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Compute the difference quotient $\frac{f(x+h)-f(x)}{h}, \quad h \neq 0$ Whenever possible, simplify the expression so that the resulting expression is defined when $h=0$.$f(x)=x^{3}$

$$3 x^{2}+3 x h+h^{2}$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Compute the difference quo…

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Determine the difference q…

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Find the indicated differe…

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Find the difference quotie…

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Assume $h \neq 0 .$ Comput…

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for this problem. We have been given a function f of X equals execute, and we want to compute the difference quotient. Now the difference quotient is f of X plus h minus F of X over h. This is a quote that we're going to see often in calculus, so let's apply our function to the difference quotient. First, we have f of X plus H. That means I'm going to take X Plus H and substituted in for X in my original function. So that means I will have X plus h cute minus ffx. Well, that's just gonna be subtracting X cubed for each. Let's expand this out. Let's get rid of my parentheses when we have something. A binomial that were cubing that's going to give us X cubed plus three. Bring down the extra now is X squared. Times H three. Bring it down one more time. Now it's a three x h squared and then we finish with H cute and we subtract. Execute over H. Well, we have some things that cancel. I haven't x cubed in the negative x cubed, which is nice and you can see I have three terms left in the numerator. Every single one of these has an h, so I can cancel the ages. So this two will become one. And the H to the third becomes H h squared. And that cancels with that h in the denominator. And that leaves me with three x squared plus three x h plus h squared.

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