Real World Problem

Task A Real World Problem:   A particle P travels from a point O along a straight line and comes to instantaneous rest at a point A. The velocity of P at time t s after leaving O is vms−1, where . Find (i) the distance OA, (ii) the maximum velocity of P while moving from O to A.

  1. Description:

 

  1.  The dependent variable is v, representing velocity of the particle. The velocity of the particle is represented by an equation involving independent variable t, the time.
  2. The independent variable is t, representing time in seconds.
  3. The domin is all nonnegative real numbers, that is, This represents the time (in seconds)the particle takes to achieve the velocity.
  4. The range is all real numbers, that is, This range represents velocity (m/s) which may either positive or negative.

 

  1. Specific Issue:

The problems above is involved finding the distance and its maximum velocity of the the particle P between two points. We are given velocity equation involving the independent variable t for time and we aer supposed to use this equation and find distance and velocity of the particle under given conditions.

  1. To solve this particular real world problem we have to take its firsr and second derivatives. The first derivative  represents the rate of velocity with respect to time which is in fact acceleration of the particle. The second derivatives  will help us in determining the nature of the stationary points whether it is maximum or minimum.
  1. Solution:

To find the acceleration of the particle P at  between O and A, we differentiate the given equation with respect to t So the acceleration of the particle after 5 second is . Now to find maximum velocity we need to find the critical values: For critical values So we have two different cricalal points. To see the nature of these critical points we will differentiate it again: Therefore the velocity of the aprticle is maximum at t = 6.67 seconds. To find the maximum velocity we will substituite this value in the given function: Thus the maximum velocity is 4 m/s.   Task B Real World Problem: Find the volume of the solid generated when the area bounded by the curve  the x-axis and the line x = 2 is revolved about the x-axis.

  1. Description:

a. The dependent variable is y. b. The independent variable is x. c. The domin is all posative real numbers, that is, d. The range is all real numbers, that is,  The volume will be greater than zero.

  1. Specific Issue:

The problems above is involved finding Find the volume of the solid generated when the area bounded by the curve given.

  1. a.  Disk Method:

Here we will solve the problem, that is, find the volume of the solid by disk method.     b.  Shell Method: Here we will solve the problem, that is, find the volume of the solid by disk   method.

  1.  The solution is given below.

Disk method: Here we have   Shell Method: Here we have     Task C                                          Differentiation and Integration both are main roots of mathematics and even mathematics cannot be imagined without differentiation and integration. Many important branches of mathematics are based on these two terms. Differentiation is the method of finding the derivative of a function. The reverse method is termed as anti-differentiation. According to the Calculus, the derivative can be stated as how a function changes as its input changes. A derivative can be explained as how much one quantity is changed  in response to changes in some another quantity. In the first problem we used differentiation to find the rate of change of velocity and then we used it to find the maximum point on the velocity curve. The reverse method  of differentiation is termed as anti-differentiation or integration. Integration is used in many ways, to find the velocity or distance treveled by some particle if acceleration function is given. In the above problem, we used integration to find the volume of surface of a given curve about some some straight line. (MITOPENCOURSEWARE, 2013)     Task D Real World Problem: Bob leaves for a trip at 3pm (time t = 0) and drives with  velocity  miles per hour, where t is measured in hours. Find

  1. The distance Bob travelled in first 2 hours.
  2. Description:

 

  1. The dependent variable is v, representing velocity.
  2. The independent variable is t, representing time in hours.
  3. The domin is all nonnegative real numbers, that is,
  4. The range is

 

  1. Specific Issue:

The problems above is involved finding the distance Bob travelled in first 2 hours.

  1. Fundametal Theorem of Calculus (First Part)

If  is continuous on and  on , then

  1. The first part of the above stated problems can be solved by using the first fundamental Theorem of Calculus. We just need to apply the basic definition of the theorem.
  2. Fundametal Theorem of Calculus (Second Part)

If  is continuous on and  on , then

  1. The second part of the above states problesm can be solved by using Fundamental theorem of calculus part two, where we just need to the integrate the velocity function from 0 to 2.
  2.  Solution by First Fundamental theorem of calculus:

Solution by First Fundamental theorem of calculus: Thus the distance Bob drove from time t = 0 hours to time t = 2 hours is 119 miles.

  1. Answer Key:
  2.   119 miles.

Task E In previous problem we employed Fundamental Theorem of Calculus to solve it. The importance of Fundamental Theorem of Calculus cannot be ignored. The importance of the Fundamental Theorem of Calculus is that it connects the notion of integral and derivative, which wasn’t something obvious back then. Fundamental Theorem of Calculus is saying that the process of integration and the process of integration are inverses of each other. (Mathematics, 2012)   Task F

  1.  

Therefore

  1.   
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144         
13 233
14 377
15 610
16 987
17 1597
18 2584
19 4181
20 6765

                                                         

  1.  
  2.  
  • The graph is given below:

 

  • The above graph shows that the sequence is converging.
  • It seems that it is converging to a number 1.6  as n increase to positive infinity.

 

1 2 3 4 5 6 7 8 9 10
1 2 1.5 1.67 1.6 1.625 1.6154 1.61905 1.6176 1.6182

Task G   Refrences:

  1. Mathematics. (2012, July). How do I explain the Fundamental Theorem of Calculus. Retrieved from http://math.stackexchange.com/questions/168876/how-do-i-explain-the-fundamental-theorem-of-calculus-to-my-teacher.
  2. MITOPENCOURSEWARE. (2013). Application of differentiation. Retrieved from http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-2-applications-of-differentiation/part-b-optimization-related-rates-and-newtons-method/session-29-optimization-problems/.

3. AceMyHomework (2014)