06.30.2019: Calculus Made Easy Ch. 5; Systematic Theology Ch. 4

Originally, the Randomizer had generated the day’s album as Belladonna: Spells of Fear, but after listening to a few unbearably bad tracks, I was forced to admit that despite giving it multiple chances over the years, it isn’t one of those albums that eventually grows on you; rather, it is objectively a really bad album, the worst album in my collection. So I deleted the album from my playlist and reloaded the Randomizer.


So today’s soundtrack is Daath: The Hinderers, a very solid debut album from this industrial melodic death metal band.


Calculus Made Easy (Silvanus Thompson)
Chapter 5: What to do With Constants

We already know that y is dependent on x. A change in the latter produces a change in the former. But there are some values which don’t change even when x and y do. We call these constants, because they are constant. How do these factor into our solutions when we derive an equation?

If we add or subtract a constant from x, it disappears entirely. If we multiply or divide a constant by x, our derivative will just be greater or smaller by the product or quotient of the constant than x‘s derivative would have alone.

On the right, you’ll see an example of differentiation of an equation that includes both multiplication of a constant (a) and addition of a constant (6). Let me walk through the process that I used to solve it:

  • y = ax³+6
    • The initial equation
  • y = ax³
    • Remove any added or subtracted constants, as they are negligible
  • y+dy = a(x+dx)³
    • Grow both variables by d
  • y+dy = a(x+dx)(x+dx)(x+dx)
    • Break apart the cube to solve
  • y+dy = a(x²+x[dx]+x[dx]+[dx]²)(x+dx)
    • Using distributive property, FOIL out the first two brackets
  • y+dy = a(x²+2x[dx]+[dx]²)(x+dx)
    • Combine like terms
  • y+dy=a(x³+2x²[dx]+x[dx]²+x²[dx]+2x[dx]²+[dx]³)
    • Multiply by the remaining bracket, using distributive property
  • y+dy=a(x³+3x²[dx]+3x[dx]²+[dx]³)
    • Combine like terms
  • y+dy=a(x³+3x²[dx])
    • Remove dx² and dx³ as they are negligible
  • dy=a(3x²[dx])
    • Since we know from the initial equation that y=ax³, we can remove both of those terms
  • (dy=3ax² ⋅ dx)/dx
    • Divide entire equation by dx to isolate dy/dx
  • dy/dx=3ax²
    • This is the solution

Of course, as we learned in Chapter 4, we can also use the power rule to greatly simplify this process. Let’s go through another example, this time using the power rule:

  • y = 13x³ᐟ² -c
    • The initial equation
  • y = 13x³ᐟ²
    • Remove any constants that are added or subtracted
  • y = 13(x³ᐟ²)
    • Isolate the multiplying constant
  • dy/dx = 13(3/2[x¹ᐟ²])
    • Use the power rule (xⁿ=nxⁿ⁻¹)
  • dy/dx = 39/2 ⋅ x¹ᐟ²
    • Simplify; this is the solution

Using the power rule is certainly the easier way to go about all this, but it’s important to understand what’s going on when we use that shortcut.


Systematic Theology (Wayne Grudem)
Chapter 4: The Authority of Scripture

In this chapter, Grudem addresses the question, “How do we know that the Bible is God’s Word?” (p. 73).

We start by investigating the bible’s own claims about itself. We want to know whether the Bible is authoritative – whether “all the words in Scripture are God’s words in such a way that to disbelieve or disobey any word of Scripture is to disbelieve or disobey God” (p. 73). Now we need to ask whether the Bible claims to be authoritative. After all, if we claim that the Scripture is God’s Word, but the Scriptures don’t claim that they are, we’ve countered our own claim.

Paul tells us that “[a]ll Scripture is God-breathed” (2 Timothy 3:16). In 2 Peter 3:16, Peter counts the New Testament books among the Scriptures. Since we already know that all of the books in the Bible belong in the Bible and are Scripture, then we can say safely that the Bible claims to be authoritative as God’s Word.

So the claim is made: The Bible says that it is God’s Word. We are assured of this by the Holy Spirit who “speaks in and through the words of the Bible to our hearts and gives us an inner assurance that these are the words of our Creator speaking to us” (p. 77). Only those who are filled with the Holy Spirit can recognize the authority of Scripture, for without His indwelling, our natural minds would reject it as “folly” (1 Corinthians 2:14). In addition, of course, the Bible is “historically accurate[…and] internally consistent” (p. 78). But these proofs mean little when the Bible is properly recognized as being our ultimate authority, since the fact that God tells us in the Bible that these are His words is the very best evidence. Is this a circular argument? Yes. Does that make it invalid? No. Because if one truly believes that the Bible is God’s Word, what higher evidence can he provide than the Scriptural claims of authority?

We know that we can believe the words of the Bible, because they are God’s words, and we know from Titus 1:2, Proverbs 30:5, and John 17:17 that God does not lie; His Word is truth.

Now we come to the crux of the matter. If God is our ultimate authority, and the Bible is the Word of God, then “to disbelieve or disobey any word of scripture is to disbelieve or disobey God” (p. 81). Grudem points out that the greatest preachers have always been those who could pick up the Bible and say, “This is what the Scripture says, and ‘[t]his is what this verse means. Do you see that meaning here as well? Then you must believe it and obey it with all your heart, for God himself, your Creator an your Lord, is saying this to you today!'” (p. 82).


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