Tangent to exponential curve

A question is this type if and only if it requires finding the equation of a tangent line to an exponential curve at a given point.

7 questions · Moderate -0.5

Sort by: Default | Easiest first | Hardest first
Edexcel C3 2008 June Q1
6 marks Moderate -0.5
  1. The point \(P\) lies on the curve with equation
$$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .
  1. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
  2. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.
Edexcel C3 Q5
11 marks Moderate -0.3
5. $$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 7\).
  4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).
OCR MEI C2 2013 June Q5
5 marks Moderate -0.8
  1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
  2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\). \(6 S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  3. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
  4. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).
OCR MEI C2 Q6
5 marks Moderate -0.8
  1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
  2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
Edexcel C3 Q17
10 marks Standard +0.3
17. The curve \(C\) with equation \(y = p + q \mathrm { e } ^ { x }\), where \(p\) and \(q\) are constants, passes through the point \(( 0,2 )\). At the point \(P ( \ln 2 , p + 2 q )\) on \(C\), the gradient is 5 .
  1. Find the value of \(p\) and the value of \(q\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
  2. Show that the area of \(\triangle O L M\), where \(O\) is the origin, is approximately 53.8. \section*{18.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-08_487_695_259_683}
    \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { - x } - 1\).
  3. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac { 1 } { 2 } | x - 1 |\). Show the coordinates of the points where the graph meets the axes. The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).
  4. Show that \(x = \alpha\) is a root of the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
  5. Show that \(- 1 < \alpha < 0\). The iterative formula \(x _ { \mathrm { n } + 1 } = - \ln \left[ \frac { 1 } { 2 } \left( 3 - x _ { n } \right) \right]\) is used to solve the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
  6. Starting with \(x _ { 0 } = - 1\), find the values of \(x _ { 1 }\) and \(x _ { 2 }\).
  7. Show that, to 2 decimal places, \(\alpha = - 0.58\).
OCR MEI C2 2008 June Q5
4 marks Moderate -0.8
5 In Fig. 5, A and B are the points on the curve \(y = 2 ^ { x }\) with \(x\)-coordinates 3 and 3.1 respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-2_700_728_1197_705} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the gradient of the chord AB . Give your answer correct to 2 decimal places.
  2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2 ^ { x }\) at A .
AQA AS Paper 2 2020 June Q6
6 marks Moderate -0.3
6 A circle has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 4 y - 71 = 0$$ 6
  1. Find the centre of the circle.
    6
  2. Hence, find the equation of the tangent to the circle at the point (1, 10), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.