1.08a Fundamental theorem of calculus: integration as reverse of differentiation

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

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }\). It is given that the curve passes through the point \(\left( \frac { 1 } { 2 } , 4 \right)\). Find the equation of the curve.

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - \frac { 8 } { x ^ { 2 } }\). It is given that the curve passes through the point \(( 2,7 )\). Find \(\mathrm { f } ( x )\).

11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).

1. Find the value of \(a\).
2. Determine the nature of this stationary point.
3. Find the equation of the curve.
4. Find the coordinates of any other stationary points on the curve.

6 A curve passes through the point \(\left( \frac { 4 } { 5 } , - 3 \right)\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 20 } { ( 5 x - 3 ) ^ { 2 } }\).

1. Find the equation of the curve.
2. The curve is transformed by a stretch in the \(x\)-direction with scale factor \(\frac { 1 } { 2 }\) followed by a translation of \(\binom { 2 } { 10 }\). Find the equation of the new curve. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-08_2716_38_143_2009}

10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Determine the nature of the stationary point.
3. Find the equation of the curve.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }\) and \(A ( 1 , - 3 )\) lies on the curve. A point is moving along the curve and at \(A\) the \(y\)-coordinate of the point is increasing at 3 units per second.

1. Find the rate of increase at \(A\) of the \(x\)-coordinate of the point.
2. Find the equation of the curve.

1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { - \frac { 1 } { 3 } } - x ^ { \frac { 1 } { 3 } }\). It is given that \(\mathrm { f } ( 8 ) = 5\).
Find \(\mathrm { f } ( x )\).

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.

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
2. Find the coordinates of the stationary points on the curve.
3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
4. Hence, or otherwise, determine the nature of each of the stationary points.

4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 3 x + 2 ) ^ { 2 } }\). The curve passes through the point \(\left( 2,5 \frac { 2 } { 3 } \right)\).
Find the equation of the curve.

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

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 2 } - 5\). Given that the point \(( 3,8 )\) lies on the curve, find the equation of the curve.

5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find

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

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.

12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).

1. Find the equation of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. Find the coordinates of the stationary point and determine its nature.

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

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 5 - 2 x ) ^ { 2 } }\). Given that the curve passes through ( 2,7 ), find the equation of the curve.

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

1. Find the equation of the curve.
2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
3. Find the set of values of \(x\) for which the gradient of the curve is positive.

2 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 2 } { x ^ { 3 } }\) passes through \(( - 1,3 )\). 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 The gradient at any point \(( x , y )\) on a curve is \(\sqrt { } ( 1 + 2 x )\). The curve passes through the point \(( 4,11 )\). Find

1. the equation of the curve,
2. the point at which the curve intersects the \(y\)-axis.

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.