Roots Of Equation From Graph

Question

Given a funny graph that resembles no usual function unfortunately I don't have a picture There are roots where the graph crosses the x-axis If for instance the function is f(x) which is unknown how can I determine the number of roots of f'(x) and F(x) the derivative and integral respectively from the graph Someone explained to me something about half horizontal tangent lines but I don't really understand what it means

Answer 1

Well, if all you have is a graph, then you have to interpret the derivative and the integral of a function $f(x)$ graphically.

The derivative of $f(x)$ at a point $a$ can interpreted as the slope of the line tangent to $f(x)$ at $a$. What does this mean? Take your curve, pick a point on that and draw tangent to your curve going through that point. Then $f'(a)$ is the slope of that line. (By slope, I mean the $m$ in $y=mx+c$). Now let me ask you, When will the f'(x) be zero? What is the 'graphical requirement'? Once your figure that out, you have the roots of $f'(x)$.

Let's look at the integral. I assume that $F(x) = \int_0^x f(x)dx$. This is quite easily interpreted as the area under the graph from $0$ to $x$ for $x>0$, and (although it doesn't matter in this case), the negative area under the curve from $x$ to $0$ for $x < 0$. If the area is over the x-axis, then we take it to be positive, and if the area is under the x-axis, we take it to be negative. Therefore $F(x) = 0$ when the areas "cancel out" or, more trivially, when we take the integral from $0$ to $0$.

As an example, look at the graph of $\sin(x)$ below. Can you see why the integral from $0$ to $2\pi$ is $0$? And where are the zeros of the derivative?

This information should be enough to get you started I hope.