Find curve equation from derivative (find unknown constant in derivative first)

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.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + \frac { k } { x ^ { 3 } }\) and passes through the point \(P ( 1,9 )\). The gradient of the curve at \(P\) is 2 .

1. Find the value of the constant \(k\).
2. Find the equation of the curve.

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.

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

\includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-31_2255_50_314_34}

6. 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 $$\mathrm { f } ( 2 ) = 2 \mathrm { f } ( 1 ) ,$$
2. find \(\mathrm { f } ( 4 )\).