Retaining the Concepts. If f(x) = 4x^2 - 5x - 2, find [f(x + h) -...

Question

Retaining the Concepts. If f(x) = 4x^2 - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0

Accepted Answer

Hey everyone here we are asked to use the difference quotient formula which is F of X plus H minus F of X. All divided by H. Where H cannot be equal to zero. And we need to use this formula to calculate the difference quotient for the given function F of X is equal to three X squared plus seven X minus nine. And so now to start solving we first need to notice that from our formula, we already have this F of X function since it is our given function. But we need to find this F of X plus H. Well to do this, we just need to use our given function. So we'll have F of X plus H. And instead of all of our X terms being just X, we need to substitute in X plus H. So we'll have three times the quantity of X plus H squared plus seven times the quantity of X plus H minus nine. And so now we can start simplifying by first squaring this first expression here, so we'll have three times the quantity of X squared plus two X H plus H squared. And next we can factor in the seven. So we'll have plus seven X plus seven H and minus nine. And the last simplification we can do is a factor in this three here. So we'll have three X squared plus six X H plus three H squared plus seven X plus seven H minus nine. So now we have found the function F. Of X plus H. And so we have both functions for our formula. So we can go ahead and just plug them in to the formula. So in the numerous our first function is F. Of X plus H. So we'll have three X squared plus six X H plus three H squared plus seven X plus seven H minus nine. And then we have minus the function F. Of X will have minus the quantity of three X squared plus seven X minus nine. And again all of these terms are divided by H. So now we can just begin by factoring in this negative sign here. So our equation is now going to be three X squared plus six X. H plus three H squared plus seven X plus seven H minus nine minus three X squared minus seven X plus nine. And once again all of these terms are divided by H. And so now we can start simplifying the terms so we can see that we have a positive three X squared. As well as a negative three X squared. So both of these terms cancel. We also have a negative nine and a positive nine. So again both of these terms cancel. And lastly we have positive seven X. And negative seven X. So both of those terms cancel as well. So now we can rewrite our expression here we have six X. H plus three H squared plus seven H. All divided by H. And now since we have a choice in the denominator we can factor out H from each term in the numerator. So we'll have a church times the quantity of six X plus three H plus seven. Again all divided by H. And with this we see that the H. Is in the numerator and the denominator cancel. And we're left with our final simplified expression of six X plus three H plus seven, which is our difference quotient and it is also answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Retaining the Concepts. If f(x) = 4x^2 - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0

Key Concepts

Function Evaluation

Difference Quotient

Limit Concept

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