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

(a) Find the slope of the tangent line to the curve $y=x-x^{3}$ at the point $(1,0)$ (i) using Definition 1 (b) Find an equation of the tangent line in part (a). (c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $(1,0)$ until the curve and the line appear to coincide.

(a) Find the slope of the tangent line to the curve
$y=x-x^{3}$ at the point $(1,0)$
(i) using Definition 1
(b) Find an equation of the tangent line in part (a).
(c) Graph the curve and the tangent line in successively
smaller viewing rectangles centered at $(1,0)$ until the
curve and the line appear to coincide.

Key Concepts

Equation of the Tangent Line

Once the slope at a point is determined from the derivative, the equation of the tangent line can be obtained using the point-slope form of linear equations. This process creates a linear approximation of the function near the point of tangency.

Local Linear Approximation

Local linear approximation is based on the idea that a function can be approximated by its tangent line in a small neighborhood around a point. This is a key concept in calculus that connects differential calculus to the practical estimation of function values close to a known point.

Slope of the Tangent Line

The slope of the tangent line at a point on a curve is exactly the derivative of the function at that point. It represents the instantaneous rate at which the function is changing, and it is essential for understanding the local behavior of the function.

Derivative as a Limit

The derivative is defined as the limit of the difference quotient as the interval approaches zero. This concept captures the idea of an instantaneous rate of change and provides a formal method to determine the slope of a curve at any point where the function is differentiable.

Transcript

00:01 So in this problem we have y equals x minus x cubed and we need to find the derivative first, which we will do using the definition of a derivative, which says that the derivative, and we're going to say that our function is also called f of x.

00:21 The derivative of f with respect to x equals the limit as some variable.

00:28 We're going to say h goes to zero of, and then our basically slope formula, where h is the change in x, and the change in y is f of x plus h minus f of x.

00:46 So you'll see this means that we have x plus h minus x plus h cubed minus, and then we have x minus x, and then we have x minus x minus x, cubed and so now we need to expand this polynomial which we're going to do but i'm not going to do it right here i'm going to just bring in the solution and rewrite what i had here and so expanding that polynomial gives us equals or expanding that binomial of x plus hq gives us this which is x plus h and then minus h cubed plus 3h squared x plus 3h squared x plus 3h x squared plus x cubed oh and i need parentheses here and then moving to a new line because this is so long or actually i'll just shrink and then we have minus x plus x cubed and now we just need to cancel terms so we have x here and minus x here and this is all over h and then we've got minus x cubed in here and plus x cubed here and now we have canceled everything we can cancel so now we're going to divide the h out which will give us limit h goes to zero h again this is all negative h squared plus three h x plus three x squared and then we had plus h on the outside, which goes to 1 plus h on the outside.

05:02 And h divided by h is just one.

05:05 And now we can just get rid of any terms that have h in it because h is going to zero.

05:13 So multiplying by zero cancels the term.

05:15 So we get 3x squared, negative 3x squared plus 1 as our derivative.

05:26 And now we need to, find a tangent line to our graph at the point x equals one...

Please give Ace some feedback