I woutould like to know how to find the equation of a quadratic function from its graph, including when it does not cut the x-axis.
Rules of calculus - functions of one variable Derivatives: There are many different ways to indicate the operation of differentiation, also known as finding or taking the derivative.
The choice of notation depends on the type of function being evaluated and upon personal preference. Suppose you have a general function: All of the following notations can be read as "the derivative of y with respect to x" or less formally, "the derivative of the function.
Suppose we have the function: After applying the rules of differentiationwe end up with the following result: How do we interpret this? For example, suppose you would like to know the slope of y when the variable x takes on a value of 2. Now for the practical part. How do we actually determine the function of the slope?
Almost all functions you will see in economics can be differentiated using a fairly short list of rules or formulas, which will be presented in the next several sections.
How to apply the rules of differentiation Once you understand that differentiation is the process of finding the function of the slope, the actual application of the rules is straightforward. First, some overall strategy.
The rules are applied to each term within a function separately.
Then the results from the differentiation of each term are added together, being careful to preserve signs. Coefficients and signs must be correctly carried through all operations, especially in differentiation.
The rules of differentiation are cumulative, in the sense that the more parts a function has, the more rules that have to be applied. Take the simple function: The derivative of any constant term is 0, according to our first rule. This makes sense since slope is defined as the change in the y variable for a given change in the x variable.
Suppose x goes from 10 to 11; y is still equal to 15 in this function, and does not change, therefore the slope is 0. Note that this function graphs as a horizontal line. The next rule states that when the x is to the power of one, the slope is the coefficient on that x.
This continues to make sense, since a change in x is multiplied by 2 to determine the resulting change in y. We add this to the derivative of the constant, which is 0 by our previous rule, and the slope of the total function is 2.
Now, suppose that the variable is carried to some higher power. The power rule combined with the coefficient rule is used as follows:Engaging math & science practice!
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