Sketching with the Derivative

If the second derivative is negative, then original function is ____

concave down

If the second derivative is positive, then the original function is ___

concave up

If a function is continuous, but there is an undefined point on the derivative, then there is either a ______, ________, or ________ on the original graph.

corner, cusp, or go-vertical

If the first derivative is negative, then the original function is _____

decreasing

If the first derivative is positive, then the original function is _______

increasing

A double root on the first derivative graph means that there is a zero at that point, but does not change signs around the zero point. This means that there is a huge ____ on the original graph and also indicates that there will be a ___ ____ ____ at this point.

layout point of inflection

If the first derivative is is zero, then there is either a _______, _______ or ______ on the original function.

maximum, minimum, or layout

If a function is even, then the signs of the first derivative to the left of the zero will be _______ of the signs on the right side of zero.

opposite

If a function is odd, then the signs of the second derivative to the left of zero will be ____ of the signs on the right side of zero.

opposite

If the second derivative is zero, there is a ______ ____ _____ on the original function. To determine if this point is a ________ _______ you must check to see if the second derivative changes signs around the point. If the concavity changes around the point, then it is an ______ _____.

possible inflection point Inflection point inflection point

If a function is even, then the signs of the second derivative to the left of zero will be ___ of the sings on the right side of zero.

same

If a function is odd, then the signs of the first derivative to the left of zero will be ______ of the signs on the right side of zero.

same

If the slopes of the first derivative are negative then the _____ ____ negative. As stated above if the second derivative is negative, then the original is ____ _____. Therefore, if the slopes of the first derivative are negative then the original is _____ ______.

second derivative concave down concave down

If the slope of the first derivative are zero, then the ____ _____ is zero. If the second derivative is zero then the original has a ___ ___ ___. If the point in question is a relative max. or a relative min. on the first derivative, then the slope changes from positive to negative or from negative to positive around that point and the value of the second derivative changes. This change in sign allows the point to be an ____ _____. Therefore, the relative max. and min. points on the first derivative are ___ ____ on the original function.

second derivative possible inflection point Inflection point inflection points

If the slopes of the first derivative are positive then the ____ _____ is positive. As stated above, if the second derivative is positive, then the original is ____ ___. Therefore, if the slopes of the first derivative are positive then the original is ____ _____

second derivative concave up concave up

If there is an undefined point on the first derivative, then there will be a _______ ____ on the second derivative at the same x- value.

undefined point