The following sample exam from the UP Mathematics Club features topics from the third part of UP Diliman Mathematics 21. You will find below its questions and the corresponding markscheme.

Questions

Do as indicated. This examination is good for 1.5 hours. This exam contains 8 questions and will be graded 35 points. Calculators are not allowed.

(3 points) Find the absolute extrema of the function f(x) = x³ + 5x² + 8x − 10 on the interval [−2,−1].
(5 points) A container box is to be made from a 7 × 7 inches galvanized flat sheet by cutting identical squares out of the four corners and turning up the sides. Find the volume of the largest box that can be made in this way.
(4 points) Christian, who owns a store, sells 500 xphones a week for $450 each. His data analyst for market research shows that each decrease of $5 in the price will result in the sale of an additional 10 xphones per week. If Christian is deciding on whether to decrease or keep his prices, what price would result in the maximum revenue?
(8 points) The position of a particle along a straight line path is defined by
s(t) = t³ − 6t² − 15t + 7 [ft]
where t is in seconds. Answer the following questions where t = 10.
1. What is the particle’s instantaneous velocity?
2. What is the particle’s acceleration?
3. (c) What is the particle’s displacement? (i.e. its position relative to its initial position.)
4. Determine the particles’s total distance travelled.
(2 points) Using a local linear approximation, approximate 8.1^(1/3).
(3 points) Suppose a cubic tank with 7m sides is to be coated on all sides with a special coating of thickness 5 × 10⁻³ [m] to protect it from leakage. Use differentials to estimate the needed volume of special coating.
(6 points) An airlane “Tango” is directly above a barn while maintaining a constant horizontal velocity of 150 meters per second and a constant altitude of 1.5 kilometers from the ground. What would be the rate at which the angle of elevation changes from the barn to the plane, when the plane has traveled 800 meters horizontally?
Evaluate the following limits.