Find curve equation from derivative (extended problem with normals, stationary points, or further geometry)

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.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }\). The point \(P ( 2,9 )\) lies on the curve.

1. A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\). [2]
2. Find the equation of the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-18_611_531_262_808} The diagram shows the curve \(y = \mathrm { f } ( x )\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - \frac { 2 } { x ^ { 3 } }\) and that the curve passes through the point \(\left( 4 , \frac { 189 } { 16 } \right)\).

1. Find the \(x\)-coordinate of \(A\).
2. Find \(\mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of \(B\) and \(C\).
4. Find, showing all necessary working, the area of the shaded region.
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4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 4 x + 1\). The curve passes through the point \(( 1,5 )\).

1. Find the equation of the curve.
2. Find the set of values of \(x\) for which the gradient of the curve is positive.

10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 } { x ^ { 3 } }\), and \(( 1,4 )\) is a point on the curve.

1. Find the equation of the curve.
2. A line with gradient \(- \frac { 1 } { 2 }\) is a normal to the curve. Find the equation of this normal, giving your answer in the form \(a x + b y = c\).
3. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 - x\) and the point \(P ( 2,9 )\) lies on the curve. The normal to the curve at \(P\) meets the curve again at \(Q\). Find

1. the equation of the curve,
2. the equation of the normal to the curve at \(P\),
3. the coordinates of \(Q\).

10 A curve is defined for \(x > 0\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x + \frac { 4 } { x ^ { 2 } }\). The point \(P ( 4,8 )\) lies on the curve.

1. Find the equation of the curve.
2. Show that the gradient of the curve has a minimum value when \(x = 2\) and state this minimum value.

12. The curve with equation \(y = \mathrm { f } ( x ) , x > 0\), passes through the point \(P ( 4 , - 2 )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$

1. find the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
2. Find \(\mathrm { f } ( x )\).

1. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , \quad x > 0\), passes through the point \(P ( 4,1 )\).

Given that \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }\)

1. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
   (4)
2. Find \(\mathrm { f } ( x )\).
   (5)
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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 \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4\)
• the point \(P ( 9,8 )\) lies on \(C\)
  1. find, in simplest form, \(\mathrm { f } ( x )\)

The line \(l\) is the normal to \(C\) at \(P\)

Find the coordinates of the point at which \(l\) crosses the \(y\)-axis.

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.

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 3 x\). The curve passes through the point (6, 20).

1. Determine the equation of the curve.
2. Hence determine \(\int _ { 1 } ^ { p } y \mathrm {~d} x\) in terms of the constant \(p\).

6 The gradient of a curve is given by the equation \(\frac { d y } { d x } = 6 x ^ { 2 } - 20 x + 6\). The curve passes through the point \(( 2,6 )\).

1. Find the equation of the curve.
2. Verify that the equation of the curve can be written as \(y = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }\).
3. Sketch the curve, indicating the points where the curve meets the axes.

1. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { x ^ { 2 } } , \quad x \neq 0 .$$

1. Using integration, find \(\mathrm { f } ( x )\).
2. Sketch the curve \(y = \mathrm { f } ( x )\) and write down the equations of its asymptotes.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point \(A\) on \(C\) has coordinates (2, 6),

1. find an equation for \(C\),
2. find an equation for the tangent to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers,
3. show that the line \(y = x + 3\) is also a tangent to \(C\).

5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$

1. Find \(\mathrm { f } ( x )\).
2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).

The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),

1. find \(f(x)\). [5]
2. Verify that \(f(-2) = 5\). [1]
3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The curve \(C\) has equation \(y = f(x)\), \(x \neq 0\), and the point \(P(2, 1)\) lies on \(C\). Given that $$f'(x) = 3x^2 - 6 - \frac{8}{x^3},$$

1. find \(f(x)\). [5]
2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers. [4]