What is the formula for the difference quotient?
Given a function f(x), and two input values, x and x + h (where h is the distance between x and x + h), the difference quotient is the quotient of the difference of the function values, f(x + h) – f(x), and the difference of the input values, (x + h) – x.
How do you interpret the difference quotient?
Let’s start with the definition: The difference quotient is used to calculate the slope of the secant line between two points on the graph of a function, f. Just to review, a function is a line or curve that has only one y value for every x value. It’s like an input/output machine.
What does simplify the quotient mean?
The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
What is the difference of a quotient?
The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.
What does the H mean in the difference quotient?
h. – represents the change in x or (x2 – x1) or ∆x. f (x+h) – f (x) – represents (y2 – y1)
Why is the difference quotient important for calculus?
One of the cornerstones of calculus is the difference quotient. The difference quotient — along with limits — allows you to take the regular old slope formula that you used to compute the slope of lines in algebra class and use it for the calculus task of calculating the slope (or derivative) of a curve.
How to find the difference quotient for a function?
The calculator will find the difference quotient for the given function, with steps shown. Find the difference quotient for f ( x) = x 2 + 3 x + 5. The difference quotient is given by f ( x + h) − f ( x) h. To find f ( x + h), plug x + h instead of x: f ( x + h) = ( x + h) 2 + 3 ( x + h) + 5.
Why is the difference quotient called the mean value theorem?
This name is justified by the mean value theorem, which states that for a differentiable function f, its derivative f′ reaches its mean value at some point in the interval. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates ( a, f ( a )) and ( b, f ( b )).
Who is the creator of the difference quotient?
The difference quotient is sometimes also called the Newton quotient (after Isaac Newton) or Fermat’s difference quotient (after Pierre de Fermat).
How is difference quotient used to find slope?
It’s like an input/output machine. For any number x that you plug into the function, you will get an output value for f ( x ). In simple terms, the difference quotient helps us find the slope when we are working with a curve. In the case of a curve, we cannot use the traditional formula of: