(a) Find the slope of the tangent to the curve y=1 / √(x) at the point where x=a . (b) Find equa

step 1 we want to find this up of the tangent line to the curve. Why is he was one of the square root o

(a) Find the slope of the tangent to the curve y=1 / √(x) at...

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which expresses the rate at which the function's value changes with respect to changes in the input. It is a fundamental tool in calculus that allows us to analyze and understand the behavior of functions.

Derivative

The derivative of a function at a particular point gives the instantaneous rate of change or slope of the function at that point. It is essential in determining the slope of the tangent line to the curve and is calculated using various rules such as the power rule, product rule, and chain rule.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. It approximates the curve locally and is important in both theoretical and practical applications for understanding the behavior of the function.

Point-Slope Formula

The point-slope formula is used to write the equation of a line when a point on the line and the slope are known. It is particularly useful for constructing the equation of a tangent line to a curve by using the derivative (slope) and the coordinates of the point of tangency.

Graphing Functions and Tangents

Graphing the function and its tangent lines involves plotting the original curve and the corresponding tangent lines on the same coordinate system. This visual representation helps in interpreting the relationship between the curve and its tangents, and in understanding the behavior of the function around specific points.

Transcript

00:01 We want to find this up of the tangent line to the curve.

00:03 Why is he was one of the square root of x at the point x is you go to a then we want to find the equations of the tangent lines at the points 11 and for what? hap.

00:13 And then lastly, we want to graft the curb along with our tangent lines just to make sure they actually line up with each other.

00:22 Now recall to find the soap with sanja line.

00:26 What we're going to do is do the limit as x approaches a oh f of x minus f a all over x minus a.

00:40 Now let's plug everything in.

00:42 What? we have the limit as x approaches a of one over the square root of x minus one over the square root of a all over x minus a.

00:56 Now we can go ahead and combined.

00:59 Our numerator is there or the fractions in the numerator and me to give us square root of a minus root ex all over that was going to be root ex times route a times x minus a.

01:25 Now, if we were to plug in a directly.

01:29 That should be a not infinity right there.

01:32 Now, if we were to plug a and directly, we get 0/0 so we can plug it in quite yet because i don't know, it's there over zero means, but notice that we can factor x minus a in the following fashion.

01:49 It's supposed to be a minus root ex all over root ex route a and that is going to be rude x minus a route x plus a cause that's really just the difference of squares and notice that and both of those should have for someone.

02:12 So let me go ahead and rewrite that again.

02:16 So should be route x minus root a times rue becks plus route a and now notice that this and this are just off by a negative one so we can cancel those out and then just multiply by negative one.

02:36 And now, moving this up here, that should give us the limit as x approaches a of negative one over root ex route a route x plus route a.

02:56 And if we were to go ahead and plug this in now, we would have no issues, and doing that would give us the following so he plugged in a so negative one over.

03:10 So route eight times were a is just a and then route a plus roue would be to route a and now we can go ahead and simplify that down even more to give us negative 1/2 times a to the three halfs powder.

03:32 So this here would be our slope for any point a.

03:36 That this function is to find her now to find the tangent lines at each...

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