MATH100_Final_Prep

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University of British Columbia *

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100

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Mathematics

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Feb 20, 2024

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A and B, two people of identical height h, stand beneath a street lamp of height L. A walks in a straight line and at a constant speed away from the street lamp. One second later, B walks in a straight line and at the same speed, but in the opposite direction, away from the street lamp. As A and B move away from the lamp, their shadows grow longer. (a) Let a be the length of A’s shadow, and b be the length of B’s shadow. Let x be the distance A has walked, and y be the distance B has walked. Draw and label a picture that illustrates the scenario two seconds after A begins to walk away from the street lamp. Your picture should indicate all relevant lengths and the associated variables. (b) As A and B walk away from the lamp, both their shadows are getting longer. Whose shadow is changing length faster, two seconds after A left the lamp? Justify your answer. | Answer: https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH100/December_2016/kQuestion—ti—(a) https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH100/December_2016/Question_11_(b}) Consider a cone with height 2 metres and whose circular base has radius 1 metre. Find the dimensions of the circular cylinder of largest volume that can be contained in the cone. (The base of the cylinder lies at the base of the cone.) Answer: https:/iwiki.ubc.ca/Science:Math_Exam Resources/Courses/MATH100/December_2016/Question_12 A bottle of soda pop at room temperature (70°F) is placed in a refrigerator where the temperature is 40°F. After half an hour the soda pop has cooled to 60°F. How long does it take for the soda pop to cool to 50°F? Answer: https:/iwiki.ubc.ca/Science:Math_Exam Resources/Courses/MATH100/December_2013/Question_05 . No marks will be given for the use of differential rules, but you may use T Using the definition of the derivative, compute f'(z) if f(z) = 22 1 z them to check your answer. Answer: https:/iwiki.ubc.ca/Science:Math_Exam Resources/Courses/MATH100/December_2013/Question_06 Find the area of the largest rectangle which has two vertices on the x-axis and two vertices lie on the graph of the function y = 8 — z? with —4/8 < < /8. Please justify your answer. Answer: https:/iwiki.ubc.ca/Science:Math_Exam Resources/Courses/MATH100/December_2013/Question_09 Prove that there exists at most one positive real number x such that 4° = 3 cos . Answer: https:/iwiki.ubc.ca/Science:Math_Exam Resources/Courses/MATH100/December_2013/Question_11

Give the 16th order Maclaurin polynomial for f(x) = sin x 4 cos x. Calculate the 3rd order Taylor Polynomial for f(x) = x* about x = 1. by interpreting it as a Maclaurin polynomial. Q[11]: Estimate the sum below QI5]: A circle with radius 3 has a sector cut out of it. It’s a smallish sector, no more than a quarter of the circle. You want to find out the area of the sector. (a) Suppose the angle of the sector is 6. What is the area of the sector? (b) Unfortunately, you don’t have a protractor, only a ruler. So, you measure the chord made by the sector (marked 4 in the diagram above). What is 6 in terms of d? (c) Suppose you measured d = 0.7, but actually d = 0.68. Estimate the absolute error in your calculation of the area removed.

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