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Sharon K. Robbert, Professor of Mathematics
Christianity and Mathematics
Devotionals connected to mathematical contentSingle-variable Calculus | Multivariable Calculus | Discrete Structures
Linear Algebra | Differential Equations | Statistics
Single Variable CalculusShould We Be Surprised?
Devotional Title Course Topic Scripture References Should We Be Surprised? Mathematical Modeling of Physical Phenomena Psalm 8:3, Gen 1:31, 1 Corin 13:12 Tools to Prevent Deception Machine graphics flaws & trick images Matt 24:24
2 Corin 11:13-14
Eph. 6:10-17God's Surgical Improvements of our Actions Function Operations Genesis 50:15-2, Romans 3:9-10, 21-24 Does God Change? Development of the Derivative Ps 11:3, Ps. 107:1, Ps 118:1, Ps. 117:2, Lam 5:19, Dan 4:34 Indicators for a Person's Heart Derivative as a Rate of Change James 2:12-18 Secant Lines and Sanctification Limit definition of derivative Ps. 119:33-40 Infinite Qualities of God Infinite Limits Ps. 139:1-18 Transformation under Christ Symbolic Differentiation Eph. 2:1-7, Eph. 4:22-24 Chain Reactions Chain Rule Inversions in the Bible Inverse Functions and Derivatives 2 Kings 20:1-11 Faith vs. Proof Three BIG Theorems: MVT, EVT, IVT Matt. 8:5-13 Blessed Exponentially Differential Equations: Growth Functions Matt 5:43--47, Gen. 12:2-3 Measurement in the Bible Integration as a measurement tool 1 Kings 7:23, Gen. 6:15,
1 Kings 6:2-3God's Zero Tolerance for Error Numerical Integration and Error Bounds Philippians 3:1--9 Psalm 8:3, Gen 1:31, 1 Corin 13:12Wow! Look at that! How amazing that mathematical equations can be effective at modeling the physical world. But is is truly amazing? If we believe that God created order out of something formless and empty, then what is surprising about finding order in our world? Maybe we should be surprised (but also grateful) that God included people in His creation who are able to appreciate the beauty of his world. However, we should not be so arrogant that we think our mathematical models can capture the entire essence of His complex creation. We are told that now we know in part, a poor reflection; but then we shall know fully! What will be amazing is knowing fully at the time of Christ's return!
Tools to Prevent DeceptionMatt 24:24, 2 Corin 11:13-15, Eph. 6:10-17In calculus we use technology freely; in particular to produce graphical images with graphing calculators and computer algebra systems. Technology is not perfect, however, and those who use technology must be aware of times when the graphical images we see are not representative of the true nature of the object. We use mathematical experience and developed intuition to judge whether an image is flawed or deceptive.
Satan is the angel of light and his disciples masquerade as "servants of righteousness." [2 Corinthians 11:13--15] But we read in Matthew 24:24 that it is impossible for false Christs to deceive the elect. We must follow the example of Jesus and use scripture as a standard against which to measure truth. We must also put on the full armor of God to protect ourselves from Satan's attacks. In both situations, knowledge helps prevent deception.
God's Surgical Improvements of our ActionsGenesis 50:15-21, Romans 3:9-10, 21-24back to topOne of the more horrible images in the book of Genesis is that of Joseph being sold by his brothers into slavery. This type of hate turned into evil act is a common occurrence in our world, too. In the Genesis situation, though, we are given the gift of 20-20 hindsight because we know the end of the story. God used the brothers' evil action to prevent starvation of the descendants of Abraham. Joseph says, "You intended to harm me, but God intended it for good. . ." [Gen. 50:20] In the same way, God makes our unrighteous actions righteous through Christ. He surgically improves our actions to his own purpose.
This idea of twisting something from one form into another is what happens when function operations work on elementary functions. You can start with two ordinary benign functions, the reciprocal function 1/x and sin(x), say, and put them together. Depending on how you put them together, you can create something interesting and easily understood, like sin(x)/x, or something with wild behavior, like sin(1/x). Either way, you have twisted one object into something very different.
Ps 11:3, Ps. 107:1, Ps 118:1, Ps. 117:2, Lam 5:19, Dan 4:34To understand the concept of derivative, it can be helpful to begin with simple functions. Linear functions, in particular, are very good ones with which to start because their rate of change is always constant (i.e., the slope of the line). Constant functions also have a constant rate of change--the rate of change of any constant function is zero. So what is God's rate of change function with respect to time? Does he have volatile rate function, one which changes wildly with small increments of time? NO! We are told that God is steadfast and endures forever. His rate of change is zero--in righteousness, in renown, in love, in faithfulness, in his reign. What a comfort to us to know that God is constant, that He will always be the same from generation to generation, everlasting!
