is \(\frac { 1 } { 2 } ( 1 - \ln 2 )\)