Find curve equation from derivative

A question is this type if and only if it gives the derivative f'(x) with unknown constants and uses conditions such as the curve passing through given points or having a stationary point to determine the constants and hence find f(x).

2 questions · Moderate -0.2

1.08a Fundamental theorem of calculus: integration as reverse of differentiation
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OCR C2 2009 June Q6
8 marks Moderate -0.8
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a\), where \(a\) is a constant. The curve passes through the points \(( - 1,2 )\) and \(( 2,17 )\). Find the equation of the curve.
Edexcel AS Paper 1 2023 June Q16
6 marks Standard +0.3
  1. A curve has equation \(y = \mathrm { f } ( x ) , x \geqslant 0\)
Given that
  • \(\mathrm { f } ^ { \prime } ( x ) = 4 x + a \sqrt { x } + b\), where \(a\) and \(b\) are constants
  • the curve has a stationary point at \(( 4,3 )\)
  • the curve meets the \(y\)-axis at - 5
    find \(\mathrm { f } ( x )\), giving your answer in simplest form.