Indicators for a Person's Heart
James 2:12-18back to topWhich comes first--the function or the derivative? If you are given a function in a graphical, symbolical, or numerical representation, you can find all the information you need about the derivative of that function--at least in the same representation. On the other hand, if you are given a derivative of a function, you can once again find out information about the original function. Is the original function increasing or decreasing? Is it concave up or down? But you will not know everything--you cannot know the exact location of the function vertically. That information is lost when the derivative is calculated.
Now compare a person's heart to their actions. If you know a person's heart, you should be able to predict actions completely, assuming consistency that is. But, suppose you only know a person's actions? Will you know that person's heart completely? Of course, only God can see the true nature of the person's heart. Human observers can only observe the behavior and then predict the nature of the heart. What conclusions will observers make when they see actions that do not model Christ? Their conclusions will inevitably be that the heart is not faithful to God. James says that "faith by itself, if it is not accompanied by action, is dead." So we must model Christ faithfully in our actions. This is the natural outcome of faith!
Secant Lines and SanctificationPs. 119:33-40In 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 1.5
Figure 3: Tangent line to f at x=1
Infinite Qualities of GodPs. 139:1-18Infinite limits are a wonderful extension of the mathematical world studied in calculus. In a precalculus context, behavior of a function at a value where the function is not defined is excluded from consideration (no division by zero nor even roots of negatives allowed!). But in a calculus context, evaluation of function behavior near these same input values numbers yields interesting characteristics of the function. Do function values approach infinitely large positive results from both slightly larger input values and slightly smaller input values? If so, we say that the limit of the function at this excluded input value is infinite. Disagreement between the two one-directional limits still gives interesting information, but not quite as general as in the case where the two agree.
Limits help finite creatures such as ourselves understand the amazing qualities of our Creator God a little better. Where we have finite knowledge, God has complete knowledge--exceeding the limit of all accumulated human knowledge for all time! Where we are limited to a single presence, God is not--he is everywhere present simultaneously! Where we are weak, God is all powerful; he can control time (see 2 Kings 20) and weather (see Matt 8:23-27) and physical laws (see Matt 14:22--36), and the hearts of people (see Exodus 7:3-4). Where we have a life-span of at most 100 years, God is not bounded by time; God transcends time (see 2 Peter 3:8). God exceeds the limits of our understanding in so many ways.
Transformation under ChristEph. 2:1-7, Eph. 4:22-24Differentiation is an operator on functions that takes one functions and transforms it into another form. The new form is related to the old form--the derivative tells interesting information about how the original function behaves graphically--but it is a completely new function. When someone accepts Jesus as Lord of their life and gives themselves wholly to God as one of His creatures, a similar transformation occurs. The person is transformed through Jesus and the Holy Spirit into a new self. This new self is redeemed, purified of unrighteousness, and claimed by God for eternal life. The new self is truly new, yet it still retains characteristics of the original self. God does not want his people to be identical to one another. God's creatures are distinct individuals with unique thoughts, unique gifts, unique appearances, and unique contributions to the body of Christ! Let us rejoice in our diversity in the community of the redeemed!
Chain Reactions1 Corin 5:16-21Once students have seen the chain rule for differentiation of composed functions, it is natural to extend the chain rule to nested functions, where there is more than two functions that are composed. Fun problems to investigate are ones that are repeated applications of the same function. Try differentiating tan(tan(tan(tan(tan x)))) or ln(ln(ln(ln(ln(ln x))))), for example. Working your way from the outside to the inside yields a derivative which is product chain of related functions.
In a similar way, when we interact with other people there is a chain reaction to our behavior. Most people believe that abused children are more likely to become abusers themselves someday, for example. Less dramatic behavior also can have a reaction that extends beyond the initial engagement. A popular Warner Brothers film of 2000 "Pay it Forward" (based on a novel with the same name by Catherine Ryan Hyde) depicts how a chain of reactions to an initial act of kindness can change an entire community. Christians need to be specially mindful of this chain reaction, since we are ambassadors for Christ. Our verbal and nonverbal witness can yield unexpected results, especially under the influence of the Holy Spirit.
Inversions in the Bible2 Kings 20:1-11Investigation of real-valued functions on the real numbers includes some common questions. Some of these questions are about the behavior of the function:
Existence of an inverse function is a very useful property of a function--it is much easier to answer questions about where a function takes on a particular output value IF we can use an inverse function.
- Where does the function increase? decrease?
- Where does the function have an increasing rate of change (i.e., where is the function concave up?)
- Is this function monotone? If so, what is the inverse of this function? If not, can you find a smaller domain where the function is monotone with range on the modified domain the same as the original function? Under these conditions, what is the inverse?
