5.4 Tangents and normalsIB Maths: Applications and Interpretation SL: Revision notes

A tangent to a curve at a point is the straight line that touches the curve there and has the same gradient as the curve. Its gradient is the value of the derivative at that point: $m_{\text{tangent}}=f'(a)$ at $x=a$. A normal is the straight line through the same point that is perpendicular to the tangent. Perpendicular gradients multiply to $-1$, so $m_{\text{normal}}=-\frac{1}{f'(a)}.$ Take the reciprocal and change the sign. For example, a tangent gradient of $8$ gives a normal gradient of $-\frac18$; a tangent gradient of $-\frac23$ gives a normal gradient of $\frac32$. If the tangent is horizontal ($f'(a)=0$), the normal is the vertical line $x=a$.

Steps:

1. Find $y$ at the given $x$ to get the point $(x_1,y_1)$ (if not given).
2. Differentiate and substitute $x_1$ to get the gradient $m$.
3. Use $y-y_1=m(x-x_1)$ and rearrange as the question asks (for example $y=mx+c$ or $ax+by+d=0$). Example: $y=x^3-4x+1$ at $P(2,1)$. $\frac{dy}{dx}=3x^2-4=8$ at $x=2$. Tangent: $y-1=8(x-2)$, so $y=8x-15$.

Use the same point, but the perpendicular gradient. For the curve above, the normal at $P(2,1)$ has gradient $-\frac18$: $y-1=-\frac18(x-2)$, so $y=-\frac18x+\frac54$, or $x+8y-10=0$. To find where a tangent or normal meets an axis, set $y=0$ for the $x$-axis or $x=0$ for the $y$-axis. The length of a normal between two points comes from Pythagoras: $\sqrt{(\Delta x)^2+(\Delta y)^2}$.

To find where the tangent has gradient $k$, solve $f'(x)=k$, then find $y$ for each solution. A tangent is parallel to a line with gradient $k$ when $f'(x)=k$. Example: for $y=2x^2-5x+3$ the tangent is parallel to $y=3x+1$ when $4x-5=3$, so $x=2$ and the point is $(2,1)$. If $f'(x)=k$ is a quadratic equation, there may be two solutions, so give both points. Where $f'(x)=0$ the tangent is horizontal.

A GDC can graph $y=f(x)$ and draw the tangent at a chosen point, then display the tangent's equation. It can also evaluate the gradient at a point numerically, solve $f'(x)=k$, and find the intersection of a tangent or normal with the curve or an axis. Use technology to check an analytic answer or when the numbers are awkward, and quote results to three significant figures unless exact values are asked for. For 'show that' or 'find the exact' questions, show the algebra. In context questions, state what the tangent or normal represents, for example the slope of a hillside or the direction of a tunnel, and include units for lengths.