University of the Philippines, ManilaCollege of Arts and Sciences

Department of Physical Sciences and MathematicsMATH 100: INTRODUCTION TO CALCULUS

First Departmental ExamJuly 15, 2010

I- TRUE OR FALSE: For each of the following, write TRUE if the given statement is accurate. Otherwise,write FALSE. (2 pts each)

1. In a hyperbola, the distance of a vertex from its center is always greater than the distance of an extremity fromthe center.

2. The circle with equation x2 + y2 = r2 has its center at the origin and radius r units.

3. If an ellipse has one extremity at (−1, 5) and one vertex at (3, 3), then its latus recta are each 2 units long.

4. The distance of the focus of a parabola from its directrix is equal to half the length of its latus rectum.

5. If f(a) exists, then limx→a

f(x) exists.

6. For a, L ∈ <, if limx→a

f(x) = L, then the graph of y = f(x) passes through the point (a, L).

7. For m,n ∈ N, m > n, then limx→0+

(1

xm− 1

xn

)= −∞.

8. For any non-negative number C, if f(a) = C, then limx→a

√f(x) =

√C.

9. limx→0

x sin1x

= 0.

10. For any real numbers a, b 6= 0, then limx→0

sin(ax)sin(bx)

=a

b.

II - FILL IN THE BLANKS: Give only the final answer for each of the following questions: (3 pts each)

1. The center-radius equation of the circle with center at (2,−4) and tangent to the line 5x− 12y = 6 is given by.

2. The hyperbola 9y2 − 4x2 − 40x− 54y − 55 = 0 is asymptotic to the lines with equations .

3. The endpoints of the latus rectum of a parabola opening downwards are at (3,−5) and (−5,−5), then its vertexis at .

4. The foci of the ellipse 9x2 + 5y2 + 90x− 10y + 185 = 0 are at .

5. The standard equation of the hyperbola with the same center, extremities and foci as the ellipse(x + 4)2

4+

(y − 6)2

16= 1 is given by .

6. limx→4+

sgn(x2 − 10x + 24)x2 − 10x + 24

=

7. limx→−∞

2x3 + 7x2 + 2x− 32x− 1

= .

1

8. limx→0−

bx− bx− bxccc = .

9. limx→−∞

√x2 +

√x2 +

√x2 + · · ·

x + x12 + x

14 + x

18 · · ·

= .

10. limx→2

tan(x2 − 4)x− 2

=

III- PROBLEM SOLVING: Show your complete and orderly solutions in solving the following limits:

1. Find the standard equation of the ellipse, such that one of its latus recta has endpoints at (4, 10) and (4, 3) andits center is at (−2, 6.5).

2. limx→−3−

(bx + 3c

sgn(x + 3)+|x + 3|x + 3

)(4 pts)

3. limx→1

x−√

2− x

x− 3√

x(6 pts)

4. limx→−∞

(2x + 3 +√

4x2 + 25) (6 pts)

5. limx→π

tan2(9x)(x− π) sin(7x)

(10 pts)

IV- CURVE TRACING: Sketch the graph of the function f given the following properties. Determine thepoints of discontinuity and identify the type of discontinuity at each. (14 pts)

• f(−3) = 0, f(−2) = 0, f(−1) = 0, f(2) = 1

• limx→−2

f(x) = +∞

• limx→−3

f(x) = 2

• limx→0−

f(x) = −∞, limx→0+

f(x) = +∞

• limx→2−

f(x) = 1, limx→2+

f(x) = 0

• limx→−∞

f(x) = 1, limx→+∞

f(x) = −1

-END OF EXAM-(90 PTS)

“Stubbornly persist, and you will find that the limits of your stubbornness go well beyond the stubbornness of yourlimits”. ∼Robert Brault

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