Tangents & Normals Flashcards

All 6 cards in this deck

What are the steps to find the equation of the tangent to a curve at a given point, in the form $y = mx + c$?
e.g. the tangent to $y = x^2$ at $(3, 9)$

1. Differentiate: $\frac{dy}{dx} = 2x$
2. Substitute the $x$-coordinate to get the gradient: $m = 6$
3. Use $y - y_1 = m(x - x_1)$: $y = 6x - 9$

What is the normal to a curve at a point?

The straight line through that point perpendicular to the tangent there.

What are the steps to find the equation of the normal to a curve at a given point?
e.g. the normal to $y = x^2$ at $(1, 1)$

1. Differentiate and substitute $x$ for the tangent gradient: $m = 2$
2. Normal gradient $= -\frac{1}{m} = -\frac{1}{2}$
3. Use $y - y_1 = m(x - x_1)$: $y = -\frac{1}{2}x + \frac{3}{2}$

What are the steps to find the point on a curve where the tangent is parallel to a given line?
e.g. on $y = x^2 - 4x$, tangent parallel to $y = 2x + 1$

1. Differentiate: $\frac{dy}{dx} = 2x - 4$
2. Set $\frac{dy}{dx}$ equal to the line's gradient: $2x - 4 = 2$, so $x = 3$
3. Substitute into the curve for $y$: $(3, -3)$

What are the steps to find the area of the triangle a tangent forms with the coordinate axes?
e.g. the tangent $y = 2x - 6$

1. Put $y = 0$ for the $x$-intercept: $x = 3$
2. Put $x = 0$ for the $y$-intercept: $y = -6$
3. Area $= \frac{1}{2} \times$ base $\times$ height $= \frac{1}{2} \times 3 \times 6 = 9$

True or false? To find the gradient of the tangent at a point you take the negative reciprocal of $\frac{dy}{dx}$ at that point.

False. Substituting into $\frac{dy}{dx}$ gives the tangent gradient directly; the negative reciprocal gives the normal gradient.