Equation of tangent or normal

AQA C2 2005 June Q7

9 marks Moderate -0.8

7 A curve is defined, for $x > 0$, by the equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$

Express $\frac { x ^ { 8 } - 1 } { x ^ { 3 } }$ in the form $x ^ { p } - x ^ { q }$, where $p$ and $q$ are integers.
1. Hence differentiate $\mathrm { f } ( x )$ to find $\mathrm { f } ^ { \prime } ( x )$.
2. Hence show that f is an increasing function.
Find the gradient of the normal to the curve at the point $( 1,0 )$.

Edexcel C3 Q8

14 marks Standard +0.8

8. The curve $C$ has the equation $y = \sqrt { x } + \mathrm { e } ^ { 1 - 4 x } , x \geq 0$.

Find an equation for the normal to the curve at the point $\left( \frac { 1 } { 4 } , \frac { 3 } { 2 } \right)$. The curve $C$ has a stationary point with $x$-coordinate $\alpha$ where $0.5 < \alpha < 1$.
Show that $\alpha$ is a solution of the equation $$x = \frac { 1 } { 4 } [ 1 + \ln ( 8 \sqrt { x } ) ]$$
Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 4 } \left[ 1 + \ln \left( 8 \sqrt { x _ { n } } \right) \right]$$ with $x _ { 0 } = 1$ to find $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving the value of $x _ { 4 }$ to 3 decimal places.
Show that your value for $x _ { 4 }$ is the value of $\alpha$ correct to 3 decimal places.
Another attempt to find $\alpha$ is made using the iteration formula $$x _ { n + 1 } = \frac { 1 } { 64 } \mathrm { e } ^ { 8 x _ { n } - 2 }$$ with $x _ { 0 } = 1$. Describe the outcome of this attempt.

Edexcel C3 Q8

14 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d17a1b86-d758-4470-834a-b32a41f90c89-4_478_937_251_450} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = 2 x - 3 \ln ( 2 x + 5 )$ and the normal to the curve at the point $P ( - 2 , - 4 )$.

Find an equation for the normal to the curve at $P$. The normal to the curve at $P$ intersects the curve again at the point $Q$ with $x$-coordinate $q$.
Show that $1 < q < 2$.
Show that $q$ is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
Use the iterative formula $$x _ { n + 1 } = \frac { 12 } { 7 } \ln \left( 2 x _ { n } + 5 \right) - 2 ,$$ with $x _ { 0 } = 1.5$, to find the value of $q$ to 3 significant figures and justify the accuracy of your answer.

Edexcel C3 Q8

13 marks Standard +0.3

8. A curve has the equation $y = x ^ { 2 } - \sqrt { 4 + \ln x }$.

Show that the tangent to the curve at the point where $x = 1$ has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with $x$-coordinate $\alpha$.
Show that $0.3 < \alpha < 0.4$
Show that $\alpha$ is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with $x _ { 0 } = 0.35$, to find $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving your answers to 5 decimal places. END

Edexcel C3 Q4

8 marks Standard +0.2

4. The curve $C$ has the equation $y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0$.

Show that the normal to $C$ at the point where $x = 3$ has the equation $$3 x + 5 y + 21 = 0$$
Find the $x$-coordinates of the stationary points of $C$.

Edexcel C4 Q7

12 marks Standard +0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a1b078fe-96e3-4d62-bf0d-415294ba022f-6_805_1445_269_230}

\end{figure} The curve $C$ with equation $y = 2 \mathrm { e } ^ { x } + 5$ meets the $y$-axis at the point $M$, as shown in Fig. 3 .

Find the equation of the normal to $C$ at $M$ in the form $a x + b y = c$, where $a , b$ and $c$ are integers. This normal to $C$ at $M$ crosses the $x$-axis at the point $N ( n , 0 )$.
Show that $n = 14$. The point $P ( \ln 4,13 )$ lies on $C$. The finite region $R$ is bounded by $C$, the axes and the line $P N$, as shown in Fig. 3.
Find the area of $R$, giving your answers in the form $p + q \ln 2$, where $p$ and $q$ are integers to be found.

AQA C2 2007 June Q5

12 marks Moderate -0.8

5 A curve is defined for $x > 0$ by the equation $$y = \left( 1 + \frac { 2 } { x } \right) ^ { 2 }$$ The point $P$ lies on the curve where $x = 2$.

Find the $y$-coordinate of $P$.
Expand $\left( 1 + \frac { 2 } { x } \right) ^ { 2 }$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence show that the gradient of the curve at $P$ is - 2 .
Find the equation of the normal to the curve at $P$, giving your answer in the form $x + b y + c = 0$, where $b$ and $c$ are integers.

AQA AS Paper 1 2020 June Q8

8 marks Standard +0.3

8

Find the equation of the tangent to the curve $y = \mathrm { e } ^ { 4 x }$ at the point ( $a , \mathrm { e } ^ { 4 a }$ ).
8
Find the value of $a$ for which this tangent passes through the origin.
8
Hence, find the set of values of $m$ for which the equation
has no real solutions. $$\mathrm { e } ^ { 4 x } = m x$$ has no real solutions.
[0pt] [3 marks]

AQA Paper 2 2023 June Q4

7 marks Moderate -0.8

4 A curve has equation $$y = \frac { x ^ { 2 } } { 8 } + 4 \sqrt { x }$$ 4

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$
4
The point $P$ with coordinates $( 4,10 )$ lies on the curve.
Find an equation of the tangent to the curve at the point $P$
□
4
Show that the curve has no stationary points.