Calculus Thought Questions
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Describe some real life things that can be modelled as vectors.

Some real life vectors that come to mind include objects like cranes. When a crane picks somthing up, a force is applied pulling down due to gravity along with the arm at some angle to the ground. Also, the weight of the entire crane itself has forces pushing down into the ground with the same forces pushing upwards on the crane from the ground. Second, a street sign hanging off a building has vector forces pulling down with its gravitational weight. Then there are forces that push against the wall and an opposite force pulling from the wall. -Justin Crook

There are many real life vectors. A submarine diving to the bottom of the ocean is a vector. A plane descending to land is a vector. Also, an object sliding down an inclined plane by the force of gravity and a horizontal force is a vector. Heck, I'm sure a man walking down the street can be considered a vector. Vectors are all around us and you just don't know it. -Terron Schoonerman

Describe five real life uses for arc length.

There are many real life uses for arc length. If you were to design a chair (like the ones in the MAPLE lab room), then you would need to know the length of materials to use over all of the curved surfaces. When you drive a car over hills/through valleys, you need to know arc length, for gas mileage, etc. If you are covering furniture with fabric, you need to know how much fabric is needed to fit around the curve of the cushions. Another thing is to determine how much covering you need to use on a tent because from the center pole to the edges is an arc. Finally, you use arc length when wrapping a present - if it is a round gift, you need to know how much paper to use! -Becky Stover

There are many uses for the arc length formula. If you wanted to measure the bend in the hood of a car you would need arc length. You could use it to find the distance across and arched surface if you followed the path. If you were measuring a vaulted wall you would also need this formula. If you set off a model rocket, and you wanted to figure out the length of the arc you would need to use the formula. Finally, if you were stringing lights around your christmas tree and you wanted to know how many lights you would need you could figure out the arc length. -Mark Ohlrich

Think of a real life 3-D structure that has radial symmetry. How could you model its volume?

A real life structure that has radial symmetry is the Pantheon in Rome. One enters the building through the center axis and can stand at the center moment of the dome. A clay pot handmade off the wheel also displays radial symmetry. Technically, a blownup balloon (hot air or cold) also displays radial symmetry. You could model its volume by taking the outside curve and putting it into our formula. -Jeffrey Pronovost

A 3-D structure with radial symmetry is the big golf ball at Disney World. You can model its volume by plotting it on a graph. All you have to do is find the curve of the golf ball, which is a perfect circle. So, to model the volume, all you really need to have is the radius of the golf ball. Then you can make a pretty model of it to have forever to remember your time at Disney. -Jessica Barnoski

What is the antiderivative of 3x^2+2x-2+3/x? What rules did you use to find this?

I want to find the antinderivative of the equation above. First, I need to change it to the integral of 3x^2 plus the integral of 2x minus the integral of 2 plus the integral of 3/x. I need to use (x^(n+1))/(n+1) + C for the first two. Then I get 3x^3/3 + 2x^2/2 minus the integral of 2 plus the integral of 3/x + C. Then, I need to use the the integral of dx is x + C. Then I get x^3 + x^2 -2x plus the integral of 3/x + C. For the last one, I need to use the integral of 1/x=ln|x| + C. Then I get the answer, x^3 + x^2 - 2x + 3ln|x| + C. -KyungSeok Yoon

The antiderivative of 3x^2 + 2x - 2 + 3/x is x^3 + x^2 - 2x + 3ln|x| + C. I used the power rule for the 3x^2 and 2x parts. I used the LN rule for the 3/x part. Also, I used the Fundamental Theorem of Calculus to solve this problem. Lastly, I remembered to put + C at the end. -Edmund Yu

When might it be important to know an approximation for an area under a curve? When is it necessary to know the exact area under a curve?

A good time to know the approximate area under a curve would be when you are painting a room. All you need is an approximate amount. You might even want to over estimate. A time that you would want an exact area is when you are finding the area under an arch of a building in order to fill the area. If the arch was to be filled in, you would want to know the exact area so you know the stability of the arch. -Kristen Kubera

The estimate of the area under a curve can be handy when absolute precision is not required. Like purchasing grass seed, you don't need to know exactly how much area since you are buying by the bag anyway. The same idea applies to things like paint, roof shingles and stucco. It is much more important to be precise with things like, oh calculus, when you want to know the exact answer. Acceleration, velocity and position are all related by integration and differentiation. If you operate an expensive NASA probe, you should be pretty accurate with your calculations of velocity. -David Fannon

Describe five real life optimization problems.

Optimization problems can be used in architecture to determine how to maximize space in a room. In building, you can figure out how to best use the building materials. In the art world, optimization can be used to figure out the best use of the canvas. You can figure out how to most cost efficiently wire a house by figuring out where to put the wires. Chefs could figure out how to optimize the ingredients that they use when they cook. -Julie Adams

To take an example from our problem, at my old college, entire courses were devoted to how best to fence your horses, so they get the best grazing, best fieldmates, best budget for your money. Camping clothes and sleeping bags are optimized for the warmest coverage yet the lightest weight. Computers are made for greatest power yet smallest size at lower temperature. People ourselves are optimized for greatest speed and ease of movement. Computer science programs are optimized for efficiency and greatest ease of modification. -Kathleen Brown

What types of things have absolute extrema?

