MHF4U
Culminating Activity
Rates of Change In Career
POLYNOMIAL FUNCTION
LINEAR FUNCTION
QUADRATIC FUNCTION
CUBIC FUNCTION
The Architect
QUARTIC FUNCTION
QUINTIC FUNCTION
BUILD THE LARGEST HOUSE
The linear function is only one function which has the constant rate of change at any points. Therefore, the graph goes towards the negative and positive infinity at constant speed on the graph.
Therefore, the shape of the graph is just a line since the x and y values are proportional. Furthermore, the shape of a line is never changed because the rates of change is always constant.
The quadratic function has one turning points, so there are both of negative and positive rates of change on the graph. In the function opening upward, both of the end behaviors of y value on the function are positive infinity. In the both sides, the function goes towards the bigger numbers more and more faster. If the function opens downward, the function goes down more and more faster. Also, the rate of change at the turning point is zero, so at this point the line is horizontal at a moment.
Therefore, the shape of graph is curved, and the line gets more and more steep because the rates of change gets more and more bigger at each point.
The cubic function has two turning points, so the rates of change is changed + to - or - to + twice on the graph. In the left side of the left turning points, the function goes down to this turning point slower and slower to 0. After that, te rates of change is increasing faster and faster to the point of inflection. After that, the slope gets gradually flatter to the right turning point. After the function reaches that point, the rate of change changes their symbol from + to -, and then goes down faster and faster.
Therefore, the shape of the function becomes like 1 wave since there are two points that the rate of change is zero, and then the slope of the line gets more and more steep outside of the turning points. Inside between these two points, the line gets steeper in the middle way, but after that it gets more and more flat to be horizontally.
The quartic function has three turning points on the graph, and it is the even-degree function, so the end behavior is like quadratic functions. Therefore, the shape of the function is opening upward or downward. There are three points that the rate of change is 0. Therefore, the positive and negative symbols of the rate of change is changed at these three points. Between the two turning points, the function goes up or down faster and faster in the middle of the way: point of inflection. After that,in the another half way, it gets flatter gradually until zero. Outside of the most right and left turning points, the surved line of the graph gets steeper and steeper since the rates of change gets bigger and bigger.
The quintic function has four turning point, so this function changes the direction of the line at each points which have the rte of change 0. Also, this function is the odd degree function, so if the left end behabior for y-value is positive infinityanother side one is negative. Outside of the most left and right turning points, the slope of the graph gets steeper and steeper. On the other hand, the inside of the most and second most left turning points, the graph's increasing speed gets faster and faste until reaching to the point of inflection. Therefore, in the left side of this point, the graph gets steeper. On the other hand, in the right side of it, the graph gets flatter to nect turning points.
....... It continuses much higher degrees
The architect is the career that is for buildings, and residential areas. The people in this career oftem receives orders of the appearence of the dream house. Based on the customers' orders, they prepare materials they would use to build the house, and then build them. Therefore, it is very important for them to make a various plans for their orders. Unfortunately, they cannot always get enough materials becuse the customer cannot pay a lot of money to order of one big house. Consequently, the people building the residential place must maximize the area. Then, architects use the cubic rates of change to calculate the maximum volume of the house because when the f(x)' is the maximum volume, the rates of change is 0 at that point. Also, if the architect knows the cubic rates of change, they can notice how we build the maximum volume's house quicker since they understand how the function increases or decreases on the graph.
APPLICATION QUESTIONS
One architect, John prepares only 20m length and 30m width of wodden board, to create the rooms. These wooden board would be used for only base of the room and wall. When we substitute x=height of the room, determine the equation of volume of this room, and then determine instantaneou rates of change at x=5.
