(a) Find the slope of the tangent line to the parabola y=4 x-x^2 at the point (1,3) (i) using D

step 1 Today, we'll be finding the slope with the line tangent to the curve, y equals 4x minus x square

(a) Find the slope of the tangent line to the parabola y=4...

(a) Find the slope of the tangent line to the parabola $$ y=4 x-x^{2} \text { at the point }(1,3)$$ (i) using Definition 1 (ii) using Equation 2 (b) Find an equation of the tangent line in part (a). (c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point $(1,3)$ until the parabola and the tangent line are indistinguishable.

(a) Find the slope of the tangent line to the parabola $$
y=4 x-x^{2} \text { at the point }(1,3)$$
(i) using Definition 1 (ii) using Equation 2
(b) Find an equation of the tangent line in part (a).
(c) Graph the parabola and the tangent line. As a check on
your work, zoom in toward the point $(1,3)$ until the
parabola and the tangent line are indistinguishable.

Key Concepts

Graphical Visualization

Graphical visualization is a technique used to plot functions and their tangent lines to provide a visual understanding of their behavior. By zooming in at specific points, one can confirm how closely the tangent line approximates the curve locally, reinforcing concepts such as instantaneous rate of change and the geometric interpretation of derivatives.

Tangent Line

A tangent line to a curve at a point is the straight line that locally touches the curve at that point without crossing it. It represents the instantaneous direction of the curve and is characterized by having a slope equal to the derivative of the curve at that specific point.

Limit Definition of Derivative

The derivative of a function at a given point is defined as the limit of the difference quotient as the increment approaches zero. This rigorous approach involves computing the limit of the ratio of the change in the function value to the change in the input value, providing the instantaneous rate of change at that point.

Differentiation

Differentiation is the process of finding a function's derivative. This essential concept in calculus involves applying rules and definitions to compute the rate at which the function's value changes with respect to changes in the input, and it is foundational for determining slopes of tangent lines to curves.

Equation of a Line (Point-Slope Form)

The point-slope form of a linear equation is a method to write the equation of a line when a point on the line and the slope are known. It establishes the relationship between any point on the line and the given point, making it a useful tool for expressing the equation of a tangent line once its slope has been determined.

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