math (0422) - GeoDer: point of inflection

the point of inflection has a tangent with gradient = 0, ie. is a stationary point.

it occurs for x^3 graphs, where on both sides of the stationary point, y increases or decreases depending on the formula.

to test for this, we draw up a table and test for the number that results in f'(x) = 0, and then for a number higher than the x value that resulted in that, and lower than that.

the result will give us the shape of the graph

math (0421) - GeoDer: increase/decrease

this is essentially a continuation of yesterday's post.

just a few things to add.

maximum point is generally the staionary point or turning point tht is of highest y value. called "local maximum" if the graph is ongoing (ie. with arrows) but if it terminates then it's the absolute maximum.

same goes for minimum, which is the lowest.

interesting thing is that if one differentiates a straight line, the resulting graph would be a straight horizontal line, and it's y value would be equal to the gradient of the line, which was differentiated.

math (0420) - GeoDer: stationary point

GEOMETRY OF DERIVATIVE

or

GEOMETRICAL APPLICATIONS OF CALCULUS

yay..

at point where curve slopes upwards, tangent has positive gradient and y increases as x increases, therefore f(x) increases

at point whre curve slopes downwards, tangent has negative gradient and y decreases as x increases, therefore f(x) decreases

at the turning point or stationary point, the tangent's gradient = 0 and f(x) is stationary

also, differentiation a function twice gives the concavity of the graph at that point