If an equation of the tangent line to the curve y=f(x) at the point where a=2 is y=4 x-5, find f

If an equation of the tangent line to the curve y=f(x) at...

Key Concepts

Function Evaluation at a Point

Evaluating a function at a specific point involves substituting the given x-value into the function to determine the corresponding y-value. This is fundamental when identifying the coordinates of the point where the tangent line touches the curve.

Tangent Line

A tangent line to a curve is a straight line that touches the curve at one specific point. At that point, the line has the same instantaneous direction or slope as the curve, making it a useful tool for approximating the curve locally.

Derivative as the Slope of the Tangent

The derivative of a function at a given point is defined as the slope of the tangent line to the function at that point. This concept links the rate of change of a function to its geometric representation through the tangent line.

Transcript

00:03 I guess so.

00:03 Given the functional, i call for x minus five and we're told the dysfunction is an equation off tangent line for f of x and x equal to grass defined f do and that kind of do so we're giving an equation for a tangent to function that were not told what it is.

00:22 It zach kovacs and we were asked to find that function if rebecca's evaluated it too, and the slope of that function evaluated it to, given that that's tangent is a tangent for x equal to.

00:37 So before we go ahead and try to evaluate these things, let's interpret what we're given first.

00:43 So why call for x minus five? that's this line right here, and we'll totally it's a tangent for fx.

00:48 Well, what does that mean? that means that if i haven't curve right here, it doesn't have to be pretty.

00:54 Um, at x equal to this curve has this tangent, but only an ex people too.

01:03 So that also means that that detained in line at the function they both share this point right here.

01:12 And that point is the value of why we're x is equal to two.

01:19 So if the f of x and the line did not have the same value at x equal to than that line is not injured.

01:28 But we're told it is so since the line itself has an equation, we can use that information to find this point, even though we don't know if kovacs so we'll go ahead and do that.

01:39 So we have why? well, four.

01:44 And we're told that this is a tangent of x equal to and we want to find f at two...

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