Chord gradient with h (algebraic)

A question is this type if and only if it requires finding and simplifying an algebraic expression for the gradient of a chord between points with x-coordinates a and a+h, then explaining how this relates to the gradient at point a.

7 questions · Moderate -0.6

Sort by: Default | Easiest first | Hardest first
OCR MEI C2 2009 June Q12
12 marks Moderate -0.8
12
  1. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
  2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
  3. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
  4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
  5. 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.
OCR C1 2016 June Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).
  1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
  2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
  3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).
AQA FP1 2009 June Q2
8 marks Moderate -0.8
2 A curve has equation $$y = x ^ { 2 } - 6 x + 5$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates 2 and \(2 + h\) respectively.
  1. Find, in terms of \(h\), the gradient of the line \(A B\), giving your answer in its simplest form.
  2. Explain how the result of part (a) can be used to find the gradient of the curve at \(A\). State the value of this gradient.
AQA FP1 2012 June Q2
7 marks Standard +0.3
2 A curve has equation \(y = x ^ { 4 } + x\).
  1. Find the gradient of the line passing through the point \(( - 2,14 )\) and the point on the curve for which \(x = - 2 + h\). Give your answer in the form $$p + q h + r h ^ { 2 } + h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
  2. Show how the answer to part (a) can be used to find the gradient of the curve at the point ( \(- 2,14\) ). State the value of this gradient.
CAIE P1 2024 November Q3
5 marks Moderate -0.8
The equation of a curve is \(y = 2x^2 - 3\). Two points \(A\) and \(B\) with \(x\)-coordinates 2 and \((2 + h)\) respectively lie on the curve.
  1. Find and simplify an expression for the gradient of the chord \(AB\) in terms of \(h\). [3]
  2. Explain how the gradient of the curve at the point \(A\) can be deduced from the answer to part (a), and state the value of this gradient. [2]
AQA FP1 2014 June Q5
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x(x + 3)\).
  1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
  2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]
AQA FP1 2016 June Q2
5 marks Moderate -0.8
A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).
  1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
  2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]