Many things in real life have absolute extrema. A motorcycle can only go so fast. A container can only hold so much liquid. A board can only bend so far before it breaks. When driving, the driver will eventually have to take a rest stop. A person cannot work forever and will eventually fall asleep. How many people can we stuff into a phone booth? -Josh Bowlby

In real life an example of something that has an absolute extrema is the top of a hot air balloon. Also, the lowest point on a mixing bowl represents an absolute minimum. When a midget is fired out of a canon at the circus, his highest point is an example of an absolute maximum. The top of Amit's head also looks like an absolute maximum. One more example of an absolute minimum would be the ends of the path of travel of a bottle rocket. -Jeffrey Pronovost

Create a function for the position of your dream car. What type of function is it? What does the velocity function look like? Is the velocity usually continuous in a car? What about the acceleration?

My function just has lines and is a piecewise function. So, the velocity is just composed of horizontal lines. The velocity is not continuous because it goes from slow to stop to fast to stop. The speed is slow at the beginning and faster later. The acceleration in my car is always zero. Velocity is usually continuous when on the highway. However, when driving around Troy, velocity is very discontinuous because of all the stop signs. -Amanda Levine

The position function of my car represents a piecewise function. The function for velocity is also a piecewise function. The velocity in a car is not usually continuous (unless you are driving on the highway). You stop and go so many times, it can't be continuous. The acceleration in a car is not always continuous either because there are sharp turns in the velocity function and there the acceleration is undefined. -Marie Grieco

In a previous thought question, you discussed the things in real life that can be modelled exponentially (or logarithmically). What do the derivatives of the exponential or logarithmic functions that you discussed represent?

The derivative of exponential or logarithmic functions (and all other functions) is the rate of change. So, for example, population growth is an exponential function. Taking the derivative at a point in time will tell us how fast the population is growing at that point. Another example would be radioactive decay, the derivative gives us information on how quickly the sample is decaying at any point in time. The information from the derivative gives us a new level of data at each point, not just how much, but how much how fast. -David Fannon

The derivative is equivalent to velocity, rate of change and the tangent slope. The derivative of population growth is the rate at which the population is increasing at a specific point in time. Another example is how fast/slow a disease spreads. The derivative can be seen at a specific point as the rate at which the disease is spreading. Derivatives are so cool! -Marcel Perez-Pirio

What is the derivative of f(x)= x^3-2*x+3? Do it by hand and then check your answer with MAPLE. What differentiation rules did you use to do this problem?

d(x^n)/dx=n*x^(n-1) => d(x^3)/dx=3*x^(3-1)=3*x^2
d(c*x)/dx=c => d(-2*x)/dx=-2
d(c)/dx=0 => d(3)=0
I used these three rules while doing it by hand. But, it is much easier with MAPLE. Because normally, you don't think about all these rules. You just enter the function and then you enter D(f); to see the derivative of that function. -Oya Baykal

The derivative of x^3-2*x+3 is 3*x^2-2. There are many differentiation rules used in this process. To take the derivative of x^3, use the power rule. To take the derivative of -2*x, you would drop the x. Lastly, the derivative of a constant is 0. -Randy Beringer

What things in real life can be modelled as tangent lines because they touch a curve at only one point?

There are many examples of tangent lines in real life. For example, a basketball sitting on the floor forms a tangent line. Other examples include poolballs on a pooltable. Or, a soccer ball on a field. Hmmm, sense a pattern here? Another example is skis on the top of a hill or jump or a mountain bike tire on a trail. -Lindsay Bennett

There are a couple of examples of tangent lines in real life. One example is a see-saw. The straight board of the see-saw touches a round curved pipe, therefore only touching one point on the curve and therefore being a tangent line. Another example might be the curved structure of a dome. If a tree falls on that dome, and if the tree is straight and has no branches, before the dome gets crushed and destriyed, the tree is a tangent line. Let's hope none of us are under that dome. -Ben Knapp

What things in real life can be modelled as secant lines because they touch a curve at two points?

Real life holds a variety of examples of secant line modelling. Hikers and mountaineers find such information useful in determining access trails. Skiers might also find these lines useful to plot a line of descent. In the sport of football, the secant lines would be useful for a coach in comparing the tosses and kicks of his players. By analyzing the slope, the coach could teach his athletes which of their methods would be most effective in moving the ball from player to player at the greatest speed, while still avoiding interference from the outstretched arms of the opponents. -Charles Dollard

In real life the axis of the earth could be modelled as a secant line. The axis through the center of the earth touches at the top and bottom only. Though this is an imaginary line, the two poles touch the earth's curve at only those two points. Another example is a straw through an orange, assuming the straw is a straight line and only the outer layer of the orange is considered. The straw would only go through two points of the orange. -Melissa Harvish

What are limits and why are they important?

