Find curve equation from derivative

7. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$

1. find an expression for \(\mathrm { f } ( x )\). Given also that
   f(2) = 2f(1),
2. find \(\mathrm { f } ( 4 )\).

2 At the point \(( x , y )\) on a curve, where \(x > 0\), the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$

1. Write \(\sqrt { x ^ { 5 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
2. Find \(\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x\).
3. Hence find the equation of the curve, given that the curve passes through the point \(( 1,3 )\).

6 The gradient of a curve is \(6 x ^ { 2 } + 12 x ^ { \frac { 1 } { 2 } }\). The curve passes through the point \(( 4,10 )\). Find the equation of the curve.

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 \(4 \frac { 1 } { 2 }\).

3

1. Find \(\int \left( 2 x ^ { 3 } + \frac { 8 } { x ^ { 2 } } \right) \mathrm { d } x\) 3
2. A curve has gradient function \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + \frac { 8 } { x ^ { 2 } }\)
   The \(x\)-intercept of the curve is at the point \(( 2,0 )\)
   Find the equation of the curve.
   Fully justify your answer.
   \(4 \quad\) Find the exact solution of the equation \(\ln ( x + 1 ) + \ln ( x - 1 ) = \ln 15 - 2 \ln 7\)

6

1. Find \(\int \left( 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } } \right) \mathrm { d } x\) 6
2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } }$$ The curve passes through the point \(( 4,90 )\). Find the equation of the curve.