The Tangent Line Problem

Question 1

What is the tangent line problem source?

Answer 1

How do we determine the slope of a curved line. We know slope is m = delta_y/delta_x but that’s for a straight line.

Question 2

How many sets of ordered pairs to calculate straight line slope?

Answer 2

Two.
[x1,y1]
[x2,y2]

Question 3

What is a tangent line?

Answer 3

Given function f(x) and a specific ordered pair within that function [x,y] a tangent line is a straight line with the same slope as f(x) @ [x,y]. It also factors in an intercept so the tangent line passes through the point [x,y].

Question 4

What is a secant line?

Answer 4

Given a function f(x) and two ordered pairs within that function [x1,y1] and [x2,y2], the secant line is the straight line y = mx + b that passes through both ordered pairs.

Question 5

m_sec formula

Answer 5

      delta_x

Question 6

What are other names for the secant slope formula?

Answer 6

m_sec
Difference Quotient
Average Rate of Change

Question 7

How is the derivative different from the secant slope formula in definition?

Answer 7

The secant slope is the “Average Rate of Change” while the derivative is the “Instantaneous Rate of Change”

Question 8

Theory behind secants finding tangents.

Answer 8

The slope of the secant line approaches the tangent line slope. The limit of the slopes of the secant lines is equal to the tangent line slope.

Question 9

Define Limit

Answer 9

getting arbitrarily close to a point