MATH EXPLORATION
General solutions of cubic functions.
I’m going to explore the roots (solutions) of cubic functions. General solutions of functions are, in other words, roots of a function. A cubic function may have one root, two roots, or three roots. An easy way to find out the roots of a function is by the use of the factor theorem. I will make very good use of this theorem throughout my essay.
My aim is to find the equations of the tangent lines, to the average of two and three roots, by taking two roots at a time. To find out where the tangents line insects the curve again, in order to state a conjecture concerning the roots of the cubic function, and the tangent line at the average value of these roots.
Let us consider
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therefore my conjecture is true.
Hence , The equation of a tangent at the average of any two roots will intersect the x-axis at the third root.
Real life example :
Air time is the feeling of weightlessness or negative gravity force experienced on a roller coaster or other amusement ride. On roller coasters this feeling is usually achieved by the train going over a hill. Air-time is caused by inertia of the train and the riders. As the train goes over a hill, the mass of the riders tends to resist the downward motion of the ride vehicle, resulting in riders being briefly lifted out of their seats. The amount of airtime caused by a ride is dependent on the velocity of the train, gravity, and the radius of the track transition. Zero-G (which means zero G-force) is achieved when the downward acceleration is equal to that of gravity, while negative G-forces are caused when the downward acceleration is greater than that of
Overall, as the concentration of the substrate increases, the enzyme activity increases up to a 70% of solution, where the enzyme activity starts to level off. The curve is polynomial because of the fact that the enzyme activity exponentially increases as the concentration of substrate increase; additional evidence for this is the fact that the gradient graph is constantly changing. The polynomial curve is shown because until 70% (the saturation point); this is because there are more casein substrate molecules that can successfully collide with the renin enzyme molecule, therefore increasing the rate of reaction.
Gravity is the force that attracts a roller coaster to the Earth and determines how far along the track it was pulled. When a roller coaster crests a hill, the gravity takes over and pulls it along the track at a “constant rate of 9.8 meters per second squared”(1) according to the website Wonderopolis’ article titled “How Do Roller Coasters Work?”. This numerical value, (or concept), is called the acceleration of gravity. It means that no matter the shape, size or mass of an object on Earth, gravity will pull it down at a rate of 9.8 meters every second, assuming there are no other interfering factors to mess with the decimal. In the article “How does Gravity work?” Tom Harris describes gravity and height’s relationship by stating, “As the coaster gets higher in the air, gravity can pull it down a greater distance” (1). This means that if a roller coaster were on top of a hill one thousand feet high, it would be pulled a lot further along the track by gravity than a coaster on a hill with a crest one hundred feet. Why? Because the coaster at one thousand feet has a stronger pull towards the Earth and can go farther because of it. The aspects of gravity, the acceleration of gravity and its relationship with height, are all important aspects of the force gravity. In conclusion, gravity is a vital, while fascinating, type of phenomena to observe in roller
I have plotted graphs from both sets of calculated gradients however I will concentrate on the graph plotted from the results show above as
Carowinds is compiled of many gravity-defying rides. Top Gun: The Jet Coaster is the Carolinas’ only inverted steel roller coaster. While on the ride, you are hurled through six swirling inversions while in the air. The Vortex is a stand-up roller coaster that takes you on a 50 m.p.h. series of loops and drops. Drop Zone Stunt Tower is a ride where you can experience the rush of gravity as you descend sixteen stories in seconds
a passenger train, a roller coaster has no engine or power source of its own. For most of the ride,
[-2, -1], [0, 1] and [1, 2]. Looking at the root in the interval [1,
This flow of air reduces the high pressure and increases the low pressure systems, thus reducing lift and increasing induced drag a great deal. However, once the plane nears the ground (usually half of the distance from the wingtip to fuselage) this flow is significantly reduced. Therefore, the lift is significantly increased. This is the ground effect.
In conclusion, since the earliest versions of roller coasters sprang up in the 16th century they have been a staple of thrill and amusement for people of all ages. But, like anything else on this Earth, they are governed by a simple yet complex set of physics principles and concepts including kinetic and potential energy, g-forces,
1. Describe the two methods of TRIZ that can be used to approach a problem.
Change (y = x2 – 6x + 11) to (x = x2 – 6x + 11) by substituting x by y and solve the quadratic equation. The solution for the equation one will give the points (x2, y2) and (x3, y3)
gradient of this graph = - (Ea / RT) which can be used to calculate
“A roller coaster is essentially a gravity-powered train (2).” Gravity is the weakest of the four physical forces, but when it comes to roller coasters, it is the dominant one. It is the driving force and what accelerates the train through all the turns and twists. Gravity is what applies a constant downward force on the cars. The deceleration or acceleration mostly depends on the inclination of the angle relative to the ground. The steeper the slope is, the greater the acceleration, and vice versa.
Raising my hand, “We just need to move the graph up three unit, then we can figure out the integral under the curve of the function”, I effortlessly answered the tricky question that American teacher was struggling with on a math class. I am an international student in the United States. My educational experience in China and the U.S. has taught me the differences on educational focus and influence on students of two countries’ educational systems.
Amusement parks are by far one of the most thrilling places on earth. As you wait in a long line to get in park, you can hear numerous kids, adults, and tourist shouting off the top of their lungs due to a tremendous jaw-dropping drop on their beloved roller coasters.
The equivalence point is halfway up the steep curve and for a strong acid- strong base titration curve is