Find curve from gradient

Given dy/dx and a point on the curve, find the equation y = f(x) by integration.

48 questions · Moderate -0.7

Sort by: Default | Easiest first | Hardest first
Edexcel C1 Q10
12 marks Moderate -0.3
The curve \(C\) with equation \(y = \text{f}(x)\) is such that $$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that \(C\) passes through the points \((0, -2)\) and \((2, 18)\),
  1. show that \(k = 2\) and find an equation for \(C\), [7]
  2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact. [5]
Edexcel C1 Q8
9 marks Moderate -0.3
Given that $$\frac{dy}{dx} = \frac{x^3 - 4}{x^2}, \quad x \neq 0,$$
  1. find \(\frac{d^2y}{dx^2}\). [3]
Given also that \(y = 0\) when \(x = -1\),
  1. find the value of \(y\) when \(x = 2\). [6]
Edexcel C1 Q7
8 marks Moderate -0.3
Given that $$\text{f}'(x) = 5 + \frac{4}{x^2}, \quad x \neq 0,$$
  1. find an expression for \(\text{f}(x)\). [3]
Given also that $$\text{f}(2) = 2\text{f}(1),$$
  1. find \(\text{f}(4)\). [5]
OCR C2 Specimen Q2
6 marks Easy -1.2
  1. Find \(\int \frac{1}{x^2} dx\). [3]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = \frac{1}{x^2}\). Find the equation of the curve, given that it passes through the point \((1, 3)\). [3]
OCR MEI C2 2006 June Q5
4 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 3 - x^2\). The curve passes through the point \((6, 1)\). Find the equation of the curve. [4]
OCR MEI C2 2008 June Q6
4 marks Moderate -0.8
A curve has gradient given by \(\frac{\text{d}y}{\text{d}x} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
OCR MEI C2 2010 June Q6
5 marks Moderate -0.5
The gradient of a curve is \(6x^2 + 12x^{\frac{1}{2}}\). The curve passes through the point \((4, 10)\). Find the equation of the curve. [5]
OCR MEI C2 2013 June Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x^3} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR C2 Q3
7 marks Moderate -0.3
Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{4}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]
OCR C2 Q9
13 marks Moderate -0.3
  1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
  2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]
OCR C2 Q8
12 marks Moderate -0.3
  1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
  2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]
OCR MEI C2 Q11
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6x^{\frac{1}{2}} - 5\). Given also that the curve passes through the point \((4, 20)\), find the equation of the curve. [5]
OCR MEI C2 Q5
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6\sqrt{x} - 2\). Given also that the curve passes through the point \((9, 4)\), find the equation of the curve. [5]
OCR MEI C2 Q8
5 marks Moderate -0.8
The gradient of a curve is \(3\sqrt{x} - 5\). The curve passes through the point \((4, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q9
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
AQA AS Paper 1 2024 June Q10
6 marks Standard +0.3
It is given that $$\frac{\mathrm{d}y}{\mathrm{d}x} = (x + 2)(2x - 1)^2$$ and when \(x = 6\), \(y = 900\) Find \(y\) in terms of \(x\) [6 marks]
AQA AS Paper 2 2018 June Q5
4 marks Standard +0.3
\(f'(x) = \left(2x - \frac{3}{x}\right)^2\) and \(f(3) = 2\) Find \(f(x)\). [4 marks]
AQA Paper 3 2024 June Q6
5 marks Easy -1.2
\begin{enumerate}[label=(\alph*)] \item Find \(\int \left(6x^2 - \frac{5}{\sqrt{x}}\right) dx\) [3 marks] \item The gradient of a curve is given by $$\frac{dy}{dx} = 6x^2 - \frac{5}{\sqrt{x}}$$ The curve passes through the point \((4, 90)\). Find the equation of the curve. [2 marks]
SPS SPS SM 2022 February Q5
6 marks Easy -1.2
The gradient of a curve is given by \(\frac{dy}{dx} = 2x^{-\frac{1}{2}}\), and the curve passes through the point \((4, 5)\). Find the equation of the curve. [6]
SPS SPS SM 2025 February Q5
7 marks Standard +0.3
A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]