Derivative Functions Monotonicity

A function that is always

increasing or decreasing on

an interval is said to be

monotonic there. Function is increasing

when the first derivative is

positive Function is decreasing

when the derivative

is negative or below

the x-axis A critical point occurs when the first derivative equals zero or does not exist To find local extremas use the First Derivative Test.

1. Find the Critical Point

2. Use a sign chart to see if the derivative changes from positive to negative for a local max, if it changes from negative to positive it is a local min, if it doesn't change signs it is not an extrema

3. When the drivative is positive, the original graph is increasing, and when the derivative is negative the original graph is decreasing To find concavity use the second derivative test:

1. Set the second derivative equal to zero and solve for x to find possible inflection points.

2. Set up a sign chart. If the second derivative is negative the original function is concave down on that interval. If the second derivative is positive the original function is concave up on that interval. A point of inflection is when the original function changes concavity. How to determine if an endpoint is a max or min and whether an extrema is absolute or relative:

1. Create an tabular model for x and y values.

2. Plug in all potential max and mins, found using the first derivative test, to the original equation and solve for the y-values

~If the interval is close, don't forget to check the endpoints!

3. The largest and smallest y-values are the absolute max and mins. The others are relative on that interval. If a function, f, is continuous and differentiable on the interval (a,b) then there is some point where the slope of the tangent line is equal to the slope of the secant line on that interval. Mean Value Theorem Extreme Value Theorem If f is continuous on a closed interval [a,b] then f has both a maximum value and a minimum value on the interval. Sketch the graph of f given the graph of f'

1. Find where the function is increasing by finding the interval on the derivative graph where it is positive

2. Find where the function is decreasing by finding the interval on the derivative graph where it is negative.

3. Where f' crosses the x-axis are the max and mins of the original function

4. Use this information to plot a rough sketch. Sketch the graph of f' given the graph of f:

1. When the original is increasing, the graph of the derivative is above the x-axis.

2. When the original is decreasing, the graph of the derivative is below the x-axis.

3.When there is a max or a min on the original graph, the derivative will cross the x-axis Sketch the graph of f given the graph f":

1. When the graph of the second derivative is positive the original function is concave up

2. when the graph of the second derivative is negative the original function is concave down.

3. When the second derivative crosses the x-axis there is an inflection point on the original By Rachel and Frankie If is continuous on a closed interval [a,b], and d is any number between f(a) and f(b) inclusive, then there is at least one number, c, in the closed interval such that f(c)=d. Intermediate Value Theorem

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# Derivative Functions-Frankie Z and Rachel K

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