what is the difference between tangent and slope of tangent?
I need your help, my question is what is the difference between tangent and slope of tangent ?
A clear example would be appreciated.
Thank you.
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$\begingroup$The tangent to a curve at a point is a straight line just touching the curve at that point; the slope of the tangent is the gradient of that straight line.
Here's a picture to help.
The green line is the tangent line to the point $(1,1)$. It is a geometric object.
The slope at $(1,1)=2$ (if you want to know why, it's to do with differentiating, or finding the gradient of the tangent at a point). This is a number.
So, to conclude: the tangent's a line; the slope is a number.
Qualitatively, your question is equivalent to the following:
"What's the difference between a hill and the steepness of a hill?".
To answer your second question (in the comments):$$\tan(\theta)=\frac{\textrm{change in y}}{\textrm{change in x}}=\frac{\Delta y}{\Delta x}=\rm{gradient}$$ (we can see this from the right-angled triangle, if we use some basic trigonometry).
$\endgroup$ 6 $\begingroup$The English word 'Tangent' also means the tangent line (at a point of a circle, say), and its slope is the numeric value $\tan(\alpha)$ where $\alpha$ is the angle of the tangent line and the right wing and (any parallel to) the $x$-axis.
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