A **tangent line** to a function at a point is a line that is in contact with the graphical representation of the function only in that particular point.

The **tangent line** is only touching the function (graph) in the specified point. In other words the tangent line is barely in contact with function (graph).

The line which is crossing (intersecting) a function graph it’s called a **secant line**.

In the example above the red line is the tangent. It’s tangent to the *f(x)* function in the point *P(x _{1}, y_{1})*. The blue line is the secant and as you can see it’s crossing the function

*f(x)*in two points.

Given a function *f(x)* and a point *P _{1}(x_{1}, y_{1})*, how do we calculate the tangent? Finding the tangent means finding the equation of the line which is tangent to the function

*f(x)*in the point

*P*.

_{1}(x_{1}, y_{1})**Example**: Find the tangent of the function *f(x)* defined below, in the point *x _{1}*.

**Step 1**: Calculate the *(x, y)* coordinates of the tangent point

The first thing to do is to evaluate the function in the tangent point *x _{1}*.

We know that:

\[y_{1}=f(x_{1})\]This means that the result of the function evaluation in the point *x _{1}* is the

*y*coordinate. We now have defined the point

_{1}*P*. Let’s plot the function

_{1}(1, 1)*f(x)*for

*x = 0 … 2*and the point

*P*with

_{1}*x*and

_{1}*y*coordinates.

_{1}**Step 2**: Define two points *P _{0}* and

*P*, on the left and right of

_{2}*P*.

_{1}We’ll chose a point *P _{0}* at

*x*and

_{0}= 0.6*P*at

_{2}*x*. These point must be equally distributed on the left and right of the

_{2}= 1.4*P*tangent point.

_{1}**Step 3**. Calculate the *y* coordinates of the *P _{0}* and

*P*points

_{2}First let’s calculate the y coordinate of the *P _{0}* point.

y_{0}&=f(x_{0})\\

f(0.6)&=2-{0.6}^{3}=1.784\\

y_{0}&=1.784

\end{split} \end{equation*} \]

We now have defined the point *P _{0} (0.6, 1.784)*.

Now we calculate the y coordinate of the *P _{2}* point:

y_{2}&=f(x_{2})\\

f(1.4)&=2-{1.4}^{3}=-0.744\\

y_{2}&=-0.744

\end{split} \end{equation*} \]

We now have defined the point *P _{2} (1.4,-0.744)*.

Let’s now update our function graph with both *P _{0}* and

*P*points and their coordinates.

_{2}**Step 4**. Calculate the slopes of the lines defined between the points *P _{0}-P_{1}* and

*P*

_{1}-P_{2}Having two points defined we can easily draw a straight line between them. Also with the coordinates of the two points we can calculate the slope of the line.

In our case we’ll have two lines, one between point *P _{0}* and

*P*and the second between

_{1}*P*and

_{1}*P*. For both lines we can calculate their slope.

_{2}The usual notation of the slope of a line is the letter *m*.

For the first line, the slope *m _{0}* is calculated as:

m_{0}&=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}=\frac{1-1.784}{1-0.6}=-1.96\\

m_{0}&=-1.96

\end{split} \end{equation*} \]

For the second line, the slope *m _{2}* is calculated as:

m_{2}&=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-0.744-1}{1.4-1}=-4.36\\

m_{2}&=-4.36

\end{split} \end{equation*} \]

Below is the graphical representation of the two lines and the *f(x)* function. Both lines are secant to our *f(x)* function.

**Step 5**. Calculate the slope of the line tangent in the point *P _{1} (1, 1)*

**The slope of the line tangent in the point P_{1} will be the arithmetic mean of the slopes of the two secant lines**. This method of calculation is possible because we have chosen the

*x*and

_{0}*x*points at equal distance from

_{2}*x*.

_{1}m_{1}&=\frac{m_{0}+m_{2}}{2}=\frac{-1.96-4.36}{2}=-3.16\\

m_{1}&=-3.16

\end{split} \end{equation*} \]

**Step 6 (final step)**. Calculate the equation of the tangent line in the point *P _{1}*.

Now that we know the coordinates of the tangent point P1 and the value of the slope, we can easily calculate the equation of the tangent line:

\[ \begin{equation*} \begin{split}y(x) &= y_{1} + m_1 \cdot (x – x_1)\\

y(x) & = 1 – 3.16 \cdot (x – 1)

\end{split} \end{equation*} \]

Having the tangent line equation we can plot on the same graph the function *f(x)* and the tangent line *y(x)*.

As you can see the tangent line is “touching” the function *f(x)* in the point *P _{1} (1, 1)*.

Using the same method we can calculate the tangent line in the point (1, 1) for the following functions:

\[f(x)=x^2 + 2 \cdot x + 1\] \[f(x)=10 \cdot \sqrt{2 \cdot x} + 3 \cdot x + 2\]Apply the same method to your function and let us now the result!

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