Find curve from gradient

1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that

$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer.

7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\). Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find \(\mathrm { f } ( x )\).

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point ( 3,6 ). Given that $$f ^ { \prime } ( x ) = ( x - 2 ) ( 3 x + 4 )$$

1. use integration to find \(\mathrm { f } ( x )\). Give your answer as a polynomial in its simplest form.
2. Show that \(\mathrm { f } ( x ) \equiv ( x - 2 ) ^ { 2 } ( x + p )\), where \(p\) is a positive constant. State the value of \(p\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the curve touches or crosses the coordinate axes.

A curve has equation \(y = \mathrm { f } ( x )\) and passes through the point (4, 22). Given that $$\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x ^ { \frac { 1 } { 2 } } - 7 ,$$ use integration to find \(\mathrm { f } ( x )\), giving each term in its simplest form.

6

1. Find \(\int x \left( x ^ { 2 } + 2 \right) \mathrm { d } x\).
2. 1. Find \(\int \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
   2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } x }\). Find the equation of the curve, given that it passes through the point \(( 4,0 )\).

3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point (4,5). Find the equation of the curve.

9 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 8 x\). The curve passes through the point \(( 1,5 )\). Find the equation of the curve.

8 The gradient of a curve is \(3 \sqrt { x } - 5\). The curve passes through the point ( 4,6 ). Find the equation of the curve.

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

6. (i) Evaluate $$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$ (ii) Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).

8 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.

2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 4\). The curve passes through the distinct points ( 2,5 ) and ( \(p , 5\) ).

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

2

1. Find \(\int \left( 6 x ^ { \frac { 1 } { 2 } } - 1 \right) \mathrm { d } x\).
2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 1\) and which passes through the point \(( 4,17 )\).

2

1. Find \(\int \left( x ^ { 2 } - 2 x + 5 \right) \mathrm { d } x\).
2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 2 x + 5\) and which passes through the point \(( 3,11 )\).

6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2\). Given also that the curve passes through the point \(( 9,4 )\), find the equation of the curve.

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

The equation of a curve is such that \(\frac{dy}{dx} = \frac{4}{(x-3)^2}\) for \(x > 3\). The curve passes through the point \((4, 5)\). Find the equation of the curve. [3]

The equation of a curve is such that \(\frac{dy}{dx} = 4x - 3\sqrt{x} + 1\).

1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac{11}{2}\). [3]
2. Given that the curve passes through the point \((4, 11)\), find the equation of the curve. [4]

The function f is such that \(\mathrm{f}'(x) = 5 - 2x^2\) and \((3, 5)\) is a point on the curve \(y = \mathrm{f}(x)\). Find \(\mathrm{f}(x)\). [3]

A curve with equation \(y = \text{f}(x)\) passes through the point \((4, 25)\) Given that $$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$ find \(\text{f}(x)\), simplifying each term. [5]

1. Show that \(\frac{(3 - \sqrt{x})^2}{\sqrt{x}}\) can be written as \(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\). [2]

Given that \(\frac{dy}{dx} = \frac{(3 - \sqrt{x})^2}{\sqrt{x}}\), \(x > 0\), and that \(y = \frac{2}{3}\) at \(x = 1\),

1. find \(y\) in terms of \(x\). [6]

\(\frac{dy}{dx} = 5 + \frac{1}{x^2}\).

1. Use integration to find \(y\) in terms of \(x\). [3]
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]

$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$

1. Use integration to find \(y\) in terms of \(x\). [3]
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]

$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$

1. Use integration to find \(y\) in terms of \(x\). [3]
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]

Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{2}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]