Finding the slope of a curve at a point is one of two fundamental problems in calculus. This abstract concept has a variety of concrete realizations, like finding the velocity of a particle given its position and finding the rate of a reaction given the concentration as a function of time.
A tangent is a straight line that touches a curve at a single point and does not cross through it. The point where the curve and the tangent meet is called the point of tangency. We know that for a line its slope at any point is . The same applies to a curve. When we say the slope of a curve, we mean the slope of tangent to the curve at a point.
To find the slope of a curve at a particular point, we differentiate the equation of the curve. If the given curve is we evaluate or and substitute the value of to find the slope.
For a line of the form (or any other form) we can find its slope by simply taking any two values of and and their respective values, and . We find the slope by the formula . In the case of curves our approach is somewhat different. In the above case, we had and . Now we need to find the slope of tangent to a curve at some point. To do this we again need
but this time and tend to zero, which means the interval is very small because it is a tangent at a point.
Notice that as the colored pairs of and come closer, the tangent shifts to a point on the graph.
When this happens we replace
and therefore find
What is the slope of curve at
The given curve is Evaluating we have
Substituting into this gives
Therefore the slope of the given curve at is
If the curve passes through and its slope at is then what are the values of and
The given curve is Evaluating , we have
Since the slope of the curve at is we have
Now, substituting into gives