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Do They Get It? The Instantaneous Rate of Change Exactly

Continuous Everywhere but Differentiable Nowhere

Today in calculus I wanted to check if students really understood what they were doing when they were finding the instantaneous rate of change. (We haven’t learned the word derivative yet, but this is the formal definition of the derivative.)

So I handed out this worked out problem.

And I had them next to each of the letters write a note answering the following individually (not as a group):

A: write what the expression represents graphically and conceptually

B: write what the notation $latex \lim_{h\rightarrow0}$ actually means. Why does it need to be there to calculate the instantaneous rate of change. (Be sure to address with h means.)

C: write what mathematical simplification is happening, and why were are allowed to do that

D: write what the reasoning is behind why were are allowed to make this mathematical move

E: explain what this number (-1) means, both conceptually and graphically

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No Connection?

I recently had a conversation with a young lady who claims to “love” math.  I was very excited for the young lady.  Then she proclaimed “I do not see the connection between precalculus and calculus.”  I wanted to ask her to explain herself, but I wasn’t sure I wanted to hear her explanation.  I asked myself how someone could have such affection towards math, but not see the connection between the two subjects.

Precalculus is the study of functions, their graphs, and their properties.  It builds the foundation for calculus.  I spend several hours each week preparing my students for calculus and higher level math courses via precalculus.  Some of my former students stop by my office to thank me for preparing them for calculus.  If there is no connection, what am I doing with my life!!

I am saddened that this young lady feels this way about precalculus and calculus.  I wanted to interrupt her and explain the connection right then and there, but time would not allow.  The most unnerving part is that she aspires to teach math and claims to be “good at it.”  What?!

I hope we one day cross paths again and I can revisit this conversation so I can get a better understanding of her proclamation that there is no connection between precalculus and calculus.  This statement ignited within me concern for all the students who encounter teachers with this same attitude towards math.

What are your thoughts?  How do you explain the connection between precalculus and calculus?   



Practical Uses for Calculus

The question finally escaped from the lips of one of my precalculus students.  The conversation went something like this:

Student: What is all this stuff useful for anyway?

Me: Precalculus is useful for preparing you for calculus.

Student: No, I mean what is calculus useful for?  Do we ever really use calculus for real life situations?

Me: Of course!  One example is the headlight used in cars.  Calculus was used to help maximize the light emitted from the headlight.

Student: As long as it’s useful for something.

He seemed satisfied.  I smiled.

As he walked away, I thought about the numerous times my students ask about the purpose of math, but this was a first.  This student wanted to know that calculus was good for something and useful in the “real world.”  So I did a little “research” and posted several sites in case someone reading this blog shares the same curiosity.

Some uses for calculus (very brief):

1. Growth rate of tumors (medicine)

2. Minimum payments due on credit cards (finance)

3. Curves for bridges, tunnels, and more (engineering)

4. Area under a curve (engineering)

5. Marginal cost (economics)

Websites to visit for more details (just a few):

NOTE: Photo borrowed from:

What are some other practical uses for calculus? 


Preparing for Calculus

Calculus is a specified field of mathematics.  Limits, derivatives, integrals, and infinite series are all applied to various types of functions (i.e., linear, quadratic, exponential, trigonometric, etc.).  They each depend on the understanding of basic arithmetic as well as specific concepts of other branches of secondary level mathematics such as algebra, geometry, and trigonometry.  Prior to taking a calculus course students should develop specific skills in each of these areas of study.

Below, I briefly discuss key skills and concepts required from three branches of mathematics (algebra, geometry, and trigonometry) to prepare students for calculus.  I discuss this material as it relates to differentiation, a fundamental component of calculus.

Algebra (Functions)

Differentiation converts formulas into other formulas.  These formulas are functions.  Algebra is the study of functions.  Since functions are the object of differentiation, it is imperative that students entering into a course of study in calculus have a thorough understanding of functions.

Many secondary students can follow instructions to substitute a value into an equation, but the study of functions will provide a greater understanding of why the method of substitution works and why a relationship exists between the parameters.

Geometry (Graphs)

The result of differentiation is a derivative.  Derivatives are used to calculate the slope of a line tangent to a point on the graph of a function.  A student can find an estimated value using the slope of a line secant to the same graph, but calculus makes it possible to find the exact value.  Students must have a thorough understanding of graphs, slopes, and functions to understand the use of derivatives with respect to graphs of functions.