Applying inverse functions in nature is something environmentalists know is very difficult to do. Think about how difficult it is to restore a harvested rain forest or to undo the ill effects of an oil spill! However, our God is able to restore nature and he will to do it completely at the second coming of Christ. There are a few examples in the Bible of God's power over nature. God can reverse or stop the spin of the earth (see 2 Kings 20 or Joshua 10). God can make iron float on water (see 2 Kings 6) or stop the flow of a river (see Joshua 3).
Faith vs. ProofMatt. 8:5-13Do you believe that the big theorems (EVT, IVT, MVT, FTC, etc.) of Calculus are true? Or do you need to see a proof? Some are willing to believe everything a perceived authority says is true, and we might disparage them as gullible people. Each person is different in what they are willing to believe on faith--this depends upon experience, how the new information fits with current knowledge, and their trust in the authority espousing the information. In mathematics, there is tension between what must be proven and what can be accepted on faith. In Calculus, some of the proofs are too complex for the context of learning the new information while others can be very instructive in understanding a new topic.
In the Christian tradition, much of what we believe must be taken on faith. Jesus was impressed by those who were willing to accept his status as son of God without proof. The centurion identified with the authority of Jesus because of his own experience in the military. We can speculate that he was witness to some of the miracles of Jesus, giving evidence to support his faith. But his willingness to accept the word of Jesus that his servant would be healed was beyond Jesus' expectations. We hope that Jesus will be impressed by our willingness to believe and that we will be blessed by our faith.
Blessed ExponentiallyMatt 5:43-47, Gen. 12:2-3Elementary functions play an important role in calculus. The rate at which those elementary functions grow for increasing input values is one characteristic we study. [Note: Applications of this growth analysis appear in algorithm complexity analysis in computer science. Exponential growth is "bad" in this instance.] The fastest growing elementary function class is the exponential function; a function which takes various powers of a fixed numerical base. The principle of exponential growth is exploited in savings plans (save early and often!) and modeled in growth of bacteria.
Christ tells us in Matthew 5:43-47 that we are to love our neighbors AND our enemies. We also read in Genesis 12:2-3 that God blessed Abraham so that "all peoples on earth will be blessed through you." Together the concepts of exponential growth and "blessed to be a blessing" tell us to "pay it forward," so that God's love for humankind and goodness can grow exponentially.
Measurement in the Bible1 Kings 7:23, Gen. 6:15, 1 Kings 6:2-3Are measurements in the Bible are to be taken literally? If so, then p is exactly three--the Great Sea in Solomon's temple was circular in shape, ten cubits in diameter and 30 cubits in circumference. Most mathematicians take exception with this value for p! Here we must assume that either the measurements or the circular shape are not exact. Other biblical measurements are also interesting. Noah's ark is 450 feet long by 75 feet wide and 45 feet high, assuming an 18 inch cubit. Compared to an NFL football field, the ark would be 1.25 the length, a little less than half as wide, and 1.5 times the height of the goal post. So the ark wouldn't fit completely on the marked field, but would fit in the central arena of Soldier Field in Chicago. Solomon's temple was much smaller, only 90 feet long and 30 feet wide with a height of 45 feet. Approximately ten copies of this temple would fit in the inside of Noah's ark. Intriguing, isn't it?
In general, I believe that numbers in the Bible are to be taken figuratively, not literally. Knowing that measurement was important enough to the Israelites to be recorded is what I find most interesting. I take comfort in the fact that whether or not these numbers are accurate does not change the story of God's redemption for people and his creation.
God's Zero Tolerance for Errorback to topPhilippians 3:1--9
Analytically finding the area between a curve and the horizontal axis is a primary topic in integral calculus. We learn that some curves are resistant to exact method of computation, so geometric estimation techniques are required. In every estimation problem, it is insufficient to find an estimate without also knowing theoretically how close the estimate is to the quantity we wish to estimate---this is finding an upper bound on the error. We deal with relations that look something like this,|Desired Quantity - Estimate| £ Error bound.
In most applied situations we can allow for a small error; if we're off by 0.00001, that might be okay.
There is an equivalent error analysis when we compare our attempts to meet God's law and the perfection demanded by God's holiness. Here, God requires zero tolerance for error in order to be accepted into His kingdom. So the relation looks like this,
|Standard of God's Law - Our imperfect actions| £ 0.
We are unable to meet this zero error bound, so on our own we cannot be accepted into the kingdom. However, Christ exchanged places with us---He put his perfect self in our place in comparison to God's Law and took our punishment of death. This makes us able to satisfy the zero tolerance for error.
Devotionals connected to mathematical contentSingle-variable Calculus | Multivariable Calculus | Discrete Structures
Linear Algebra | Differential Equations | Statistics
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skr--Spring 2003Mathematics | Sharon Robbert | Triathlon | Christianity & Mathematics | Vita