The tangent line

Differentiation: The derivative

The tangent line

Consider the graph of the function $f(x)=2\sqrt{x}$ pictured to the right. We can draw a straight line $\orange{k}$ for the difference quotient at $\blue{a}$ with difference $\green{h}$ which moves through the two points $\rv{\blue{a},f(\blue{a})}$ and $[\blue{a}+\green{h},f(\blue{a}+\green{h})]$ , like we did with linear functions.

If we take a smaller $\green{h}$ , we see that line $\orange{k}$ around point $\blue{a}$ is increasingly similar to the graph.

plaatje

The tangent line is an important concept. This is a straight line through a point on a graph that has the same slope as the graph at that point.

The tangent line

We call a line that touches a graph at point $\blue{P}$ the tangent line at that point $\blue{P}$ . This means that the slope of the tangent line and of the graph at point $\blue{P}$ are equal, making them look very similar around point $\blue{P}$ .

The tangent line $\orange{k}$ is a linear function $\orange{k(x)}=\orange{ax+b}$ . Here the slope $a$ is equal to the slope of the graph at point $\blue{P}$ .

In the image to the right, you can move the point $\blue{P}$ yourself to see what the tangent line $\orange{k}$ to the graph $f(x)=\tfrac{1}{5}x^3-2x$ is at point $\blue{P}$ .

plaatje

Below you see the graph $f(x)=x^2-3x$ . Additionally, the lines $a$ , $b$ , $c$ , and $d$ are drawn.

Which of these lines is/are tangent lines of the graph $f(x)$ ?

Line $a$

Line $b$

Line $c$

Line $d$