Limits can be found nearly everywhere. The human body has natural limits. For example, a certain person can only drink alcoholic beverages to a certain degree before the body's internal systems begin to fail. This amount of alcohol is a person's limit. Once this limit has been exceeded, their health is at risk. There are also limits in design. A structure can only be built so high or so large before it becomes unstable. As a result, the maximum stable height is that structure's limit. The construction site itself also provides limits to design. For example, the top soil found at the site or the climatic conditions all offer limits in the structure's capabilities. -Matt Hains

Limits are important because sometimes technology gets ahead of humanity. Engineers can create aircraft that pull more than twenty G's of force, however, the human body can only take about nine G's at the most. Obviously, it does no good to have a jet going super fast if the pilot is knocked out. Another situation where limits are important to know would be in architecture. A building can't be so big that the ground under it will give way. -James Johnson

What kinds of things in real life have exponential curves? Can you think of anything that has a logarithmic shape?

Things which grow exponentially increase by leaps and bounds, while things which grow logarithmically grow continuously but with increasing slowness. The human population grows exponentially with each offspring adding to the increasing population. People, and all creatures, however fast we procreate, learn slowly, logarithmically, with our zest for learning (and our ability to cram more into our brains) tapering off over time. However much people want to deny it, as well, sound can only get so loud before the new $4000 amplifier makes no difference whatsoever. As mentioned, however, our learning curve (and sound measures) taper off eventually. -Kathleen Brown

In real life, an exponential curve can be found at the Eiffel Tower. Each of the columns start out at a starting point that curves down to the ground. If those curves kept on going in both directions, it would look exponential. The logarithmic shape is used in making curbs all the time, or a guard rail. The curve comes up from the ground and continues down the road. -Luke Ericson

In art and architecture, shifted or streched versions of the same design often form a motif or common pattern. Discuss an example of this.

In the pop art of Andy Warhol, an object would be taken and moved or shifted about the canvas. One example are his famous "Soup Cans." In this work, the same basic soup can is stretched and squished. Then, all of these variations are arrayed on the canvas. The same simple design then begins to evolve into a completely different form. -David Fannon

A person that I am studying is Frank Gehry, an architect. He is famous for his most recent work of the Gugenheim Museum in Bilbao, Spain. This building is made out of objects that take on this half cylinder type of form. He uses and combines larger and smaller versions of this object to create a multi-faceted building. The building has many edges and corners. This design is prevalent in a lot of his work. -Marcel Perez-Pirio

What things in real life can be modeled with linear models?

With a linear model, the cost of a certain product could be graphed against demand for that product. Different functions can be input to represent supply versus demand at various time periods. Another aspect that a linear model can help you see is supplies needed versus the number of people in a place. Certain conditions would cause the function to fluxuate here. Linear models can help solve many situations like this. -James Johnson

Linear models can be used to represent thousands of things. However, one example stands out in my mind. The number of thought question responses I have composed is proportional to the amount of repulsion I feel each time one is assigned. The model would be something like this: R=2*t+1, where R is in UOR (units of repulsion) and t is the number of thought questions assigned. This model has a y-intercept of 1 because some repulsion is felt just by knowing that thought questions exist. -Mark Schopmeyer

What do domain and range represent to you? How do you determine them? Why are they important?

Domain represents things that can't be changed by me or anyone else. These types of things include the passage of time, the temperature, the weather, etc. The range is represented by things affected by the domain. Temperature affects when it will snow versus when it will rain. I have determined these things through my own experiences growing up. For instance, I know that if it is 20 degrees Fahrenheit outside, it can't rain; it is too cold, and therefore snow will fall. These are important ideas because this application of math is pertinent to daily life. It helps you to better understand the world around you. -Becky Stover

Domain and range are important in determining my planning and plotting of my graph. The domain is determined by examining the x-values. The range is determined by examining the y-values. In each case, finding the essential endpoints of either the x or y values is key. Domain and range are also important for understanding the area of your graph as well as where a given point falls within the domain and range. Both concepts are used in many applications in calculus. -Alyssa Klem

Many times in our daily lives we see parabolas and circles. Write about when you see parabolas and circles every day.

One object that I see everyday that has a circular shape is the face of a clock or watch. The face of a clock is a circle because it is the easiest shape in which the two hands can rotate around the numbers to give the time. Time on a clock was given a circular motion because early time devices like the sun dial relied on the sun and a shadow to give the time of day. A place that I see parabolas everyday is the archways for doors or main entrances of buildings. The parabola is used to give an interesting, open feeling to an entrance and it structurally maintains the opening. -Sean Anderson

Circles and parabolas are seen every day. I use my laptop a lot, and the red mouse button is a circle. The arches of the student union and the foot bridge are parabolas. The drain of a shower has a circular shape. The golden arches at McDonald's are parabolas. The cap of a bottle is a circle. The rainbow has the shape of a parabola. There are circles on my friend's Dave Matthews poster. -Jessica Barnoski

  Last Modified on Wednesday, December 6, 2000.