Trigonometry (Angles, Periods of revolution)

Trigonometry is the study of trigonometric functions.  Trigonometric functions are used to calculate measures involving triangles, periods of revolution, etc.  During the course of a calculus class students will encounter problems involving trigonometric functions which are useful for explaining periodicity.  Students studying calculus should understand trigonometric functions, their properties, and their graphs.

As discussed above, students preparing to take calculus should have a specific set of skills relating to calculus content.  The major subjects required for the development of these skills, and the understanding of the related concepts, are algebra, geometry, and trigonometry.  More specifically, students need to have a thorough understanding of functions, graphs, slopes, and trigonometric functions, to name a few, in order to effectively prepare to enter into a calculus course.

Although students have the benefit of the use of technology to solve many problems, they must have an understanding of the relationship between the functions used, their graphs, and the derivative so they can understand the output generated from the use of technology.  Simply plugging values into a technological tool does not provide this level of understanding.  In most cases, students will not have access to technological tools in college level math courses (see “Calculators Not Allowed” on this blog site), so it’s best to learn and understand the concepts without these aids.

Do you have any other ideas about concepts students should learn prior to taking a calculus course?


The Land of Calculus… written during my youth

This is a reminder of my passion for math even in my youth.  I found this short story, written in English class, while rumbling through my high school records. “The battle was on!  Freeman against Jarvis: the two most prestigious mathematical kingdoms of the nation of Arithmetic.

Jarvis was at an advantageous point: he had Elementary Functions and Trigonometry as allies.  His other allies were at equal standing with Freeman’s allies.  Freeman had a plan.  She would find her way through the lands owned by Jarvis, incognito, starting with Algebra Township and working her way through to Algebra City.  In doing this, she would be closer to Elementary Functions.  She would start a revolt by challenging the skills of the armed forces of the Trigonometric Functions, which would cause a civil war.  She would then reveal the less complex solution to the problems, gaining their praise.  Hence, she would conquer Elementary Functions and move on to Calculus.

Algebra Town was a breeze.  Freeman was undefeated!

Unexpectedly, a little town just below Algebra City was out for revenge.  What would Freeman do?  She had not known or heard about this town in all her years of reign.  She couldn’t give up though, her kingdom depended on her.  She would find a way.  And she did.  Freeman became familiar with the area of Geometry.  She familiarized herself with Proof, the Chief in Command of the armed forces.

It wasn’t as bad as Freeman had anticipated.  She’d conquered Proof and his men, despite the few casualties she’d suffered.  Geometry was hers.

She began as planned.  She gossiped about which was the better branch of the armed forces, Elementary Functions or Trigonometry.  She started near Circle Township and spread throughout Elementary Functions.  She ran into a fork in the road: a face she remembered seeing in Geometry that was a part of Proof’s army.  She attacked him head on.  Surprisingly, she defeated him in one try.

The forces were at battle.  This went on for about one year.  The city was falling.  Freeman would rebuild Elementary Functions and fight her final battle, but first she had to train her armed forces.  She went the entire summer rejuvenating from the previous obstacles she faced.  She was determined she would win!

The day finally arrived.  The final battle began.  Jarvis and Freeman, both with equal skill, fought for several months.  Jarvis was known for his determination to win and nothing less.  Again, Freeman devised a plan.  She had come too far to give up now.

Freeman did her research, studied the battle grounds, reviewed procedures used in former victories, and thought of more strategic plans to use against Jarvis.

While in the midst of battle, Freeman learned of some very disturbing news.  She would have to go back to General Kingdom and honor the new king and queen of Geometry.  How could she be two places at the same time?  She had to attend the Honors Ball, so she assigned the most capable leaders to follow through with the battle and to keep her posted of progress.

It was the last week of the battle of Calculus.  Freeman had returned from the ceremonious events.  She learned that she had received misleading information: she had the impression that the battle was hers to win.  However, they were losing. She would have to go out herself and win the battle with flying colors.

She first defeated Differential Land, but Integral Land was the most difficult of any imaginable battle.  With great skill and intense concentration, Freeman conquered Integral Land and won Calculus!  She freed the slaves, decreased labor, rebuilt lands, and made the Freeman kingdoms proud and dignified.  Freeman overcame her obstacles and conquered Calculus!!!”

Do you have high school memorabilia that reminds you of your passions of today?