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.

3 questions · Moderate -0.5

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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 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 \(x = 2\).