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) = 7 / (x + 3)

Accepted Answer

Welcome back, everyone. For the function F of X C equals C11 divided by x minus 5, 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 of A divided by X minus A. So let's begin with the first one. We want to evaluate F X plus H minus F of X divided by H. And what we're going to do is simply evaluate the F of X plus H term, and this means that we're taking our function E11 divided by X minus 5. And replacing X with X plus H, so in the denominator we get X plush minus 5. We are then subtracting F of X, which is 11 divided by x minus 5, and all of that is divided by H. From here we are done with our substitution. We simply want to simplify this form and write a single fraction. So what we have to do is identify the common denominator. We get 11 for the first fraction, multiplied by the denominator of the second fraction, which is X minus 5. We are then subtracting 11 multiplied by the denominator of the first fraction. X plus H minus 5, and now the common denominator is the product of the two denominators. We get X plus H minus 5 multiplied by X minus 5, and all of that is divided by H. So we have a double division, which means that we can write H in the same denominator, right? From here we're going to distribute the terms. In the numerator, so we get 11 X minus 55. Minus 11 X minus 11H plus 55. And now in the denominator we have H. X plus H minus 5 multiplied by X minus 5. Combining the like terms, we can notice that we can cancel out -55 and 55, 11 X and -11 X, and then we can divide both sides by H, which cancels out the H. So for this one, we can write that our final answer is -11. Divided by X plus H minus 5 multiplied by X minus 5, and this is our first answer, so we can label it, and then we can move on to calculating the second difference quotient. Specifically, we are evaluating F of X minus F of A divided by X minus A. So first of all, we take F of X, which is 11 divided by X minus 5, and we subtract F of A, which means that instead of X, we are now writing a minus 5 in the denominator. All of that divided by X minus A. And once again, we need to simplify, so let's find the common denominator. We take 11, multiply it by a minus 5. Subtract 11 multiplied by the first denominator X minus 5. And now once again we have a double division. First of all, X minus 5. Multiplied by a minus 5, this is our common denominator, and then we are dividing by X minus A. So now let's distribute the terms. We get 11A minus 55 minus 11 X plus 55. All of that divided by X minus 5. A minus 5 and X minus A. Now we can see that 55 and 55, those can be canceled out, and then we can factor out -11. So we get -11 in C X minus A. And then we can rewrite the denominator, X minus 5, A minus 5. X minus A. The reason why this factorization is useful is because now we can cancel out the X minus a term, and this gives us our final answer -11 divided by X minus 5, A minus 5. Which is our final answer. 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) = 7 / (x + 3)

Key Concepts

Difference Quotient

Limit

Derivative

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