The Difference Quotient formula is $$ {f(x+h)-f(x)\over h} $$ which computes the red secant line slope - that connects the 2 given points - below. Show  The Difference Quotient can be looked at in 2 different ways.1) In Algebra we learn that the slope m of a line passing through the two points (x1,y1) and (x2,y2) is \[ m= {y2-y1 \over x2-x1} \] 2) Moving on to Calculus, we generalize this concept to Functions. We compute the so called Secant Line Slope as the slope through the two points (x,f(x)) and (x+h,f(x+h)) that lie on the Function f(x). Thus, the formula for the Difference Quotient is \[ {f(x+h)-f(x) \over (x+h)-x } \] which simplifies to the above Difference Quotient Formula \[ {f(x+h)-f(x)\over h} \] . The slope of the red line in the above image is the secant line slope which is exactly what the Difference Quotient finds. Example1 (simple): If \( f(x)=2x+5 \) then its Difference Quotient is \[ {f(x+h)-f(x) \over h } ={2(x+h)+5 - (2x+5) \over h} = {2h\over h} = 2 \] which is the slope between any 2 points on the line 2x+5. Example2 (more advanced): If \( f(x)=x^2+5x \) then its Difference
Quotient is \[ {f(x+h)-f(x) \over h} = {(x+h)^2+5(x+h) - (x^2+(5x)) \over h} = \] \[ {(x^2+2xh+h^2)+5x+5h - (x^2+5x) \over h} = {2xh+h^2+5h \over h} = 2x+h \] The Difference Quotient in Calculus:In Calculus, we extend the idea of a Difference Quotient. It is used to find the slope at one point instead of between two points. To accomplish this, we take the limit as h->0 of (f(x+h)-f(x))/h which moves the "dummy" point (x+h, f(x+h)) towards point (x,f(x)).By finding the slope between those two approaching points we actually find the slope at (x,f(x)). This is called "Finding the Slope at a Point by using the Limit of a Difference Quotient of a Function" in Calculus, quite a mouthful. Example: If \( f(x)=x^2 \) then \[ {f(x+h)-f(x) \over h} = {(x+h)^2 - x^2 \over h } = {2hx +h^2\over h} = 2x + h = 2x\] as limit h->0 This tells us that the slope at any point on the graph of f(x)=x^2 is 2x thanks to the Difference Quotient and taking the Limit h->0. For instance, the slope of f(x)=x^2 at x=5 is computed as 2*5 = 10. Here is another Difference Quotient calculator that you may like: https://calculator-online.net/difference-quotient-calculator/ You may also like this How-to-Find-A-Difference-Quotient Video at https://www.youtube.com/watch?v=eyn1ARLkLcA Here is how: The Difference Quotient computes the Average Rate of Change between 2 given points. Since x=2 lies between 1 and 3 we can use their Average Rate of Change, 4, as an approximation to the Instantaneous Rate of Change at x=2. The exact Instantaneous Rate of Change is found by computing the derivative of x^2 , which is 2x , and evaluating it at x=2 yielding also 4. This perfect match between the instantaneous rate of change and average rate is actually a result of the Mean Value Theorem in Calculus. For details about the Mean Value Theorem read Paul's Online Notes. Step by Step Math & Science Apps for the Ti-Nspire CX (CAS)Become a Wizard! Improve Understanding! Boost your Grade!For Math & Science Test Prep, Homework. Check your own Work.---Step by Step to Success. Apps run in minutes. Test our free trials first.---Laura Pennington Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions. View bio Robert Ferdinand Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics. View bio Used widely in math, the difference quotient is a measure of a function's average rate of change. Learn how to solve the difference quotient with steps to solve and how to apply it for geometrical interpretation. Updated: 01/05/2022 Steps to SolveThe difference quotient is an important part of mathematics, especially calculus. 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. Continuing on, we can simplify the denominator of this because the x cancels out, which you can see play out here. We see that the difference quotient is as follows: (f(x + h) - f(x)) / h Great! We've got a formula for the difference quotient. Now, let's consider the steps involved in solving this difference quotient. Any time we are given a formula, solving the formula is just a matter of finding the values of the variables and expressions involved in the formula, plugging them into the formula, and then simplifying. Therefore, the following steps are used to solve the difference quotient for a function, f(x).
Okay, three steps. We can handle this! Let's try an example. Suppose we want to find the difference quotient for the function g(x) = x2 + 3 As you can see here, we start by plugging the expression x + h into g, and simplifying. As we can see, we get that g(x + h) = x2 + 2xh + h2 + 3. Next, we plug g(x + h) and g(x) into g(x + h) - g(x) and simplify. As you can see, we end up with 2xh + h2. So far, so good! All we have left to do is plug this in for the numerator in the difference quotient and simplify. We see that given the function g(x) = x2 + 3 and two input values of x and x + h, the difference quotient is 2x + h, where h is the difference between the two input values. That's not so bad!
Geometrical InterpretationOkay, we know the formula for the difference quotient and we know how to solve the difference quotient, but why is this useful? To understand this, let's consider the geometrical interpretation of the difference quotient. When we introduced the formula for the difference quotient, (f(x + h) - f(x)) / h we defined it in terms of a given function f(x), and two input values x and x + h. Let's think about that for a second. If we have input values of x and x + h, then the corresponding function values, or y-values, are f(x) and f(x + h), so we're considering two points on the function; (x, f(x)) and (x + h, f(x + h)). Now, notice that before we simplified the difference quotient, it was the difference of the function values (or y-values) f(x + h) and f(x) divided by the difference in the x-values, x + h and x. (f(x + h) - f(x)) / (x + h) - x Practice Questions to Solve a Difference Quotient (Show All Your Work)1. Find the difference quotient for the function f(x) = 1/x (reciprocal function). 2. Find the difference quotient for the function g(x) = x^3 (cube function). Answers (To Check Your Work)1. The difference quotient for f(x) can be expressed as follows: [f(x + h) - f(x) ] / h = [1/(x + h) - 1/x ] / h Using LCD = x(x + h) gives us the difference quotient as: = [x - (x + h)] / [hx(x + h)] = [x - x - h] / [hx(x + h)] = -h / [hx(x + h)] = -1 / [x(x + h)]. 2. The difference quotient for g(x) can be expressed as follows: [g(x + h) - g(x) ] / h = [(x + h)^3 - x^3 ] / h Using (x + h)^3 = x^3 + 3 x^2 h + 3 x h^2 + h^3 leads to the difference quotient as: = [x^3 + 3 x^2 h + 3 x h^2 + h^3 - x^3] / h = [3x^2 h + 3 x h^2 + h^3] / h Pulling out an h from the numerator above gives us the difference quotient as: = h[3x^2 + 3x h + h^2] / h = 3x^2 + 3x h + h^2. Note: Passing the limit as h goes to zero gives us the first derivative of the function at x for both 1 and 2 above. Register to view this lessonAre you a student or a teacher? Unlock Your EducationSee for yourself why 30 million people use Study.comBecome a Study.com member and start learning now.Become a Member Already a member? Log In Back Resources created by teachers for teachersOver 30,000 video lessons & teaching resources‐all in one place. Video lessons Quizzes & Worksheets Classroom Integration Lesson Plans I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline. Back Create an account to start this course today Used by over 30 million students worldwide Create an account |
Evaluate the difference quotient for the given function calculator
Copyright © 2024 en.apacode Inc.
