Finding a curve from its gradient

5 minAS Level1.8.2

By the end of this lesson

By the end you can find the equation of a curve when you know its gradient $\dfrac{dy}{dx}$ and one point on it.

What they ask

A curve has $\dfrac{dy}{dx} = 6x^2 - 4$ and passes through the point $(2, 7)$ . Find the equation of the curve.

Other wordings mean the same: "the gradient of the curve is given by…", or "find $y$ in terms of $x$ ".

Integrating $\dfrac{dy}{dx}$ gives $y = \ldots + c$ . That's a whole family of curves: same shape, slid up or down.

All of them have the right gradient. The point tells you which one is yours.

Three steps:

Integrate $\dfrac{dy}{dx}$ to get $y = \ldots + c$ .
Put the point's $x$ and $y$ into $y = \ldots + c$ and solve for $c$ . Not into $\dfrac{dy}{dx}$ : the point is on the curve.
Write the curve's equation with the number in place of $c$ .

Worked example: The easiest one

A curve has $\dfrac{dy}{dx} = 2x$ and passes through $(1, 4)$ . Find its equation.

Have a go first?

Quick check

Write an expressionQuestion 1 · Recall

A curve has $\dfrac{dy}{dx} = 6x^2 - 4$ and passes through $(2, 7)$ . Find $y$ in terms of $x$ .

Type just the expression for $y$ .

Use ^ for powers, * or nothing for times, / to divide, and functions like sqrt( ), sin( ), ln( ). Any equivalent form is accepted.

Answer the 2 questions above to continue.

Every curve here has

\dfrac{dy}{dx} = 2x

. Only one goes through

(1, 4)

: the one with

c = 3

Multiple choiceQuestion 2 · Recall

The curves $y = x^3 + c$ all have $\dfrac{dy}{dx} = 3x^2$ . Which one passes through $(2, 3)$ ?

y = x^3 - 5

y = x^3 + 3

y = x^3 - 9

y = x^3 + 11