1.07i Differentiate x^n: for rational n and sums

12. $$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C\), where \(A , B , C\) and \(k\) are constants to be determined.
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\).
3. Find an equation of the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 4\) 2. LIIIII

Given \(y = \frac { x ^ { 3 } } { 3 } - 2 x ^ { 2 } + 3 x + 5\)

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying each term.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\)

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 8 } { x } + \frac { 1 } { 2 } x - 5 , \quad 0 < x \leqslant 12$$ The curve crosses the \(x\)-axis at \(( 2,0 )\) and \(( 8,0 )\) and has a minimum point at \(A\).

1. Use calculus to find the coordinates of point \(A\).
2. State
   1. the roots of the equation \(2 \mathrm { f } ( x ) = 0\)
   2. the coordinates of the turning point on the curve \(y = \mathrm { f } ( x ) + 2\)
   3. the roots of the equation \(\mathrm { f } ( 4 x ) = 0\)

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,11 )\).
Line \(l\) is the tangent to \(C\) at the point \(P\).

1. Use calculus to show that \(l\) has equation \(y = 5 x - 9\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(x = 1\), the \(x\)-axis and the line \(l\).
2. Find, by using calculus, the area of \(R\), giving your answer to 2 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. Given that

$$y = \frac { 2 x ^ { \frac { 2 } { 3 } } + 3 } { 6 } , \quad x > 0$$ find, in the simplest form,

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

3. A curve has equation $$y = \sqrt { 2 } x ^ { 2 } - 6 \sqrt { x } + 4 \sqrt { 2 } , \quad x > 0$$ Find the gradient of the curve at the point \(P ( 2,2 \sqrt { 2 } )\).
Write your answer in the form \(a \sqrt { 2 }\), where \(a\) is a constant.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \(L\)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the coordinates of \(A\).
3. Use your answer to part (b) to write down the turning point of the curve with equation
   1. \(y = \mathrm { f } ( x + 1 )\),
   2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).

12. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).

1. Show that the coordinates of \(A\) are \(( 3,0 )\).
2. Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\) The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
3. Use algebra to find the \(x\) coordinate of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
4. Find, by using calculus, the area of region \(R\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where

$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying your answer,
2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4. The curve \(C\) has equation \(y = 4 x \sqrt { x } + \frac { 48 } { \sqrt { x } } - \sqrt { 8 } , x > 0\)

1. Find, simplifying each term,
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use part (a) to find the exact coordinates of the stationary point of \(C\).
3. Determine whether the stationary point of \(C\) is a maximum or minimum, giving a reason for your answer.

11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).

1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

14. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).

1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.

4. Given that $$y = 5 x ^ { 2 } + \frac { 1 } { 2 x } + \frac { 2 x ^ { 4 } - 8 } { 5 \sqrt { x } } \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
(6)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 27 \sqrt { x } - 2 x ^ { 2 } , \quad x \in \mathbb { R } , x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The curve has a maximum turning point \(P\), as shown in Figure 2.
2. Use the answer to part (a) to find the exact coordinates of \(P\).

1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
(5)
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3. Given that \(y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0\), find in its simplest form

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

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10. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.

1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
2. Verify that the \(x\) coordinate of \(F\) is 25
3. Use algebraic integration to find the exact area of the shaded region \(R\).

4. Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form,

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

7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .

1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
3. Find the value of \(k\).

4. Given that \(y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0\),

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

1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that

$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer.

9. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.

1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
4. Find the exact coordinates of \(R\).

1. Given that

$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-02_29_45_2690_1852}

8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P ( 4,8 )\) lies on \(C\).
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20 .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
4. Find the length \(P Q\), giving your answer in a simplified surd form.

5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.