Secant Lines vs. Tangent Lines
Secant lines versus tangent lines. Suppose we have a function f(x), now one thing we could ask is. As we start at one point. Say at a. And go to an other point, say at b.
We could ask on average, how much does the function increase? In other words we could just connect the dots as it were from a to b. And we would get something called a secant line. Now, finding the slope of a secant line is relatively easy. The slope of a secant line, would just be the change in y, f(b) minus f(a), over the change in x, b minus a.
And again, this would give you the average increase of a function if this were. A position vs time graph, the secant line would give you the average velocity of the function. So that's something very easy. Now, turns out mathematicians for centuries, were trying to solve a much harder problem.
Not the secant line but rather the tangent line. See, secant line touches the function at two points. A tangent line, touches the function only at one point. And that's a much harder problem to find the slope of a tangent line. Because notice that we don't have a change in y, or a change in x.
We only have one point. So both change in y, and change in x would be 0. Which is not very helpful so, with ordinary geometric means with ordinary means that we have of thinking of lines in the x, y plane. It is impossible to find the slope of the tangent line. And in fact this whole problem, finding the slope of the tangent line, is one of the central questions that Calculus answers, so without Calculus, it is not possible to answer this question.
Also, I'll point out why is this important. Well again, on an x vs T graph the slope of the tangent line, would tell you the instantaneous velocity at a single point. And more generally, the slope of the tangent line tells you the instantaneous rate of change of the function. How much the function is changing, at that specific immediate instant in space and time.
So this is an introduction, to the problem of Calculus.
Read full transcriptWe could ask on average, how much does the function increase? In other words we could just connect the dots as it were from a to b. And we would get something called a secant line. Now, finding the slope of a secant line is relatively easy. The slope of a secant line, would just be the change in y, f(b) minus f(a), over the change in x, b minus a.
And again, this would give you the average increase of a function if this were. A position vs time graph, the secant line would give you the average velocity of the function. So that's something very easy. Now, turns out mathematicians for centuries, were trying to solve a much harder problem.
Not the secant line but rather the tangent line. See, secant line touches the function at two points. A tangent line, touches the function only at one point. And that's a much harder problem to find the slope of a tangent line. Because notice that we don't have a change in y, or a change in x.
We only have one point. So both change in y, and change in x would be 0. Which is not very helpful so, with ordinary geometric means with ordinary means that we have of thinking of lines in the x, y plane. It is impossible to find the slope of the tangent line. And in fact this whole problem, finding the slope of the tangent line, is one of the central questions that Calculus answers, so without Calculus, it is not possible to answer this question.
Also, I'll point out why is this important. Well again, on an x vs T graph the slope of the tangent line, would tell you the instantaneous velocity at a single point. And more generally, the slope of the tangent line tells you the instantaneous rate of change of the function. How much the function is changing, at that specific immediate instant in space and time.
So this is an introduction, to the problem of Calculus.