Critical points
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Calculus involves finding the gradient (or the "slope") at any point on a graph. When we study some polynomial graphs, we notice that at some points they have a positive gradient (an upwards slope) and at other points they have a downwards slope. But on some graphs like x = D, a simple vertical line through the point (D, 0), the gradient is neither positive, nor negative. It is undefined. Afterall, how could we describe or quantify the "steepness" of a vertical line?
If you look at other graphs, you notice that they rise and fall at certain points. Some parabolas look like archways because they have what's called a "turning point." Some cubics have turning points too. These points are where the graph rises and drops. If you zoomed in on a turning point enough, you would notice that it straightens out into a small space of horizontol line. And a horizontol line has zero gradient.
So some of the interesting features of a graph (the turning points, the points of inflection) all have ZERO gradient. The point on the graph where the curve "straightens out" temporarily is called a stationary point (called stationary because the graph is not increasing or decreasing at this point). If you throw a ball into the air, and it makes an arc and begins to descend, the point at which it is "stationary" in the air is the stationary point on its displacement-time graph.
Stationary points come in a few types:
Turning points - The point where the curve "dips;" ie: a curve has a certain gradient on one side of a point, and when it "bounces" or "turns" from a particular point it achieves a negated gradient. So, if it's positive on one side of a turning point, it will bounce off into a negative gradient.
Turning points: Local maximum - A local maximum point is where a curve approaches a stationary point with a positive slope and turns off that point into a negative slope, illustrated sloppily below:
You should look up the other types of stationary points. Remember, that the gradient function of y, dy/dx, or the 'derivative of y with respect to x at any point (x,y)', must be set to '0' if you want to find stationary/critical points. If you don't know how to calculate the derivative, you shouldn't be bothering about stationary points yet.
Once you find a bunch of stationary points by solving dy/dx = 0, you must "classify" them as a local maximum, local minimum, or stationary point, for example. Means of classifying stationary points involve the "second derivative test," ie, if x = D is a stationary point and y"(D) > 0, then a local minimum point lies on the line X = D. There are other tests, too.
If you look at other graphs, you notice that they rise and fall at certain points. Some parabolas look like archways because they have what's called a "turning point." Some cubics have turning points too. These points are where the graph rises and drops. If you zoomed in on a turning point enough, you would notice that it straightens out into a small space of horizontol line. And a horizontol line has zero gradient.
So some of the interesting features of a graph (the turning points, the points of inflection) all have ZERO gradient. The point on the graph where the curve "straightens out" temporarily is called a stationary point (called stationary because the graph is not increasing or decreasing at this point). If you throw a ball into the air, and it makes an arc and begins to descend, the point at which it is "stationary" in the air is the stationary point on its displacement-time graph.
Stationary points come in a few types:
Turning points - The point where the curve "dips;" ie: a curve has a certain gradient on one side of a point, and when it "bounces" or "turns" from a particular point it achieves a negated gradient. So, if it's positive on one side of a turning point, it will bounce off into a negative gradient.
Turning points: Local maximum - A local maximum point is where a curve approaches a stationary point with a positive slope and turns off that point into a negative slope, illustrated sloppily below:
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. .You should look up the other types of stationary points. Remember, that the gradient function of y, dy/dx, or the 'derivative of y with respect to x at any point (x,y)', must be set to '0' if you want to find stationary/critical points. If you don't know how to calculate the derivative, you shouldn't be bothering about stationary points yet.
Once you find a bunch of stationary points by solving dy/dx = 0, you must "classify" them as a local maximum, local minimum, or stationary point, for example. Means of classifying stationary points involve the "second derivative test," ie, if x = D is a stationary point and y"(D) > 0, then a local minimum point lies on the line X = D. There are other tests, too.
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Re: Critical points examples
For step-by-step worked examples and explanations, please review a textbook or an on-topic web site, or else please consider enrolling in a calculus course.
My best wishes to you in your studies.
Eliz.
. . . . .Visual Calculus: Maxima and Minima
. . . . .Paul's Math Notes: Applications of Derivatives
. . . . .Derivatives in Curve Sketching
. . . . .Finding Extreme Values (PDF)
The previous poster went "above and beyond" by providing you with a mini-review. (Thank you, morson!) But I'm afraid we simply cannot reasonably provide the requested classroom instruction within this environment.After a tutor posted a mini-review said:
For step-by-step worked examples and explanations, please review a textbook or an on-topic web site, or else please consider enrolling in a calculus course.
My best wishes to you in your studies.
Eliz.
. . . . .Visual Calculus: Maxima and Minima
. . . . .Paul's Math Notes: Applications of Derivatives
. . . . .Derivatives in Curve Sketching
. . . . .Finding Extreme Values (PDF)