1.07e Second derivative: as rate of change of gradient

13 The curve with equation \(\mathrm { y } = \mathrm { px } + \frac { 8 } { \mathrm { x } ^ { 2 } } + \mathrm { q }\), where \(p\) and \(q\) are constants, has a stationary point at \(( 2,7 )\).

1. Determine the values of \(p\) and \(q\).
2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
3. Hence determine the nature of the stationary point at (2, 7).

12 A function is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - x\).

1. By considering \(\frac { f ( x + h ) - f ( x ) } { h }\), show from first principles that \(f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1\).
2. Sketch the gradient function \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that the curve \(y = f ( x )\) has a single point of inflection which is not a stationary point.

4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Determine the coordinates of the stationary points on the curve.
2. Determine the nature of each stationary point.
3. Write down the equation of the vertical asymptote.
4. Deduce the set of values of \(x\) for which the curve is concave upwards.

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 \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
2. 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 \(t = 1\).
3. 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.

7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
   2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
   3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
2. The point \(P ( 1,8 )\) lies on the curve.
   1. Show that the gradient of the curve at the point \(P\) is 12 .
   2. Find an equation of the normal to the curve at \(P\).
3. 1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
   2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).

6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
   2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
   3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
2. Find the equation of the curve.

4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
   1. Find the \(x\)-coordinate of \(M\).
   2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
   3. Find the equation of the curve.

5

1. A curve has equation \(y = \mathrm { e } ^ { 2 x } - 10 \mathrm { e } ^ { x } + 12 x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (2 marks)
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      (1 mark)
2. The points \(P\) and \(Q\) are the stationary points of the curve.
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) are given by the solutions of the equation $$\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 6 = 0$$ (1 mark)
   2. By using the substitution \(z = \mathrm { e } ^ { x }\), or otherwise, show that the \(x\)-coordinates of \(P\) and \(Q\) are \(\ln 2\) and \(\ln 3\).
   3. Find the \(y\)-coordinates of \(P\) and \(Q\), giving each of your answers in the form \(m + 12 \ln n\), where \(m\) and \(n\) are integers.
   4. Using the answer to part (a)(ii), determine the nature of each stationary point.

9

1. Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$ (3 marks)
2. Given that \(\tan y = x - 1\), use a trigonometrical identity to show that $$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
3. Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$ (l mark)
4. A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
   1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   3. Hence show that the curve has a minimum point which lies on the \(x\)-axis.

8

1. 1. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t\) to obtain \(y\) in terms of \(t\).
   2. Given that \(y = 50\) when \(t = \pi\), show that \(y = 50 \mathrm { e } ^ { - ( 1 + \cos t ) }\).
2. A wave machine at a leisure pool produces waves. The height of the water, \(y \mathrm {~cm}\), above a fixed point at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t$$
   1. Given that this height is 50 cm after \(\pi\) seconds, find, to the nearest centimetre, the height of the water after 6 seconds.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) and hence verify that the water reaches a maximum height after \(\pi\) seconds.

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(T\) with equation \(y = \cos 2 x\) and the circle \(C _ { 1 }\) that touches \(T\) at \(\left( \frac { \pi } { 4 } , 0 \right)\) and \(\left( \frac { 3 \pi } { 4 } , 0 \right)\) .

1. Find the radius of \(C _ { 1 }\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_486_586_1199_841} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of \(T\) and part of a circle \(C _ { 2 }\) that touches \(T\) at the point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , - 1 \right)\) .For values of \(x\) close to \(\frac { \pi } { 2 }\) the curve \(T\) lies inside \(C _ { 2 }\) as shown in Figure 3.
2. Without doing any calculation,explain why the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(C _ { 2 }\) at \(P\) is less than the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(T\) at \(P\) . The radius of \(C _ { 2 }\) is \(r\) .
3. Use the result from part(b)to find a value of \(k\) such that \(r > k\) . Given that \(C _ { 2 }\) cuts \(T\) at the point \(( 0,1 )\) ,
4. find the value of \(r\) .

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) giving each term in simplest form.
   2. Hence determine the nature of the stationary point of \(C\), giving a reason for your answer.
4. State the range of values of \(x\) for which \(y\) is decreasing.

A curve has equation \(y = (2x - 1)^{-1} + 2x\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]

A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.

1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
3. Find \(\mathrm{f}(x)\). [4]

A curve is such that \(\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4\). The curve has a stationary point at \(P\) where \(x = 2\).

1. State, with a reason, the nature of this stationary point. [1]
2. Find an expression for \(\frac{dy}{dx}\). [4]
3. Given that the curve passes through the point \((1, 13)\), find the coordinates of the stationary point \(P\). [4]

\includegraphics{figure_8} The diagram shows the curve \(y = x\cos\frac{1}{2}x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac{dy}{dx}\) and show that \(4\frac{d^2y}{dx^2} + y + 4\sin\frac{1}{2}x = 0\). [5]
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis. [5]

A curve is defined by the equation \(y = 2x \ln(1 + x)\).

1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]

It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)

1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]

A curve C has equation \(f(x) = 5x^3 + 2x^2 - 3x\).

1. Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary. [5]
2. Determine the range of values of \(x\) for which C is concave. [2]