1.07d Second derivatives: d^2y/dx^2 notation

10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   (2)
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
   (1)
   Given that the point \(P ( 2,4 )\) lies on \(C\),
3. find \(y\) in terms of \(x\),
   (5)
4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   (5)
   1. continued

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

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\) ,length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\) ,as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick,\(S \mathrm {~cm} ^ { 2 }\) ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Find \(\frac { \mathrm { d } S } { \mathrm {~d} x }\)
3. Hence find the value of \(x\) for which \(S\) is stationary.
4. Find \(\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }\) and hence show that the value of \(x\) found in part(c)gives the minimum value of \(S\) .
5. Hence find the minimum surface area of the brick.

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)

1. Use calculus to find the coordinates of the turning point on \(C\).
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. State the nature of the turning point.

3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
2. Hence find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of each turning point of the curve \(C\).

6. (a) Differentiate with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
   (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).

5. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$

1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \left( a x ^ { 2 } + b \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\) and \(b\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\)
3. use your series to estimate the value of \(y\) at \(x = - 0.2\), giving your answer to four decimal places.

2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$

1. Show that $$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined. Given that \(y = 1\) at \(x = 2\)
2. determine the Taylor series expansion for \(y\) in ascending powers of \(( x - 2 )\), up to and including the term in \(( x - 2 ) ^ { 3 }\), giving each coefficient in simplest form.

10. $$\left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$ At \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1\).

1. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\).
2. Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
   (Total 7 marks)

5. $$y = \sec ^ { 2 } x$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\).
2. Find a Taylor series expansion of \(\sec ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).

2. The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$ When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).

3. $$f ( x ) = \ln ( 1 + \sin k x )$$ where \(k\) is a constant, \(x \in \mathbb { R }\) and \(- \frac { \pi } { 2 } < k x < \frac { 3 \pi } { 2 }\)

1. Find f \({ } ^ { \prime } ( x )\)
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { - k ^ { 2 } } { 1 + \sin k x }\)
3. Find the Maclaurin series of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

7. $$y = \tan ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\)
2. Hence show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \sec ^ { 2 } x \tan x \left( A \sec ^ { 2 } x + B \right)\), where \(A\) and \(B\) are constants to be found.
3. Find the Taylor series expansion of \(\tan ^ { 2 } x\), in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\)

3. Given that $$y = x - \operatorname { artanh } \left( \frac { 2 x } { 1 + x ^ { 2 } } \right)$$

1. show that $$1 - \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 - x ^ { 2 } }$$ where \(k\) is a constant to be found.
2. Hence, or otherwise, show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \left( 1 - \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 0$$

5. Given that \(y = \tan ^ { 2 } x\)

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \tan x \sec ^ { 2 } x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of \(\tan ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
   
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Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. \(\int y \mathrm {~d} x\).

3 Given that \(y = 3 x ^ { 5 } - \sqrt { x } + 15\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),

1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
2. find \(\mathrm { f } ^ { \prime } ( x )\),
3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).

8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$

1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
3. Determine whether the turning point is a maximum or minimum point.

1. Given that

$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence verify that the origin is a stationary point of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that the origin is a minimum point.
3. Using the substitution \(u = 1 + x\), show that \(\int \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x = \int \left( u - 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x\), giving your answer in an exact form.
4. Using integration by parts and your answer to part (iii), evaluate \(\int _ { 0 } ^ { 1 } 2 x \ln ( 1 + x ) \mathrm { d } x\).

6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).