(a) Find the slope of the tangent line to the curve y=x-x^3 at the point (1,0) (i) using Defini

step 1 Today, we'll be finding the slope of a line tangent to the curve y equals x minus x cubed at the

(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 $$ y=x-x^{3} \text { at the point }(1,0)$$ (i) using Definition 1 (ii) using Equation 2 (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 $$
y=x-x^{3} \text { at the point }(1,0)$$
(i) using Definition 1 (ii) using Equation 2
(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.

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Key Concepts

Graphical Analysis and Zooming

Graphical analysis involves plotting the function along with its tangent line to visually inspect their behavior near the point of tangency. Zooming in on the region where the tangent touches the function helps illustrate the concept of local linearity, wherein the tangent line and the function appear nearly identical in a small neighborhood around the point of contact.

Derivative

The derivative of a function at a particular point represents the instantaneous rate of change of the function with respect to its variable. It coincides with the slope of the tangent line to the function's graph at that point and is foundational for understanding how small changes in the input affect the output.

Limit Definition of Derivative

This key concept involves finding the derivative by computing the limit of the difference quotient as the increment approaches zero. It formalizes the process of determining the slope of the tangent line by examining the behavior of the function over infinitesimally small intervals.

Tangent Line Equation

Once the slope of the tangent line is established, the tangent line can be expressed using the point-slope form of a line. This equation is used to approximate the function near a specific point and is essential in linear approximations and differential calculus.

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