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University of the Philippines, ManilaCollege of Arts and Sciences

Department of Physical Sciences and MathematicsMATH 100: INTRODUCTION TO CALCULUS

Third Departmental ExamSeptember 25, 2009

I-TRUE OR FALSE: Write True if the given statement is accurate. Otherwise, write False. (2pts each)

1. If a continuous function f has an absolute extremum at some c in the interval [a, b], then f has arelative extremum at c.

2. Iff is continuous on an interval (a, b), then f has an absolute extremum in (a, b).

3. Iff(x) is increasing on (a, b), and g(x) is decreasing on (a, b) then the function F(x) = (f g)(x)is also increasing on (a, b).

4. The polynomial function y = ax3 + bx2 + cx + d is concave up on the interval

b

3a,

5. A twice differentiable continuous function f(x) is concave down on an interval I, if and only iff(x)is decreasing on I.

6. IfF(x) is an antiderivative of f(x), then1

aF(ax) is an antiderivative of f(ax) for a a , a = 0.

7. The antiderivatives of equal functions are equal.

8. The indefinite integral x

2x2 + 2dx may be solved by substituting u = x2 + 2 to get

u

1

2 du.

9. The antiderivative ofxn for any non-zero integer n is given byxn+1

n + 1+ C.

10.

43dx = 4 + C, where C is any real number.

II- Solve the following indefinite integrals. SHOW YOUR COMPLETE SOLUTIONS.

1. 12x5 + 4

4x3 7

3

x dx (4 pts)

2.

8x

5x4 6x2 + 9dx (5 pts)

3.

t4 + 1

t2

53t4 1t3

dt (5 pts)

4.

x5

5

x2 + 5dx (7 pts)

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III- Solve the following differential equations. SHOW YOUR COMPLETE SOLUTIONS.

1.dy

dx

=3x3 5

5y + 5

(x4

4)4

(6 pts)

2.d2y

dx2= 3

3x 1 given that y = 1 and y = 2 when x = 3. (8 pts)

IV- PROBLEM SOLVING: Show your complete solutions for the following problems. (6 pts each)

1. The number of people entering an auditorium to watch the Biogyugan at time t is given by R(t) =1000 36t + 45t2 12t3, where R(t) is the number of people entering the auditorium t hours afterthe doors have been opened. Noone is in the auditorium at t = 0 when the doors are opened, andthe doors are closed when the program begins at t = 3. Find the time when the number of peopleentering the auditorium is a maximum.

2. Find the radius of the closed cylinder having a volume of 108 cubic inches such that its surface areais a minimum. (Volume of a Cylinder: r2h; Surface Area: 2r2 + 2rh)

3. An island is at a point A 4 km from a point B on a straight shore. Point C is on the shore, 10 kmfrom B. A man on the island wishes to go to point C. If he can row at 3kph and walk at 5kph,how far from B should he land his boat in order to reach C in the shortest possible time?

V-CURVE TRACING: Graph the given curve. Determine the critical points and points ofinflection of the function and the intervals where it is increasing/decreasing and concave up/down. Makesure you show concavity on the graph. LABEL THE COORDINATES OF ALL IMPORTANT POINTS.(12 pts)

1. y = (x + 1)3(x 3)

-END (85 pts)-

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