Find tangent at given point (polynomial/algebraic)

8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
2. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\) Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
3. find the value of \(\alpha\)

1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
(5)
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6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)

10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),

1. find \(\mathrm { f } ( x )\).
2. Verify that \(f ( - 2 ) = 5\).
3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

11. The curve \(C\) has equation $$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point \(P\) has coordinates (2, 7).

1. Show that \(P\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The point \(Q\) also lies on \(C\).
   Given that the tangent to \(C\) at \(Q\) is perpendicular to the tangent to \(C\) at \(P\),
3. show that the \(x\)-coordinate of \(Q\) is \(\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )\).

11. The curve \(C\) has equation \(y = 2 x ^ { 3 } + k x ^ { 2 } + 5 x + 6\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\), where \(x = - 2\), lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the line with equation \(2 y - 17 x - 1 = 0\) Find
2. the value of \(k\),
3. the value of the \(y\) coordinate of \(P\),
4. the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. The curve \(C\) has equation

$$y = ( 2 x - 3 ) ^ { 5 }$$ The point \(P\) lies on \(C\) and has coordinates \(( w , - 32 )\).
Find

1. the value of \(w\),
2. the equation of the tangent to \(C\) at the point \(P\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.

1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .

Find an equation for the tangent to the curve at \(A\).

4 Find the equation of the tangent to the curve \(y = x ^ { 3 } + 2 x - 7\) at the point where it cuts the \(y\) axis.

2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).

1. Find the equation of the tangent to the curve \(y = x ^ { 4 }\) at the point where \(x = 2\). Give your answer in the form \(y = m x + c\).
2. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 4 }\) where \(x = 2\) and \(x = 2.1\).
3. (A) Expand \(( 2 + h ) ^ { 4 }\).
   (B) Simplify \(\frac { ( 2 + h ) ^ { 4 } - 2 ^ { 4 } } { h }\).
   (C) Show how your result in part (iii) (B) can be used to find the gradient of \(y = x ^ { 4 }\) at the point where \(x = 2\).
4. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
5. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
6. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
7. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
8. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
9. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8ff8b67d-1489-4cb1-bcd2-b32db674e29f-3_651_770_242_737} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
10. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.

2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).

1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4e8d7217-61f7-4ae4-96dd-d34e37c4d623-2_1020_940_244_679} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the cubic curve \(y = \mathrm { f } ( x )\). The values of \(x\) where it crosses the \(x\)-axis are - 5 , - 2 and 2 , and it crosses the \(y\)-axis at \(( 0 , - 20 )\).
4. Express \(\mathrm { f } ( x )\) in factorised form.
5. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
6. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
7. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4e8d7217-61f7-4ae4-96dd-d34e37c4d623-3_768_1023_223_598} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
8. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
9. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
10. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
11. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
12. Sketch the curve. A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
13. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
14. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).

3 A curve has an equation which satisfies \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k x ( 2 x - 1 )\) for all values of \(x\). The point \(P ( 2,7 )\) lies on the curve and the gradient of the curve at \(P\) is 9 .

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

1. A curve has equation

$$y = 2 x ^ { 3 } - 4 x + 5$$ Find the equation of the tangent to the curve at the point \(P ( 2,13 )\).
Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
Solutions relying on calculator technology are not acceptable.
(5)

5 In this question you must show detailed reasoning.
This question is about the curve \(y = x ^ { 3 } - 5 x ^ { 2 } + 6 x\).

1. Find the equation of the tangent, \(T\), to the curve at the point ( 0,0 ).
2. Find the equation of the normal, \(N\), to the curve at the point ( 1,2 ).
3. Find the coordinates of the point of intersection of \(T\) and \(N\).

15

1. Show that, for the curve \(y = a x ^ { 2 } + b x + c\), the equation of the tangent at the point with \(x\)-coordinate \(t\) is \(\mathrm { y } = ( 2 \mathrm { at } + \mathrm { b } ) \mathrm { x } - \mathrm { at } ^ { 2 } + \mathrm { c }\).
2. Hence show that for the curve with equation \(y = a x ^ { 2 } + b x + c\), the tangents at two points, \(P\) and Q , on the curve cross at a point which has \(x\)-coordinate equal to the mean of the \(x\)-coordinates of points P and Q , as given in lines 11 to 14 .

18 A student is investigating the intersection points of tangents to the curve \(y = 6 x ^ { 2 } - 7 x + 1\). She uses software to draw tangents at pairs of points with \(x\)-coordinates differing by 5 . Find the equation of the curve that all the intersection points lie on.

4 The curve sketched below passes through the point \(A ( - 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{889639d6-0a31-4569-8370-1e72291a0c47-3_538_734_365_662} The curve has equation \(y = 14 - x - x ^ { 4 }\) and the point \(P ( 1,12 )\) lies on the curve.

1. 1. Find the gradient of the curve at the point \(P\).
   2. Hence find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 14 - x - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve \(y = 14 - x - x ^ { 4 }\) and the line \(A P\).
      (2 marks)

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

3 A curve has equation \(y = 7 - 2 x ^ { 5 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation for the tangent to the curve at the point where \(x = 1\).
3. Determine whether \(y\) is increasing or decreasing when \(x = - 2\).

3 A curve has equation \(y = 2 x ^ { 5 } + 5 x ^ { 4 } - 1\).

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. The point on the curve where \(x = - 1\) is \(P\).
   1. Determine whether \(y\) is increasing or decreasing at \(P\), giving a reason for your answer.
   2. Find an equation of the tangent to the curve at \(P\).
3. The point \(Q ( - 2,15 )\) also lies on the curve. Verify that \(Q\) is a maximum point of the curve.
   [0pt] [4 marks]

6. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2. Find an equation for the tangent to the curve at \(A\).

14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).

5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where \(x = 0\)

6 The curve \(y = x ^ { 3 } + a x ^ { 2 } + b x + 1\) has a gradient of 11 at the point \(( 1,7 )\). Find the values of \(a\) and \(b\).