2.1 The Difference Quotientap Calculus

Difference quotient is used to find the slope for a curved line provided between the two points in a graph of a function 'f'. Use this online difference quotient calculator to find f(x+h) - f(x) / h by entering the equation.

f(x+h) - f(x) / h Calculator

Difference quotient is used to find the slope for a curved line provided between the two points in a graph of a function 'f'. Use this online difference quotient calculator to find f(x+h) - f(x) / h by entering the equation.

In single-variable calculus, difference quotient is used to compute the slope for the secant line (i.e., the line passes between two points on a curve) between any two points in a graph for the function f. For a function f (x), the difference quotient would be f(x+h) - f(x) / h, where h is the point difference and f(x+h) - f(x) is the function difference. The difference quotient formula helps to determine the slope for the curved lines. The f(x+h) - f(x) / h calculator can be used to find the slope value, when working with curved lines. Feel free to use this online difference quotient calculator to find the difference quotient by providing an input equation.

• Consider the difference quotient formula. F (x+h)−f (x) h f (x + h) - f (x) h Find the components of the definition. Tap for more steps.
• Difference quotient is used to find the slope for a curved line provided between the two points in a graph of a function 'f'. Use this online difference quotient calculator to find f(x+h) - f(x) / h by entering the equation.
• Homework: Difierence Quotient Practice Page 1 This worksheet is homework to be included in your homework notebook. Odd-Numbered Answers on Back.

Calculus‎ Differentiation. Jump to navigation Jump to search. 1 Find the Derivative by Definition; 2 Prove the Constant Rule; 3 Find the Derivative. Math AP®︎/College Calculus AB Differentiation: definition and basic derivative rules The quotient rule Differentiate quotients AP.CALC: FUN‑3 (EU), FUN‑3.B (LO), FUN‑3.B.2 (EK).

Example:

Find the difference quotient of 12x + 4

Step 1 :

f(x+h) = 12(x+h) + 4

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Step 2 :

f(x) = 12x+4

Step 3 :

12(x+h) + 4 - (12x+4)
=12x + 12h +4 - 12x - 4
=12h

Step 4 :

12h / h = 12
Hence 12 is the difference quotient.

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The difference quotient is a formula that finds the average rate of change of any function between two points. In calculus, the difference quotient is the formula used for finding the derivative, which is the limit of the difference quotient between two points as they get closer and closer to each other (this limit is also the rate of change of a function at a single point). The difference quotient was formulated by Isaac Newton.

Definition[change | change source]

Introduction[change | change source]

The difference quotient can be described as the formula for finding the slope of a line that touches a curve at only two points (this line is called the secant line). If we are trying to find the slope of a perfectly straight line, then we use the slope formula, which is simply the change in y divided by the change in x. This is very accurate, but only for straight lines. The difference quotient, however, allows one to find the slope of any line or curve at any single point. The difference quotient, as well as the slope formula, is merely the change in y divided by the change in x. The only difference is that in the slope formula, y is used as the y-axis, but in the difference quotient, the change in the y-axis is described by f(x). (For a detailed description, see the following section.)

Mathematical definition[change | change source]

The difference quotient is the slope of the secant line between two points.

If one uses the notation ${\displaystyle x_2=x_1+\mathrm{\Delta }x}$, then this becomes ${\displaystyle m=\frac{f(x+\mathrm{\Delta }x)-f(x)}{(x+\mathrm{\Delta }x)-x_1}=\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}}$.

The Difference Quotient Calculator

(here, ${\displaystyle \mathrm{\Delta }x}$ is called the limiting variable of the difference quotient. Other variables, such as ${\displaystyle \delta x}$ and ${\displaystyle h}$, are used as well.^[1][2][3])

The difference quotient can be used to find the slope of a curve, as well as the slope of a straight line. After we find the difference quotient of a function and take the limit as two points get closer and closer to each other, we have a new function, called the derivative. To find the slope of the curve or line, we input the value of x to get the slope. The process of finding the derivative via the difference quotient is called differentiation.

Applications of the difference quotient (and the derivative)[change | change source]

The derivative has many real life applications. One application of the derivative is listed below.

Physics[change | change source]

In physics, the instantaneous velocity of an object (the velocity at an instance in time) is defined as the derivative of the position of the object (as function of time). For example, if an object's position is given by x(t)=-16t²+16t+32, then the object's velocity is v(t)=-32t+16. To find the instantaneous acceleration, one simply takes the derivative of the instantaneous velocity function. For example, in the above function, the acceleration function is a(t) = -32.

Related pages[change | change source]

2.1 The Difference Quotientap Calculus Solutions

References[change | change source]

2.1 The Difference Quotientap Calculus Worksheet

1. ↑'List of Calculus and Analysis Symbols'. Math Vault. 2020-05-11. Retrieved 2020-10-14.
2. ↑'Mathwords: Difference Quotient'. www.mathwords.com. Retrieved 2020-10-14.
3. ↑'Difference Quotient'. www.analyzemath.com. Retrieved 2020-10-14.

2.1 The Difference Quotientap Calculus Pdf