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

2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the point (4,7) lies on the curve. Find the equation of the curve.

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }\). It is given that the curve passes through the point \(\left( \frac { 1 } { 2 } , 4 \right)\). Find the equation of the curve.

11 The equation of a curve is \(y = 2 \sqrt { 3 x + 4 } - x\).

1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
2. Find the coordinates of the stationary point.
3. Determine the nature of the stationary point.
4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).

11 \includegraphics[max width=\textwidth, alt={}, center]{aaba3158-b5be-464e-bea3-1a4c460f9637-16_622_1091_260_525} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + k ^ { 2 } x ^ { - \frac { 1 } { 2 } }\), where \(k\) is a positive constant.

1. Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).
   The tangent at the point on the curve where \(x = 4 k ^ { 2 }\) intersects the \(y\)-axis at \(P\).
2. Find the \(y\)-coordinate of \(P\) in terms of \(k\).
   The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = \frac { 9 } { 4 } k ^ { 2 }\) and \(x = 4 k ^ { 2 }\).
3. Find the area of the shaded region in terms of \(k\).
   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 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 ( 4 x - 7 ) ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(\left( 4 , \frac { 5 } { 2 } \right)\). Find the equation of the curve.

9 The equation of a curve is \(y = 3 x + 1 - 4 ( 3 x + 1 ) ^ { \frac { 1 } { 2 } }\) for \(x > - \frac { 1 } { 3 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point of the curve and determine its nature.

8 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-12_577_1088_260_523} The diagram shows the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + 4 x ^ { - \frac { 1 } { 2 } }\). The line \(y = 5\) intersects the curve at the points \(A ( 1,5 )\) and \(B ( 16,5 )\).

1. Find the equation of the tangent to the curve at the point \(A\).
2. Calculate the area of the shaded region.

10 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find the coordinates of the maximum point of the curve.
2. Verify that the line \(A B\) is the normal to the curve at \(A\).
3. Find 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.

5 The equation of a curve is \(y = 2 x ^ { 2 } - \frac { 1 } { 2 x } + 3\).

1. Find the coordinates of the stationary point.
2. Determine the nature of the stationary point.
3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

9 \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_764_967_292_555} The diagram shows the curve with equation \(y = \sqrt { 2 x ^ { 3 } + 10 }\).

1. Find the equation of the tangent to the curve at the point where \(x = 3\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_2716_35_141_2013}
2. The region shaded in the diagram is enclosed by the curve and the straight lines \(x = 1 , x = 3\) and \(y = 0\). Find the volume of the solid obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

1 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 x + 2 } + x ^ { 2 }\) for \(x < - 1\).
Determine whether f is an increasing function, a decreasing function or neither.

10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Determine the nature of the stationary point.
3. Find the equation of the curve.

11 \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the equation of the tangent to the curve at \(A\).
3. Find the \(x\)-coordinate of the maximum point of the curve.
4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 It is given that a curve has equation \(y = k ( 3 x - k ) ^ { - 1 } + 3 x\), where \(k\) is a constant.

1. Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.
   The function f has a stationary value at \(x = a\) and is defined by $$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
2. Find the value of \(a\) and determine the nature of the stationary value.
3. The function g is defined by \(\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x\) for \(x \geqslant 0\). Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 At the point \(( 4 , - 1 )\) on a curve, the gradient of the curve is \(- \frac { 3 } { 2 }\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { - \frac { 1 } { 2 } } + k\), where \(k\) is a constant.

1. Show that \(k = - 2\).
2. Find the equation of the curve.
3. Find the coordinates of the stationary point.
4. Determine the nature of the stationary point.

5 A curve has the equation \(\mathrm { y } = \frac { 3 } { 2 \mathrm { x } ^ { 2 } - 5 }\).
Find the equation of the normal to the curve at the point \(( 2,1 )\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

11 \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-14_467_757_262_653} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } ^ { - \frac { 2 } { 3 } } - 3 \mathrm { x } ^ { - \frac { 1 } { 3 } } + 1\) for \(x > 0\). The curve crosses the \(x\)-axis at points \(A\) and \(B\) and has a minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the region bounded by the curve and the line segment \(A B\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The equation of a curve is \(y = 2 + \sqrt { 25 - x ^ { 2 } }\).
Find the coordinates of the point on the curve at which the gradient is \(\frac { 4 } { 3 }\).

12 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-18_557_677_264_733} The diagram shows a curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - 2 x\) for \(x \geqslant 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the \(x\)-axis at \(A ( 4,0 )\) and crosses the straight line at \(B\) and \(C\).

1. Find, by calculation, the \(x\)-coordinates of \(B\) and \(C\).
2. Show that \(B\) is a stationary point on the curve.
3. Find 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.

7 The point \(( 4,7 )\) lies on the curve \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } }\).

1. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 4\).
2. Find the equation of the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-14_378_666_264_737} The diagram shows part of the curve \(y = \frac { 2 } { ( 3 - 2 x ) ^ { 2 } } - x\) and its minimum point \(M\), which lies on the \(x\)-axis.

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y \mathrm {~d} x\).
2. Find, by calculation, the \(x\)-coordinate of \(M\).
3. Find the area of the shaded region bounded by the curve and the coordinate axes.

8 The equation of a curve is \(y = 2 x + 1 + \frac { 1 } { 2 x + 1 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point and determine the nature of the stationary point.

10 A curve has equation \(y = \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } }\) where \(x > 0\) and \(k\) is a positive constant.

1. It is given that when \(x = \frac { 1 } { 4 }\), the gradient of the curve is 3 . Find the value of \(k\).
2. It is given instead that \(\int _ { \frac { 1 } { 4 } k ^ { 2 } } ^ { k ^ { 2 } } \left( \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } } \right) \mathrm { d } x = \frac { 13 } { 12 }\). Find the value of \(k\).

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
2. Find the coordinates of the stationary points on the curve.
3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
4. Hence, or otherwise, determine the nature of each of the stationary points.