Finding the Constant of Integration | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

A point selects one curve

The functions $x^2-1$ , $x^2+1$ and $x^2+3$ all have derivative $2x$ . Changing $c$ shifts the curve vertically: its height changes, but its gradient at a fixed $x$ does not.

At x = 1, all three gradients are 2, but only y = x² + 3 passes through (1, 4).

A given point supplies the missing height. The notation $f(2)=3$ means that $(2,3)$ lies on the original graph, not on its derivative graph.

Integrate before substituting the point

Worked example

Given dy/dx = 6x − 4 and the point (2, 3), find the curve.

\begin{aligned}y&=3x^2-4x+c\\3&=3(2)^2-4(2)+c\\c&=-1\end{aligned}

Answer

y=3x^2-4x-1

Check both pieces of information: the derivative is $6x-4$ and the function gives 3 at $x=2$ . Substituting 3 for the gradient would use the point incorrectly.

Your turn9709/13 May/Jun 2020 Q2 [4 marks]

A curve has $dy/dx=3\sqrt{x}-3/\sqrt{x}$ and passes through $(4,7)$ . Find its equation.

Show worked answer

\begin{aligned}y&=2x^{3/2}-6x^{1/2}+c\\7&=16-12+c\\c&=3\end{aligned}

Answer

\begin{gathered}y=2x^{3/2}-6\sqrt{x}+3\\x>0\end{gathered}

Differentiate and substitute the point to check. The original gradient expression excludes $x=0$ .

A point selects one curve

The functions $x^2-1$ , $x^2+1$ and $x^2+3$ all have derivative $2x$ . Changing $c$ shifts the curve vertically: its height changes, but its gradient at a fixed $x$ does not.

At x = 1, all three gradients are 2, but only y = x² + 3 passes through (1, 4).

A given point supplies the missing height. The notation $f(2)=3$ means that $(2,3)$ lies on the original graph, not on its derivative graph.

Integrate before substituting the point

Worked example

Given dy/dx = 6x − 4 and the point (2, 3), find the curve.

\begin{aligned}y&=3x^2-4x+c\\3&=3(2)^2-4(2)+c\\c&=-1\end{aligned}

Answer

y=3x^2-4x-1

Check both pieces of information: the derivative is $6x-4$ and the function gives 3 at $x=2$ . Substituting 3 for the gradient would use the point incorrectly.

Your turn9709/13 May/Jun 2020 Q2 [4 marks]

A curve has $dy/dx=3\sqrt{x}-3/\sqrt{x}$ and passes through $(4,7)$ . Find its equation.

Show worked answer

\begin{aligned}y&=2x^{3/2}-6x^{1/2}+c\\7&=16-12+c\\c&=3\end{aligned}

Answer

\begin{gathered}y=2x^{3/2}-6\sqrt{x}+3\\x>0\end{gathered}

Differentiate and substitute the point to check. The original gradient expression excludes $x=0$ .