Find tangent line equation

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 \(\alpha\)
   

4. The straight line with equation \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2 x ^ { 2 } + 8 x + 3\) Calculate the value of \(c\) $$\begin{aligned} & y = m x + c \rightarrow y = 4 x + c \quad ( m = 4 ) \\ & \therefore \frac { d y } { d x } = 4 x + 8 = 4 \quad \text { (Gradient equation) } \\ & 4 x + 8 = 4 \\ & x = - 1 \rightarrow y = - 3 \\ & \text { At } ( - 1 , - 3 ) - 3 = 4 ( - 1 ) + c \\ & \quad c = 1 \\ & y = 4 x + 1 \end{aligned}$$ \section*{PMT PhysicsAndMathsTutor.com}

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\)
The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
3. Use integration to find the exact area of the shaded region \(R\).

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 \(b\).
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7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .

1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
3. Find the value of \(k\).

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 \(A\).

8.
\includegraphics[max width=\textwidth, alt={}, center]{b7078372-d0d3-4563-818d-637260be5efc-3_592_727_251_493} The diagram shows the curve with equation \(y = 2 + 3 x - x ^ { 2 }\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). The line \(m\) is the normal to the curve at the point \(B\).
   Given that \(l\) and \(m\) are parallel,
2. find the coordinates of \(B\).

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.

9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).

1. State the gradient of \(C\) at \(P\).
2. Find the \(x\)-coordinate of \(P\).
3. Find an equation for \(C\).
4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.

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 \(( 2 , - 4 )\).

1 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.

3 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65dd0efe-5c99-4814-b741-16e368c3469e-3_641_791_240_714} \captionsetup{labelformat=empty} \caption{Fig. 10}
   \end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.
4. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
5. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
6. The two tangents intersect at the point D . Find the \(y\)-coordinate of D .

5 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).

4 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-3_643_1034_267_549} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).
4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
5. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
6. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.

8

1. Find the equation of the tangent to the curve \(y = 7 + 6 x - x ^ { 2 }\) at the point \(P\) where \(x = 5\), giving your answer in the form \(a x + b y + c = 0\).
2. This tangent meets the \(x\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
3. Find the equation of the line of symmetry of the curve \(y = 7 + 6 x - x ^ { 2 }\).
4. State the set of values of \(x\) for which \(7 + 6 x - x ^ { 2 }\) is an increasing function.

8 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).

11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.

6
\includegraphics[max width=\textwidth, alt={}, center]{fc7679bf-a9a1-493d-bf89-35206382787f-3_576_821_258_662} The diagram shows the curve with equation \(y = \sqrt { 3 x - 5 }\). The tangent to the curve at the point \(P\) passes through the origin. The shaded region is bounded by the curve, the \(x\)-axis and the line \(O P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 10 } { 3 }\) and hence find the exact area of the shaded region.

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.
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8. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of the curve \(C\) with equation $$y = x ^ { 3 } - 14 x + 23$$ The line \(l\) is the tangent to \(C\) at the point \(A\), also shown in Figure 3.
Given that \(l\) has equation \(y = - 2 x + 7\)

1. show, using calculus, that the \(x\) coordinate of \(A\) is 2 The line \(l\) cuts \(C\) again at the point \(B\).
2. Verify that the \(x\) coordinate of \(B\) is - 4 The finite region, \(R\), shown shaded in Figure 3, is bounded by \(C\) and \(l\).
   Using algebraic integration,
3. show that the area of \(R\) is 108

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = 3 x ^ { 2 } - 2\)
The point \(P ( 2,10 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
2. Find the gradient of the line \(P Q\), giving your answer in terms of \(h\) in simplest form.
3. Explain briefly the relationship between part (b) and the answer to part (a).

9 In this question you must show detailed reasoning.
Find the equation of the straight line with positive gradient that passes through \(( 0,2 )\) and is a tangent to the curve \(y = x ^ { 2 } - x + 6\).

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 \(N\).