What is the higher order derivative?
The higher order derivative is known as the second derivative of the function, that is, if f(x) is a function and its first derivative f´(x) exists.
What are the applications of higher order derivatives?
The topic of higher order derivatives is necessary to solve optimization problems that require the calculation of maximums or minimums, making use of the second derivative. It is also useful in graphing functions, a topic that will be taken up in later units.
What are third order derivatives?
The Criterion or test of the Third Derivative is a method of mathematical calculation in which the third derivative of a function is used to confirm or check the inflection points obtained from the second derivative. It is a particular case of the Highest Order Derivative Criterion.
What happens when a function is negative?
A linear function is increasing if its slope is positive. A linear function is decreasing if its slope is negative. A linear function is constant if its slope is zero.
When the tangent line is positive, is the function increasing or decreasing?
We also have that when the slope of the tangent line is positive, the function f grows; and when the slope of the tangent line is negative, the function decreases.
What is the second and third derivative of a function?
When it exists, the second derivative f′′ of a function f is obtained by differentiating the first derivative f′. When f′ has a derivative, it is denoted by f′′ and is called the second derivative of f. So the third derivative f′′′ of f is the derivative of the second derivative.
What criteria is used to determine the concavities?
To determine the concavity of the graph of a function, we must determine the intervals on which f»(x)<0 (concave down) and on which f»(x)>0 (concave up).
How is the concavity of a function determined?
The concavity, as a characteristic of the graph of a function, refers to the geometric condition of the region located under a curve. A function f(x) is said to be concave when the region under the curve is convex, in case the function is twice differentiable, it is concave if, and only if, f»(x) < 0.