Secant Lines and Sanctification

Limit Definition of a Derivative

Psalms 119:33-40 NIV

Teach me, lord, the way of your decrees,

that I may follow it to the end.

Give me understanding, so that I may keep your law

and obey it with all my heart. 

Direct me in the path of your commands,

for there I find delight.

Turn my heart toward your statutes

and not toward selfish gain.

Turn my eyes away from worthless things;

preserve my life according to your word.

Fulfill your promise to your servant,

so that you may be feared.

Take away the disgrace I dread,

for your laws are good.

How I long for your precepts!

In your righteousness preserve my life.

In differential calculus we study how a slope of a linear function can be generalized to the slope of a function whose graph is curved, creating the derivative of the original function. The definition of derivative uses a sequence of lines (secant lines) drawn through two points on a function that are approaching each other and a single point on the function curve. The derivative value or tangent line slope is defined to be the limiting slope value of this sequence of secant lines. (See the figures below.)

Figure 1 : Secant line between 1 and 1.8   

Figure 2 : Secant line between 1 and 1.                                   

Once a person has been called to be a Christian, we are redeemed by Christ but not released from following the law of God. We are justified once but continue with the process of sanctification for the remainder of our lives. This sanctification process is like the limit process of the secant lines approaching the tangent line.

There is one distinction between the concepts of sanctification and secant line limits, however. In the mathematical contexts, we accept results that are "sufficiently close," results that are in an epsilon-neighborhood of the desired quantity. While in our quest for perfection, the "better" we get the further we realize we are from satisfying all aspects of the law.


Figure 3: Tangent line to f at x=1