Question One
Learning Objectives:
1) Apply the understanding of the integration as the antiderivative to real-life connections.
2) Assess critical thinking and problem-solving skills. Description: In this assessment, you will act out the following instructions through a series of situations, then analyse the results to answer the given questions accordingly.
Situation A: From the roof of your house, or any floor above the first, drop a ball and record the amount of time it took to reach the ground using a stopwatch. Using the antiderivative (Integration) find: - The velocity of the ball when it hit the ground. - The height from where the ball was dropped. (Important note: Use the constant of acceleration as − 9.8 m/s2 instead of – 32 ft/s2)
Situation B: From the same height throw the ball upwards first and record the time it takes to hit the ground. Find: - The initial velocity. - The maximum height the ball reached.
Situation C: From ground level, throw a ball vertically as hard as you can and record the time it stayed in the air. What would be the initial velocity needed to reach the height of 6 meters? Note: Consider the initial height of the ball as well.
Situation D: Use a 2 Kg bag and lift it up as high as you can, record the height reached. Calculate the work done. Assessment guidelines: Please go thoroughly through this checklist, as it helps make sure you covered everything needed to meet the assessment standard. - All situations should be documented with pictures and videos. - Each student is to submit a PowerPoint presentation with a voiceover explaining their findings. - The presentation has to be submitted on Microsoft Teams, under the tab “Q4 Summative assessment”, before 23:59 on Thursday 4th of June 2020.
Question Two
Consider the cubic polynomial y=(ax+b) (x^2-3x+1) where a, b are real numbers. Dividing by the first derivative by (x-1) yields a remainder of -1, while dividing the second derivative by the same divisor yields a remainder of -6. Determine the value of a and b.
Question Three
Write a report (you can just type it in this Excel sheet, below) that addresses the following:
* how many meters, and how many floors, did the elevator move vertically?
* Explain how you did your calculations, as if writing to a calculus professor that hadn’t heard of this project.
* use some graphs to show what happened. Paste them from other sheets into the appropriate places in your report, below.
* Evaluate your model. Are the results reasonable? If not, try modifying it and reporting both the original and modified methods and results.
* how precise do you think your estimate is, in centimetres? If the elevator doors opened based on when it calculated it was in the right position based on inertial navigation, would the gap between the elevator floor and main floor be acceptably small?
* If you drove from Ypsi to Ann Arbor (about 10 km), and used an iPad-based inertial navigation system, how far off might you expect its final position estimate be, in meters?
* again for that 10-km drive, would the final position estimate be precise enough to know which cross-street you were at on a main avenue?
* Explain any assumptions that you made in doing your work.
* Explain any limitations to your work.
Question Four
This assignment comprises one extended question in eleven parts and contributes 10% towards your final grade. It should be completed, scanned and uploaded through the MATH1111 Canvas portal by 11:59 pm on Sunday 17 May. Your tutor will give you feedback and allocate an overall grade using the following criteria: A+(10): excellent and scholarly work, answering all parts, with clear and accurate explanations and working, with appropriate acknowledgement of sources, if appropriate, and at most minor or trivial errors or omissions;
A (9): excellent work, making progress on all parts, but with one or two substantial omissions, errors or misunderstandings overall; B +(8): very good work, making progress on more than six parts, but with three or four substantial omissions, errors or misunderstandings overall;
B (7): very good work, making progress on more than five parts, but making five or six substantial omissions, errors or misunderstandings overall;
C +(6): reasonable attempt, making progress on more than five parts, but with seven or eight substantial omissions, errors or misunderstandings overall; C (5): reasonable attempt, making progress on more than five parts, but with more than eight substantial omissions, errors or misunderstandings overall;
D (4): making progress on just four or five parts;
E (2): making progress on just two or three parts. Scenario: You are at a point A in a field separated from a friend at point B on the other side of a river. Your friend is in need and you have to reach him or her as soon as possible. The river is 20 m wide and flows in a direction perpendicular to the direct line of sight between you at point A and your friend at point B. The nearest point on the river bank closest to you is 80 metres away. Your friend is 50 metres away from the nearest point on the opposite side of the river. You are able to run 7 metres per second on land and swim 1 metre per second across the river, perpendicular to the flow.
