Summary and examples
We want to find an equation for the tangent line at a point of the graph of a function .
Before we go into this in more detail, we will first discuss the following general case: what is the equation of the line through a point with coordinates with slope .
The general equation of a line is:
Because the slope is equal to is, we have:
This line has to go through the point , thus we write:
from which it follows:
Now the formula for the line is:
It is most convenient to just remember this formula and this is not difficult given the simple, symmetric shape.
Now we return to the equation of a tangent line at a point of the graph of the function . When we want to determine the tangent line at the point of the graph of the function , the slope can be found by:
and thus de formula for the tangent line is:
Determine the tangent line at the point of the graph of the function:
First we calculate the derivative of the function:
In the point we have and this is the slope of the tangent line at .
According to the formula given above the equation of the tangent line is:
Where does the tangent line at the point of the graph of the function:
intersect the -axis.
First we determine the equation of the tangent line in the point .
The derivative of the function is:
In the point the derivative has the value , so the slope of the tangent line is . The tangent line goes through the point and thus the equation of the tangent line is:
The tangent line intersects the -axis in the point .
Given the function:
Determine the equation of the tangent line to the graph of this function, parallel to the line:
The slope of the tangent line to the graph of the given function is found by differentiation:
In a point with the slope of the tangent line is:
The line is parallel to the given line, so:
From this we may conclude:
The corresponding tangent points are:
The tangent lines through these points are, respectively: