For the following functions, a. use Equation 3.4 to find the slope of the tangent line m tan =f^

step 1 Good day, ladies and gentlemen. So this is problem number 15 from Section 3 .1. And it asked us

For the following functions, a. use Equation 3.4 to find the...

Key Concepts

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve only at that point, sharing the same instantaneous rate of change as the function. The slope of this line is given by the derivative of the function at that point.

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This fundamental concept captures the idea of the instantaneous rate of change or the slope of the tangent line to the function at that point, and it underpins much of differential calculus.

Differentiation Rules

Differentiation rules, including the power rule and the rules for rational functions, provide systematic methods for finding derivatives without directly resorting to the limit definition each time. These rules simplify the process of computing the derivative for a wide range of functions.

Point-Slope Form

The point-slope form of a line is a useful method for writing the equation of a straight line when a point on the line and its slope are known. It is expressed as y - y? = m(x - x?), where m is the slope and (x?, y?) is the point on the line, making it ideal for representing the tangent line once the derivative is known.

Transcript

00:01 Good day, ladies and gentlemen.

00:04 So this is problem number 15 from section 3 .1.

00:10 And it asked us to use using this formula here, which, using this formula, we're supposed to both calculate the slope of the tangent line at a and the equation of the tangent line at a.

00:30 Okay.

00:30 And for problem 13, or this is, sorry, problem number 15, i apologize.

00:38 For problem number 15, we're looking at f of x equals to 7 over x and a equals 3.

00:45 And again, sorry, this is problem number 15, just for clarity sake.

00:50 Okay, so the first step that i look at is the difference function, which in this case is f of a plus h minus f of a, divided by h.

01:03 And so when you look at the, this is a lot of algebra here, folks.

01:10 So this is, of course, equal to seven divided by, whoops, seven divided by a plus h minus seven divided by a.

01:32 Course all of that all of this divided by h actually what i want to do first is i'm going to actually just look at the numerator first so before we look at the denominator let's let's just consider the numerator okay so um now what i have to do here uh before i i have to simplify this so i want to get a common denominator and of course i'm sure all of you know how to get a common denominator by now basically we're going to get this to equal to 7 this becomes 7 a minus 7a minus 7h and of course now this is divided by and of course now this is divided by a this is divided by a times a plus h right because you just we multiply you multiply this one by a and this one by a plus h to get the denominator and then we put that together and so this is then this is equal to negative 7h now divided by divided by a times a plus h okay and then when i divide by h here so now i divide by h on both sides and then this guy's cancels here so my so this so this is then equal to just negative 7.

03:43 Oops, this is equal to negative 7 divided by a times a plus h.

03:53 Okay, so this is our, so this is the, what we might call the difference function here for h.

04:03 Okay, and the next step then that i look at is i look at the limit here so then you look at the limit as h goes to zero of the difference function so if we look at the limit now as h goes so so the limit as h goes to zero here so when we look at when you look at the limit as h goes to zero this the denominator is just going to be a times a here because the h goes to zero...

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