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Unformatted text preview: College of Engineering
University of San Carlos
Nasipit, Talamban, Cebu City 6000 Philippines College Midterm Examinations
on Engineering Mathematics Course: EM 122 Calculus 1 Approved by:
Exam Schedule: February 7, 2015; 9:00 a.m.12.‘00 Noon. Exam Engr. Ricardo L. Forms; and Engr. Dr. Evelyn B. Taboada Committee: Patrick U. Tan College Dean QUESTIONNAIRE General Instruction: Do as directed in Tests 1, 2 & 3. Present in writing the full details of your
answers or solutions in the bluebook provided The grades you earn for each item will be based on correctness comgleteness, and clarity of gresentation. Test 1. Answer the following problems as speciﬁed.
Time Allocation: 45 minutes Weight: 30% 1) 2) 3) 4) 5) 6) 7) Given f(x): 3x+2 if x<4
5x+k if 43x Find the value of k such that lim f (x) exists. x—>4 Find the point of inﬂection of the curve slope of the tangent to the curve y = x3 — x2 — x Ify=x2 3x2+2 ﬁnd Q
dx A cell is spherical in shape. Find the rate of change of the volume of the cell with respect to the
diameter when the diameter is 3 micrometer. At what point on the curve y = 2x2 — 4x — 1 would the slope of the tangent line be 4?
x + 2
Find ’ of = .
y y x2 _ 3
. 2 2 — 7 + 6
Evaluate the limit :llmH2 [x—:]
x _ 8) Given that f(x)=\/2x+x2 and g(x)=x2—16,ﬁnd (gof)(x) Test 2. Answer the following problems as specified. Time Allocation: 75 minutes Weight: 50% 1)
2) 3) 4) 5) 6) 7) dzy 2 Given x2 + 25 y2 = 100, ﬁnd when x=8. Find the maximum possible area of a rectangle whose two vertices are on the xaxis and the other two vertices are on the curve )62 + y — 9 = 0 . Water runs into an inverted conical tank at the rate of 8m3/hr. If the height of the tank is 6m and
the base diameter is 4m, how fast is the water level rising when the water is 3m deep? Two lines through the point (—1,3) are tangent to the curve x2 + 4 y2 — 4x — 8y + 3 = 0. Find the equation of each line. Find the area of the largest rectangle that can be placed inside the triangle ABC whose sides are
AB=3m, BC=4m, and AC=5m if one side of the rectangle is on side AC of the triangle. If f (x) = ax3 + bx2 + cx, determine a, b, and c so that the graph will have a point of inﬂection at (1,2) and so that the slope of the inﬂectional tangent there will be 2. A cardboard box manufacturer wishes to make open boxes from pieces of cardboard 12
inches square by cutting equal squares from the four corners and turning up the sides.
Find the length of the side of the square to be cut out in order to obtain a box of the
largest possible volume. Test 3. Answer the following problems as speciﬁed. Time Allocation: 30 minutes Weight: 20% 1) 2) Find the radius and height of the rightcircular cylinder of largest volume that can be
inscribed in a sphere of radius 200m. Tuition at small college is currently Php8,000 per semester. The college currently has 1000
students. If tuition is increasing at the rate of Php400 per semester and enrolment is declining
at the rate of 20 students per year, ﬁnd the rate at which semestral revenues from tuition will
be changing two years from now? ...
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 Fall '15

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