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

10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 } { x ^ { 3 } }\), and \(( 1,4 )\) is a point on the curve.

1. Find the equation of the curve.
2. A line with gradient \(- \frac { 1 } { 2 }\) is a normal to the curve. Find the equation of this normal, giving your answer in the form \(a x + b y = c\).
3. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 - x\) and the point \(P ( 2,9 )\) lies on the curve. The normal to the curve at \(P\) meets the curve again at \(Q\). Find

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

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k - 2 x\), where \(k\) is a constant.

1. Given that the tangents to the curve at the points where \(x = 2\) and \(x = 3\) are perpendicular, find the value of \(k\).
2. Given also that the curve passes through the point \(( 4,9 )\), find the equation of the curve.

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

1. Find the set of values of \(x\) for which f is an increasing function.
2. Given that the curve passes through \(( 1,3 )\), find \(\mathrm { f } ( x )\).

4 A function f is defined for \(x \in \mathbb { R }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 6\). The range of the function is given by \(\mathrm { f } ( x ) \geqslant - 4\).

1. State the value of \(x\) for which \(\mathrm { f } ( x )\) has a stationary value.
2. Find an expression for \(\mathrm { f } ( x )\) in terms of \(x\).

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }\). The line \(3 y + x = 17\) is the normal to the curve at the point \(P\) on the curve. Given that the \(x\)-coordinate of \(P\) is positive, find

1. the coordinates of \(P\),
2. 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.

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

9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.

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

9 The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 2,10 )\) and it is given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 12 } { x ^ { 3 } }\).

1. Find \(\mathrm { f } ( x )\).
2. Find the coordinates of the other stationary point.
3. Find the nature of each of the stationary points.

9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).

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 on the curve and, showing all necessary working, determine the nature of this stationary point.

11 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762} The diagram shows a shaded region bounded by the \(y\)-axis, the line \(y = - 1\) and the part of the curve \(y = x ^ { 2 } + 4 x + 3\) for which \(x \geqslant - 2\).

1. Express \(y = x ^ { 2 } + 4 x + 3\) in the form \(y = ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. Hence, for \(x \geqslant - 2\), express \(x\) in terms of \(y\).
2. Hence, showing all necessary working, find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).

1. Find the set of values of \(x\) for which f is decreasing.
2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). 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 \(\mathrm { f } ( x )\).

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.

7

1. Find the exact area of the region bounded by the curve \(y = 1 + \mathrm { e } ^ { 2 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 1 } { 2 }\) and \(x = 2\).
2. \includegraphics[max width=\textwidth, alt={}, center]{e3ee4932-8219-4332-9cd2-e7f835522469-3_469_719_397_753} The diagram shows the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { \sin 2 x }\) for \(0 < x < \frac { 1 } { 2 } \pi\), and its minimum point \(M\). Find the exact \(x\)-coordinate of \(M\).

7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 1 } { 2 } \mathrm { e } ^ { 3 x } + x ^ { 2 } \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 61 - 2 a ^ { 3 } \right)\).
2. Show by calculation that the value of \(a\) lies between 1.0 and 1.5.
3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.

6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x + 1\) ) the remainder is 1 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

6 A particle moves in a straight line and passes through the point \(A\) at time \(t = 0\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 2 t ^ { 2 } - 5 t + 3$$

1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
2. Sketch the velocity-time graph for the first 3 seconds of motion.
3. Find the distance travelled between the two times when the particle is instantaneously at rest.

2 A particle starts at a point \(O\) and moves along a straight line. Its velocity \(t\) s after leaving \(O\) is \(\left( 1.2 t - 0.12 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the displacement of the particle from \(O\) when its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).

1. Find A's velocity when \(t = 200\) and when \(t = 500\). \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
2. Find the distance between \(A\) and \(B\) when \(t = 500\).

4 A particle \(P\) moves in a straight line. At time \(t\) seconds after starting from rest at the point \(O\) on the line, the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.075 t ^ { 2 } - 1.5 t + 5\).

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Hence find the time taken for \(P\) to return to the point \(O\).