You run directly to a point P that you have chosen on the left bank of the river, located z metres perpendicular to, and to the left, of your line of sight with your friend, and enter the river at P. It takes 20 seconds for you to swim across the river. From the point Q, where you exit the river on the right bank, you run directly to reach your friend. Suppose the river is flowing uniformly downstream (to your right as you look from A to B towards your friend) and carries you F metres per second downstream for the time you are in the river. Let D be the distance you are carried downstream as you swim across the river, as in the diagram on the next page.
(a) Explain briefly why D = 20F. (b) Factorise the quadratic 39x 2 − 128x + 64 given that 3x − 8 is a factor. Hence solve the quadratic equation 39x 2 − 128x + 64 = 0 exactly for x. Your two answers should be simple fractions. (c) Let k be a constant. Now solve the quadratic equation 39x 2 − 128kx + 64k 2 = 0 exactly for x, as two simple expressions in terms of k.
) Explain why the total time T = T(z) seconds taken to travel from point A to point B (as a function of the choice z metres), following the path in the diagram, from A to P to Q to B, is given by the rule T(z) = √ 802 + z 2 7 + 20 + √ 502 + (z − D) 2 7 . (e)
Show that T ′ (z) = 1 7 (z √ 802 + z 2 + z − D √ 502 + (z − D) 2). (f) Show that if T ′ (z) = 0 then z satisfies the quadratic equation 39z 2 − 128Dz + 64D2 = 0. Solve this quadratic equation exactly for z in terms of D. [Hint: use the result of part (c) with z = x and D = k.] (g) Only one of your two solutions for z in the previous part in fact yields T ′ (z) = 0 when D > 0. Which one is it? Explain briefly. (h) Use a sign diagram, or otherwise, to determine whether the critical point for the function T = T(z) occurs at a maximum or at a minimum, assuming D > 0. [Hint: to test the signs of T ′ (z), look at T ′ (0) and T ′ (D).] (i) Suppose now that F = 13/2 (that is, the river carries a swimmer 6.5 metres downstream per second).
Show that the shortest possible time to reach your friend is 130√ 2 7 + 20 seconds, and this occurs when z = 80. (j) Given that F = 13/2, suppose instead you chose z = 0, that is, you first ran to the nearest point on the left bank of the river, along the line of sight between A and B. How much longer, to the nearest second, would it take to reach your friend by doing this rather than choosing the optimal solution z = 80 of the previous part? (k) Confirm, using the formula for T in part (d), that if F = 0, that is, if the river is still, then the fastest route to reach your friend is along the straight line from A to B.
Question Five
Derive the order of divergence when b → ∞ for the integration defined below as a function of b. Here, we denote that the order of divergence of I(b) as b → ∞ is s(b) if there exits nonzero constant C such that lim b→∞ I(b) s(b) = C. I(b) = ∫ 1 −1 dz ∫ b a dy ∫ b a h(x, y, z)dx, (a > 0, b > 0). Here h(x, y, z) = x 2y 2 f(x)f(y) (1/g(x) + 1/g(y))2 x 2 + y 2 + 2xyz L(x, y, z) 2(x 2 + y 2 + 2xyz + m2) 3/2 , f(x) = √ x 2 + c 2, (c > 0), g(x) = √ x 2 + m2 − m + f(x), (m > 0), L(x, y, z) = √ x 2 + y 2 + 2xyz + m2 − m + f(x) + f(y). Further, in answering this question, consulting Proof Lemma 3.6 in an article titled” Mass Renormalization in the Nelson Model” (e-print arXiv:1611.08538) might be useful.