Calculate the slope of the line passing through (a, f(a)) and (a+h, f(a+h)) for the function f g

step 1 All right, we need to find the slope of this line right here. And remember, slope is the change

Calculate the slope of the line passing through (a, f(a))...

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness, defined as the ratio of the vertical change to the horizontal change between two points. In analytic geometry and calculus, it represents the rate at which one variable changes in relation to another.

Difference Quotient

The difference quotient is an expression that calculates the average rate of change of a function over a small interval. It is given by the difference between the function values at two points divided by the difference in the x-values and is foundational for understanding derivatives.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. This concept is used to compute values such as f(a) and f(a+h), which are necessary for forming the difference quotient.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form by factoring, canceling common terms, and combining like terms. In the context of the difference quotient, it often involves canceling out the difference variable to arrive at a more concise expression for the slope.

Transcript

00:01 All right, we need to find the slope of this line right here.

00:08 And remember, slope is the change in the rising part, which is y.

00:15 The change in y over change in x, which is the running part.

00:23 So let's notice that the change in y is talking about, from here, here up that change right here so it's going from f of a to f of a plus h so we have to use this function that they provided us f of x and plug in a and a plus h and then we will take the differences of them to find that difference here okay so f of a let's start with that one you just strictly substitute a in for x.

01:06 So just another variable.

01:08 And now i have that.

01:12 F of a plus h, kind of the same thing.

01:16 But now instead of x, we've got to plug in the whole thing, a plus h squared plus 5.

01:25 Okay, so now i'm going to factor out that a plus h squared.

01:32 Okay, and remember that's a plus h times a plus h.

01:39 Here we go.

01:42 So, um, foil, right? first is a times a, a squared.

01:48 Outer is a times h.

01:54 Inner a times h again.

01:58 I'm going to keep it in that order.

02:01 And last, h times h, h squared.

02:05 Okay, i'm just going to put the ahs together and there are just two of them.

02:17 Okay, so now when i take my equation that i just had, this long one, right, this upper point up here, and i subtract my lower point here that we found earlier, well look at this, the a squared, subtracts with this a square...

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