(a) Find the slope of the tangent to the curve y=3+4 x^2-2 x^3 at the point where x=a (b) Find

step 1 we want to find the slope of the tangent to the curve. Why is it go to three plus four X squared

(a) Find the slope of the tangent to the curve y=3+4 x^2-2...

(a) Find the slope of the tangent to the curve
$$y=3+4 x^{2}-2 x^{3} \text { at the point where } x=a$$ (b) Find equations of the tangent lines at the points $(1,5)$
and $(2,3) .$
(c) Graph the curve and both tangents on a common
screen.

Key Concepts

Derivative

The derivative of a function provides the rate at which the function value changes with respect to changes in the independent variable. It is fundamental in determining the slope of the tangent to the curve at any point, as the derivative evaluated at a specific point gives the slope of the tangent line there.

Point-Slope Form of a Line

The point-slope form is used to write the equation of a line when a point on the line and the slope are known. This allows one to construct the tangent line equation by using the derivative (slope) and the coordinates of the point of tangency.

Graphing Functions and Tangent Lines

Graphing both the function and its tangent lines on the same coordinate plane helps in visualizing the point(s) at which the tangent touches the curve. This provides insight into the relationship between the curve's behavior and the slopes of its tangents at various points.

Transcript

00:01 We want to find the slope of the tangent to the curve.

00:04 Why is it go to three plus four x squared minus to execute at point x is equal to a to start.

00:12 Now the equation will need to use for this is going to be released one of the equations that i should say so f prime of a is equal to the limit as x approaches a oh alexe minus half a day all over x minus a.

00:31 Now let's go ahead and plug everything in, which would give oh so f of x was three plus four x squared minus two.

00:43 Ex cute when we have f of a that just replaces all the exes with aids and then subtracting it should give us three minus or a squared plus two a cube all over x minus a no.

01:05 Notice the threes cancel out.

01:08 And the next thing i'm going to do is group all of the cube and square terms together and factor out the coefficients for the ex terms.

01:21 So we'd have or x squared minus ace, where and then minus two times execute minus a cute all over x minus eight.

01:39 Now notice that x squared minus a squared is the difference off squares.

01:48 So that means we can divide x minus eight to it in the following way.

01:53 So it should leave us with x plus a and then likewise over here.

01:58 This is the difference of cubes and dividing that by ex mice.

02:02 A should give us x squared plus a x plus a squared.

02:15 Now, if we were to plug a in or evaluate the limit, i should say instead at a well, this here is going to give us to a we'd get a bunch of a cubes right there.

02:27 So be three a cube...

Please give Ace some feedback