Graphing Functions Using Derivatives
Derivatives could be used to graph functions. f'(x) could be used to represent a function's slope at a certain period of time. f"(x) could be used to show the concavity of the function, where it changes signs.
Sketch a possible graph for f(x).
Concavity
If y">0, the function is concave up.
If y"<0, the function is concave down.
Sketch a possible graph for f(x).
From negative infinity to x1, y' and y" are less than 0. So the curve from negative infinity to x1 will be an decreasing curve that is concave down. A x1, y'=0. That means the curve will be flat a horizontal line at that instant.
From x1 to x2, y' is greater than 0 while y" is less than 0. That means the curve is increasing on a concave down curve. At x2, the curve will be flat or becomes a horizontal line at that instant.
From x2 to x3, the y' and y" are greater than 0. That means the curve is increasing at a concave up curve. At x3, the curve will be a horizontal line at that instance because y'=0 at x3.
From x3 to positive infinity, y' and y" are both greater than zero. That means the curve is increasing at a concave up curve.
From negative infinity to x2 (open interval), y'=-3 and y"=0. So the curve will be a decreasing linear line with a slope of -3. Since, y' is already a constant, y" have to be zero because there is no acceleration.
At x2, y' and y" are undefined. That means the function is undefined at x2. There is a hole at x2.
From x2 to positive infinity, y' and y" are greater than 0. That means the curve is increasing on a concave up curve.
