Increasing and Decreasing Functions/ Concavity and Second Derivative

Standard

IN CLASS WE HAVE BEEN TALKING ABOUT INCREASING AND DECREASING FUNCTIONS. HERE ARE SOME NOTES THAT I HAVE

When we talk about increasing and decreasing, we mean whether the graph goes up or down as it moves from LEFT to RIGHT.

When a function is increasing at a particular point, the tangent line at that point is increasing. This means that the slop of the tangent line is positive, so the derivative of the function at that point is positive.

When a function is decreasing at a particular point, the tangent line at that point is decreasing. This means that the slope of the tangent line is negative, which means that the derivative at that point is negative.

When trying to find if the graph is increasing or decreasing we have to first find the derivative and set it to zero to find the critical values.

The critical values are x-values in the domain of the function where the derivative of the function is equal to zero or does not exist. Once you find your critical values you can pick points on either side of the critical values to find if the derivative is negative or positive. If the derivative is negative then the function is decreasing between those intervals and if the derivative is positive it is increasing between those intervals.

When a function changes direction from either increasing to decreasing or form decreasing to increasing, we have a turning point.

If the turning point is changing from decreasing to increasing it a relative minimum. If the turning point is changing from increasing to decreasing then it is the relative maximum.

CONCAVITY AND SECOND DERIVATIVE

It is frequently useful to analyze the derivative of the derivative of a function. This is called the second derivative of the function.

If the original functions gives you an objects position, the first derivative gives you velocity. The second derivative gives you acceleration.

If the second derivative of a function is positive at a particular point, then the original function is concave downward a that point.

If the second derivative of a function is negative at a particular point, then the original function is concave downward at that point. 

There are inflection points on graphs, which is where the second derivative changes sign. ( Remember, a function can change sign at points where it is zero, or at points where there is a discontinuity)  

To find if the graph is concave down or up, you need to find the second derivative and set it equal to zero. This will give you the critical values for the second derivative. Pick numbers to the left and right of the critical values to find if the graph is concave up ro down between those intervals.

This sums up the notes I have on increasing and decreasing functions and concavity and second derivative. We had a test yesterday on some of this, I hope I did good. I am so ready for spring though so I can have time to relax. I wish everyone a wonderful and safe spring break. :]]]

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