Tangent lines revisited

Rules of differentiation: Applications of derivatives

Tangent lines revisited

At the introduction (see introduction and the notion of difference quotient) of differentiation we started with the question: can we find the slope of a tangent? Now we know how to do it.

Let $f$ be a function that is differentiable at $a$ . The tangent line to $f$ at $a$ is given by the linear equation $y-f(a)=f'(a)\cdot (x-a)\tiny.$

The point $\rv{x,y}=\rv{a,f(a)}$ is a solution of the linear equation. This means that $\rv{a,f(a)}$ lies on the line given by the equation.

Moreover, the slope $f'(a)$ of $f$ at $a$ is equal to the slope of that line. Thus, the line given by the linear equation is the unique line through $\rv{a,f(a)}$ with slope $f'(a)$ . This implies that it is the tangent line to $f$ at $a$ .

It is not necessary to memorize this formula for the tangent line. What matters is that it is the unique line through the point $\rv{a,f(a)}$ with slope $f'(a)$ .

This ensures that we can calculate the tangent ourselves.

In the figure below, the graph $f(x)={{x^2}\over{4}}+4$ and the tangent to $f$ at the point $\rv{3,{{25}\over{4}}}$ are drawn.

Determine the function rule $l(x)$ for the tangent at the point $\rv{3,{{25}\over{4}}}$ . Enter your answer in the form $a\cdot x+b$ for suitable values of $a$ and $b$ .

$l(x)=$ ${{3\cdot x}\over{2}}+{{7}\over{4}}$

The requested function rule has the form $l(x)=a\cdot x+b$ , where $a$ and $b$ are real numbers.

The number $a$ is the slope of the tangent line, so $a=f'(3)$ . The derivative of the function $f$ equals $f'(x)={{x}\over{2}}$ . Therefore, $a=f'(3)={{3}\over{2}}$ .

We conclude that the function rule of the tangent line looks like $l(x)={{3}\over{2}} \cdot x +b$ , where $b$ is yet to be determined. We do so by using the fact that the tangent line moves through the point $\rv{3,{{25}\over{4}}}$ . This means that ${{25}\over{4}}={{3}\over{2}} \cdot 3 +b$ and, hence, $b={{25}\over{4}}-{{9}\over{2}} = {{7}\over{4}}$ .

In conclusion, the formula for the tangent is $l(x)={{3\cdot x}\over{2}}+{{7}\over{4}}$ .

New example