What is the meaning of "slope of the line at a point"

Question

I am new to calculus and until now i knew that slope of a straight line is the rate of change of the y-coordinate with respect to the change in x-coordinate of the straight line or the rise over run and to calculate it we need at least two points. Now, i recently encountered a statement where the phrase "slope of the line at a point" is used. What does this really mean? Don't we need two points to calculate the slope of a line? How come there exists a slope for a point in a line and isn't slope a property of the line and not the point?

Answer 1

See the plot:

First we have to define the slope of a straight line. Then "slope of a curved line at a point" means the slope of the tangent to the curve at that point and this is equivalent to bringing two points on the curve so close to each other that there will be negligible difference between them and then finding the slope of the line passing through the two infinitesimally close points and this line can be regarded as a tangent line.

Answer 2

Initially, we did need two points in order to calculate slope. However, many functions have this property, that if you were to take the two points closer and closer, the slope approaches some value. which we name the slope of the function at that "common" point.

For example let's take $f(x)=x^2$, taking two different points $(a,a^2),(b,b^2)$ yields a slope of $\frac{a^2-b^2}{a-b}=a+b$ if we take the second one near the first one (so close we can "assume" $a=b$), we get that the slope is $2a$. This is what we call the derivative of $f(x)$ at $x=a$.

Answer 3

Slope between two points in a Cartesian coordinate system is defined as the relationship between a change in the ordinate value and the associated change in the abscissa value ("the rise divided by the run"). That requires two points.

Slope at a single point cannot be defined that way. Indeed, slope at a point is not and cannot be defined by computation.* Instead it is defined by declaration. In any text or any answer here you will always find that the slope at a single point is described and inferred as being parallel to the slope of two nearby points between which it is centered. We speak of the slope at a point because it is useful to do so.

*The calculus derivative also arrives at the slope of a point by approximation. The rise over the run is taken as $[f(x+\Delta x)-f(x)]/\Delta x$

Approximation is introduced when $\Delta x$ is taken as infinitesimally small, but not zero. Therefore the computation is still for two points. But when the expression is expanded by applying the function, the resulting terms in the numerator which contain $\Delta x$ can be considered to be approximately zero, and it is permissible to divide by $\Delta x$ because it is not zero.