1.07i Differentiate x^n: for rational n and sums

Edexcel FP1 Specimen Q1

6 marks Moderate -0.8

1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$

1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

Edexcel P4 2021 October Q3

8 marks Standard +0.3

3. $$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$

1. Find the values of the constants \(A , B , C\) and \(D\). A curve has equation $$y = g ( x ) \quad x > 0$$ Using the answer to part (a),
2. find \(\mathrm { g } ^ { \prime } ( x )\).
3. Hence, explain why \(\mathrm { g } ^ { \prime } ( x ) > 3\) for all values of \(x\) in the domain of g .

Edexcel C1 2005 January Q2

8 marks Easy -1.3

1. Given that \(y = 5 x ^ { 3 } + 7 x + 3\), find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), (b) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Find \(\int \left( 1 + 3 \sqrt { } x - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

Edexcel C1 2005 June Q2

5 marks Easy -1.2

Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\int y \mathrm {~d} x\).

Edexcel C1 2007 June Q3

7 marks Easy -1.3

Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. \(\int y \mathrm {~d} x\).

Edexcel C1 2008 June Q9

8 marks Moderate -0.3

The curve \(C\) has equation \(y = k x ^ { 3 } - x ^ { 2 } + x - 5\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The point \(A\) with \(x\)-coordinate \(- \frac { 1 } { 2 }\) lies on \(C\). The tangent to \(C\) at \(A\) is parallel to the line with equation \(2 y - 7 x + 1 = 0\). Find
2. the value of \(k\),
3. the value of the \(y\)-coordinate of \(A\).

Edexcel C1 2011 June Q2

7 marks Easy -1.2

Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).

OCR C1 2005 January Q7

9 marks Easy -1.3

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
3. \(y = \sqrt [ 5 ] { x }\).

OCR C1 2006 January Q3

5 marks Easy -1.2

3 Given that \(y = 3 x ^ { 5 } - \sqrt { x } + 15\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

OCR C1 2007 January Q7

8 marks Easy -1.3

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases.

1. \(y = 5 x + 3\)
2. \(y = \frac { 2 } { x ^ { 2 } }\)
3. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)

OCR C1 2008 January Q10

10 marks Standard +0.3

10 Given that \(\mathrm { f } ( x ) = 8 x ^ { 3 } + \frac { 1 } { x ^ { 3 } }\),

1. find \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. solve the equation \(\mathrm { f } ( x ) = - 9\).

OCR C1 2005 June Q6

7 marks Easy -1.2

6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),

1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
2. find \(\mathrm { f } ^ { \prime } ( x )\),
3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).

OCR C1 2005 June Q10

13 marks Moderate -0.8

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

OCR C1 2007 June Q7

9 marks Easy -1.2

7

1. Given that \(f ( x ) = x + \frac { 3 } { x }\), find \(f ^ { \prime } ( x )\).
2. Find the gradient of the curve \(\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }\) at the point where \(\mathrm { x } = 4\).

OCR C1 2008 June Q5

5 marks Easy -1.2

5 Find the gradient of the curve \(y = 8 \sqrt { x } + x\) at the point whose \(x\)-coordinate is 9 .

OCR C1 2008 June Q8

10 marks Moderate -0.3

8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
3. Determine whether this stationary point is a minimum or maximum point.
4. Find the \(x\)-coordinate of the other stationary point.

OCR C1 Specimen Q4

7 marks Easy -1.3

4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = 4 x ^ { 3 } - 1\),
2. \(y = x ^ { 2 } \left( x ^ { 2 } + 2 \right)\),
3. \(y = \sqrt { } x\)

OCR C1 Specimen Q6

12 marks Moderate -0.5

6

1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.

OCR C1 Q6

8 marks Moderate -0.3

6.

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The diagram shows the curve with equation \(y = 3 x - x ^ { \frac { 3 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) and has a maximum at the point \(B\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the coordinates of \(B\).

OCR C1 Q7

9 marks Moderate -0.8

7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
4. Determine whether each stationary point is a maximum or a minimum point.

OCR C1 Q10

13 marks Standard +0.3

10. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.

OCR C1 Q8

9 marks Moderate -0.3

8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$

1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
3. Determine whether the turning point is a maximum or minimum point.

OCR C1 Q10

14 marks Moderate -0.3

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).

OCR C1 Q7

8 marks Moderate -0.8

1. Given that

$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$

OCR C1 Q3

5 marks Moderate -0.3

3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.