Evaluate and simplify the difference quotients (f(x + h) - f(x...

Question

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = x² - 2x

Accepted Answer

Welcome back, everyone, to another video. For the function F of X is equal to 4 x 2 minus 6 X, determine the simplified form of the different quotients F X plus H minus F of X divided by H and F of X minus F A divided by X minus A. So first of all, let's identify all of those terms. The first one that we need is F of X plus H. So for every X that we have, we're going to substitute X plus H. We get 4 in C X plus H squad minus 6. Multiplied by x plus H. And now we can simplify, right? So we get 4. Applying the formula, we get X2 + 2 XH. Plus H squared. And then we are subtracting 6 acts. And 6H when we distribute the negative 6. Now let's distribute the terms. We get 4 x 2 + 8 X H plus 4 H2 minus 6 X minus 6 H. And now let's see if we need to simplify further. We have X2 XH, H2 X, and H. So we cannot combine any like terms, so we can just move on. Now let's evaluate F of A. F of A is going to be equal to 4A2 minus 6A. And that's simply because we're replacing X with A. And now we can evaluate the different quotients. The first one, F of X plus H minus F of X divided by H. It's going to be equal to our result. 4 X2 + 8 X H + 4 H2 minus 6 X minus 6 H minus F of X. So we're taking minus 4 X squared. And minus -6 X, which gives us 6 X. And we're dividing that by age. So now let's Combine the like terms, negative 6 X and 6X. Those can be canceled out. Then we get 4 X squared. And -4 X squad, and now we can factor out the h because it's a repeating term. Is we get 8 acts. Plus 4 H. And finally minus 6 divided by H. And now we can cancel out the H, and this gives us the final result of 8 x + 4H minus 6. So we have the first result that we need. And now for the second one, let's evaluate F of X minus F of A divided by X minus A. Now, let's substitute. The given valleys First of all, we need F of X, that's 4 X2 minus 6 X, and we are subtracting for a squad. 6A. Let's divide that by X minus A. We're going to get 4 embrencies. X2 minus A2 because we can factor out the 4, right? And then we get X2 minus A2. Minus 6A plus 6A, right? So we can factor out -6 and this gives us X. Right? And then we take X minus A. Now, in the denominator we get X minus A. And we can simplify further, right? Because we have a difference of squares formula in the numerator. So this is 4 X minus A multiplied by X plus A. And then we take minus 6 X minus A. Divided by x minus a. Notice that all of these, they have the same common factor X minus A. So we can cancel out those. And this gives us our final result, which is 4 x + A minus 6. And if we distribute, we're going to get 4 X plus 4A minus 6. So we have our final results. Let's label them. The first one is 8X plus 4H minus 6, and the second one is 4 X minus 4A minus 6. Thank you for watching.

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.
f(x) = x² - 2x

Key Concepts

Difference Quotient

Derivative

Function Evaluation

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