(a) use any method to estimate the slope of the tangent to...

(a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given $x$ -coordinate and $(\boldsymbol{b})$ find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. HINT [See Example $2(\mathrm{~b}) .]$ $$ f(x)=\sqrt{x} ; x=4 $$

Key Concepts

Tangent Line

The tangent line to a curve at a particular point is the straight line that just 'touches' the curve at that point and has the same slope as the curve there. It is used to approximate the function near that point and provides a linear model of the function's behavior in a small neighborhood around the point of tangency.

Point-Slope Form

The point-slope form is a method used to express the equation of a line when a point on the line and the slope of the line are known. It is particularly useful in finding the equation of the tangent line to a function's graph because once the slope (derivative) and the coordinates of the point of tangency are determined, the equation of the tangent line can be easily formulated using this form.

Derivative

The derivative of a function at a given point represents the instantaneous rate of change of the function with respect to its variable. It is defined as the limit of the difference quotient as the increment approaches zero and provides the slope of the tangent line to the function's graph at that point, which is essential in understanding local behavior and performing linear approximations.

Difference Quotient

The difference quotient is a fundamental concept used to estimate the derivative of a function. It involves computing the ratio of the change in the function value to the change in the input value over a small interval. As the interval diminishes, the quotient approaches the exact derivative. This method is not only foundational in the definition of the derivative but also practical in numerical approximations.

Transcript

00:01 So being familiar with the square root function could be helpful.

00:05 So y equals the square root of x.

00:08 And also knowing that this ordered pair when x is 4, the y value is 2 because the square of 4 is 2.

00:16 So if you look at the graph of this, it does have a positive slope at that tangent line, but the slope should be pretty small, like a fraction, let's put it that way.

00:29 So i should have wrote f of x.

00:31 That's okay.

00:34 Y, f of x, same thing.

00:35 So anyway, the derivative of that is i like to do the limit definition where you do f of x plus h, so you're replacing the x and the problem with x plus h minus f of x, so the same function all over h.

00:50 Now i'm a big fan of the rationalizing technique where you multiply by the square root of x plus h plus the square root of x and do the same thing on bottom.

01:01 And the reason why this works is because if you foil in the numerator or distribute whatever you want to call it, the outside and the inside terms cancel each other out.

01:12 So you just have to worry about root something times itself.

01:17 Well, it gives it itself.

01:19 Again, the outside and the inside terms cancel and double check my math.

01:22 Root x times root x is just x.

01:25 And then leave the denominator alone as well.

01:28 Root x plus h plus root x.

01:31 So as you look, the x is will eliminate...

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