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

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).

8 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 3 } { 4 x - p }\) for \(x > \frac { p } { 4 }\), where \(p\) is a constant.

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence determine whether f is an increasing function, a decreasing function or neither.
2. Express \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(\frac { p } { a } - \frac { b } { c x - d }\), where \(a , b , c\) and \(d\) are integers.
3. Hence state the value of \(p\) for which \(\mathrm { f } ^ { - 1 } ( x ) \equiv \mathrm { f } ( x )\).

10 \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-16_942_933_262_605} Curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 1\) and \(y = \frac { 1 } { 2 } x ^ { 2 } - x + 1\) intersect at \(A ( 0,1 )\) and \(B ( 4,5 )\), as shown in the diagram.

1. Find the area of the region between the two curves.
   The acute angle between the two tangents at \(B\) is denoted by \(\alpha ^ { \circ }\), and the scales on the axes are the same.
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-18_951_725_267_703} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\). Tangents touching the circle at points \(B\) and \(C\) pass through the point \(A ( 0,10 )\).
   1. By letting the equation of a tangent be \(y = m x + 10\), find the two possible values of \(m\).
   2. Find the coordinates of \(B\) and \(C\).
      The point \(D\) is where the circle crosses the positive \(x\)-axis.
   3. Find angle \(B D C\) in degrees.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point.
3. State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

7 The curve \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = \frac { - 3 } { ( x + 2 ) ^ { 4 } }\).

1. The tangent at a point on the curve where \(x = a\) has gradient \(- \frac { 16 } { 27 }\). Find the possible values of \(a\).
2. Find \(\mathrm { f } ( x )\) given that the curve passes through the point \(( - 1,5 )\).

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

1. Find the equation of the normal to the curve at the point \(A ( 4,3 )\), giving your answer in the form \(y = m x + c\).
   A point is moving along the curve \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of increase of the \(y\)-coordinate at \(A\).
   At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
3. As the point moves down the normal, find the rate of change of its \(x\)-coordinate.

11 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-14_693_782_267_669} The diagram shows part of the curve with equation \(y = x + \frac { 2 } { ( 2 x - 1 ) ^ { 2 } }\). The lines \(x = 1\) and \(x = 2\) intersect the curve at \(P\) and \(Q\) respectively and \(R\) is the stationary point on the curve.

1. Verify that the \(x\)-coordinate of \(R\) is \(\frac { 3 } { 2 }\) and find the \(y\)-coordinate of \(R\).
2. Find the exact value of the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

12 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-20_524_972_274_548} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Calculate the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Calculate the value of \(m\).

3

1. Differentiate \(4 x + \frac { 6 } { x ^ { 2 } }\) with respect to \(x\).
2. Find \(\int \left( 4 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x\).

10 The equation of a curve is \(y = \sqrt { } ( 5 x + 4 )\).

1. Calculate the gradient of the curve at the point where \(x = 1\).
2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(x\) has the constant value 0.03 units per second. Find the rate of increase of \(y\) at the instant when \(x = 1\).
3. Find the area enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

11 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-4_862_892_932_628} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } : x \mapsto \frac { 6 } { 2 x + 3 }\) for \(x \geqslant 0\).

1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and explain how your answer shows that f is a decreasing function.
2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Copy the diagram and, on your copy, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs. The function g is defined by \(\mathrm { g } : x \mapsto \frac { 1 } { 2 } x\) for \(x \geqslant 0\).
4. Solve the equation \(\operatorname { fg } ( x ) = \frac { 3 } { 2 }\).

6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-3_791_885_1281_630} The diagram shows a curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { k } { x ^ { 3 } }\), where \(k\) is a constant. The curve passes through the points \(( 1,18 )\) and \(( 4,3 )\).

1. Show, by integration, that the equation of the curve is \(y = \frac { 16 } { x ^ { 2 } } + 2\). The point \(P\) lies on the curve and has \(x\)-coordinate 1.6.
2. Find the area of the shaded region.

11 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-4_686_805_950_669} The diagram shows the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\) for \(x \geqslant 0\). The curve has a maximum point at \(A\) and a minimum point on the \(x\)-axis at \(B\). The normal to the curve at \(C ( 2,2 )\) meets the normal to the curve at \(B\) at the point \(D\).

1. Find the coordinates of \(A\) and \(B\).
2. Find the equation of the normal to the curve at \(C\).
3. Find the area of the shaded region.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.

6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$

1. Show that \(z = 3 x + \frac { 1200 } { x }\).
2. Find the stationary value of \(z\) and determine its nature.

4

1. Differentiate \(\frac { 2 x ^ { 3 } + 5 } { x }\) with respect to \(x\).
2. Find \(\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\) and hence find the value of \(\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto 2 ( x - 1 ) ^ { 3 } + 8 , \quad x > 1 . \end{aligned}$$

1. Evaluate fg(2).
2. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
3. Obtain an expression for \(\mathrm { g } ^ { \prime } ( x )\) and use your answer to explain why g has an inverse.
4. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

5 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_636_947_1738_598} The diagram shows the curve \(y = 7 \sqrt { } x\) and the line \(y = 6 x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).

1. For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(k\) for which \(y = 6 x + k\) is a tangent to the curve \(y = 7 \sqrt { } x\).

10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).

1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.

9 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_849_565_1466_790} The diagram shows part of the curve \(y = 8 - \sqrt { } ( 4 - x )\) and the tangent to the curve at \(P ( 3,7 )\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\).
3. Find, showing all necessary working, the area of the shaded region.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { \sqrt { } ( 4 x + a ) }\), where \(a\) is a constant. The point \(P ( 2,14 )\) lies on the curve and the normal to the curve at \(P\) is \(3 y + x = 5\).

1. Show that \(a = 8\).
2. Find the equation of the curve.

8 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).