Find curve equation from derivative

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x\). Given that the curve passes through the point (4,6), find the equation of the curve.

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { x ^ { 3 } } - 1\) and the point \(( 2,4 )\) lies on the curve. Find the equation of the curve.

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.

2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).

2 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { - \frac { 3 } { 2 } } + 1\) and that \(\mathrm { f } ( 4 ) = 5\). Find \(\mathrm { f } ( x )\).

2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).

10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { a } x ^ { - \frac { 1 } { 2 } } + a x ^ { - \frac { 3 } { 2 } }\), where \(a\) is a positive constant. The point \(A \left( a ^ { 2 } , 3 \right)\) lies on the curve. Find, in terms of \(a\),

1. the equation of the tangent to the curve at \(A\), simplifying your answer,
2. the equation of the curve. It is now given that \(B ( 16,8 )\) also lies on the curve.
3. Find the value of \(a\) and, using this value, find the distance \(A B\).

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { \sqrt { } x }\), where \(k\) is a constant. The points \(P ( 1 , - 1 )\) and \(Q ( 4,4 )\) lie on the curve. Find the equation of the curve.

2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).

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

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x }\). The point \(( 0,1 )\) lies on the curve.

1. Find the equation of the curve.
2. The curve has one stationary point. Find the \(x\)-coordinate of this point and determine whether it is a maximum or a minimum point.

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 \(\mathrm { f } ( x )\).

9. (i) Find $$\int \frac { ( 3 x + 2 ) ^ { 2 } } { 4 \sqrt { x } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
(ii) A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given

• \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } + a x + b\) where \(a\) and \(b\) are constants
• the \(y\) intercept of \(C\) is - 8
• the point \(P ( 3 , - 2 )\) lies on \(C\)
• the gradient of \(C\) at \(P\) is 2
  find, in simplest form, \(\mathrm { f } ( x )\).
  

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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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9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 9,10 )\). Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find \(\mathrm { f } ( x )\), fully simplifying each term.

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.

7. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 4,25 )\) Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$ find \(\mathrm { f } ( x )\), simplifying each term. $$\begin{aligned} & \therefore F ( x ) = \int F ^ { \prime } ( x ) = \int 3 / 8 x ^ { 2 } - 10 x ^ { - 1 / 2 } + 1 d x \\ & F ( x ) = \frac { 3 x ^ { 3 } } { 8 ( 3 ) } - \frac { 10 x ^ { 1 / 2 } } { 1 / 2 } + x + c \\ & F ( x ) = 1 / 8 x ^ { 3 } - 20 x ^ { 1 / 2 } + x + c \\ & 25 = 1 / 8 ( 4 ) ^ { 3 } - 20 ( 4 ) ^ { 1 / 2 } + 4 + c \\ & 25 = 8 - 40 + 4 + c \\ & C = 53 \\ & F ( x ) = 1 / 8 x ^ { 3 } - 20 x ^ { 1 / 2 } + x + 53 \end{aligned}$$ \section*{PMT PhysicsAndMathsTutor.com}

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.

4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 35\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.

1. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,10 )\). Given that

$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$ find the value of \(\mathrm { f } ( 1 )\).

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that \(y = 7\) at \(x = 1\), find \(y\) in terms of \(x\), giving each term in its simplest form.

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve.

7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 3 } }\). The curve passes through \(( 1,4 )\).
Find the equation of the curve.

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - x ^ { 2 }\). The curve passes through the point \(( 6,1 )\). Find the equation of the curve.

7 The gradient of a curve \(y = \mathrm { f } ( x )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 10 x + 6\). The curve passes through the point \(( 2,3 )\) Find the equation of the curve.