Tangents

Two important concepts in differential calculus are the tangent (this section) and the secant (next section).

Motivation

Finding the slope of a tangent is a fundamental problem in calculus because it allows us to describe the instantaneous rate of change of a curve at a specific point.

Definition 1

Consider a smooth curve (that is, a curve without corners) and a point $A$ on this curve. The tangent to this curve at point $A$ is the straight line that touches but does not cross the curve at $A$ . Thus, close to $A$ , the straight line stays on the same side of the curve. The equation of the tangent is the linear function

Equation 1

t(x)=ax+b

The equation of the tangent is the graph of a linear function

whose graph (a straight line) is the tangent. Recall that $a$ is the slope of the linear function, that is, the slope from the tangent.

Insight

While a tangent line is defined by having a touchpoint, it can intersect the curve elsewhere many times.

Note 1

In rare cases we can determine the equation of the tangent directly, without the apparatus of calculus we will develop soon. See the example below:

Exercise 1

Let's find the tangent to a circle. Consider a circle with a radius of $1$ and a centre at $(0\vert 0)$ . Point $A$ is on this circle at angle $30^\circ$ . Determine the equation of the tangent to the circle at $A$ .

Hint: SOHCAHTOA

Solution

Note that at $A$ the radius and the tangent form a right angle. Thus we can apply SOHCAHTOA.

The equation of the tangent has the general form $t(x)=ax+b$ . For the slope $a$ it is (see figure below)

a=\frac{\Delta y}{\Delta x}=\tan(60^\circ)=1.732

But we have to take the negative value for the slope, because the straight line is decreasing.

And if we regard $b$ as the hypotenuse $H$ (see picture below), we see that

\sin(30^\circ)=\frac{R}{H}=\frac{1}{b}

and thus

b=\frac{1}{\sin(30^\circ)}=2

Thus, the equation of the tangent is

t(x)=-1.732 x +2