Expand f(a+h) algebraically

A question is this type if and only if it asks students to expand f(a+h) or (a+h)ⁿ and express the result in a specified form, as a preliminary step before finding a derivative.

4 questions · Moderate -0.6

1.07g Differentiation from first principles: for small positive integer powers of x
Sort by: Default | Easiest first | Hardest first
AQA FP1 2011 June Q6
7 marks Moderate -0.8
6
  1. Expand \(( 5 + h ) ^ { 3 }\).
  2. A curve has equation \(y = x ^ { 3 } - x ^ { 2 }\).
    1. Find the gradient of the line passing through the point \(( 5,100 )\) and the point on the curve for which \(x = 5 + h\). Give your answer in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
    2. Show how the answer to part (b)(i) can be used to find the gradient of the curve at the point \(( 5,100 )\). State the value of this gradient.
AQA FP1 2005 June Q4
7 marks Moderate -0.8
4 The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x$$
  1. Express \(\mathrm { f } ( 2 + h ) - \mathrm { f } ( 2 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
  2. Use your answer to part (a) to find the value of \(\mathrm { f } ^ { \prime } ( 2 )\).
AQA AS Paper 1 Specimen Q9
5 marks Moderate -0.3
  1. Given that \(f(x) = x^2 - 4x + 2\), find \(f(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
  2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]
SPS SPS SM Pure 2021 June Q8
5 marks Moderate -0.3
  1. Given that \(\mathbf{f}(x) = x^2 - 4x + 2\), find \(\mathbf{f}(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
  2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]