MATH 27 Midterm Exam
1st Semester, AY 2023-2024
General Instructions: Do not tear any part of the questionnaire or your blue book. When submitting your blue book,
fold and insert the questionnaire between the last page and the back cover of your blue book.
I. (Multiple Choice) Write the CAPITAL letter of the correct answer in the FIRST PAGE of your bluebook.
DO NOT show any solutions for this part. (2 points each)
For items 1 and 2, consider the graph of the function f below.
1. As the value of x decreases without bound, the function value of f
A. gets closer and closer to 4
B. decreases without bound
C. gets closer and closer to 0
D. increases without bound
2. The function f is continuous on the following intervals EXCEPT
A. (−4, −2)
B. (−3, −1)
C. (−2, 0)
D. (1, 3)
B. −3 + π
C. −7 − π
D. −3 − π
B. −4
C. 0
D. +∞
C. 1
D. +∞
C. cos θ − csc θ cot θ
D. − cos θ − csc θ cot θ
3. lim (5x3 − 2x + π) is equal to
x→−1
A. −7 + π
4.
lim
x−2
x→−2+ x + 2
is equal to
A. −∞
5.
x2 + 10x + 5
is equal to
x→+∞
x+5
lim
A. −∞
B. 0
6. If f (θ) = sin θ + csc θ, then f ′ (θ) equals
A. cos θ + csc θ cot θ
B. − cos θ + csc θ cot θ
7. Given x = Arccos t − Arccsc t, the value of
1
1
− √
t2 − 1 t 1 − t2
1
1
B. − √
+ √
2
1−t
t 1 − t2
A. − √
dx
is given by
dt
1
1
− √
t2 − 1 t t2 − 1
1
1
D. − √
+ √
2
2
1−t
t t −1
C. − √
2
8. The derivative of f (x) = ex +
A. e2x +
1
is given by
x
1
x2
2
B. ex · 2x +
C. e2x −
1
x2
2
1
x2
D. ex · 2x −
9. Using L’Hopital’s Rule, lim
ln t
is equal to
3t
B. ln 13
1
x2
t→+∞
A. 0
C. 13
D. +∞
10. Which of the following statements is/are always TRUE?
I. If f (x) = x27 , then f (28) (x) = 27 · 26 · · · 2 · 1.
II. If c is in the domain of f where f ′ (c) = 0 and f ′′ (c) = −1, then f has a relative maximum at x = c.
A. I only
B. II only
C. Both I and II
D. Neither I nor II
II. (Problem Solving) Do as indicated. Show your solutions completely, neatly and orderly. Solutions must be
written in the same order as the questions appear in the questionnaire. Use one page per item. You may write
back-to-back in the bluebook.
x−9
1. Without using L’Hopital’s Rule, evaluate lim √
.
(4 points)
x→9
x−3
(
πx
cos
, x<1
2
2. Let f be the function defined by f (x) =
. Determine if f is continuous at x = 1. If the
ln(2x − 1), x > 1
function is not continuous at x = 1, determine the type of discontinuity.
(4 points)
3. Determine y ′ if
√
3x4 + 8 x
.
x2/3 + 3
(b) y = sec5 (4x3 ) + Arccot (sin x)
(c) y = 2cot x log3 (5x)
(d) y = x tan(x + y), where y is a differentiable function of x
dy
x cos x
4. Apply logarithmic differentiation to solve for
if y =
.
dx
Arctan x
(a) y =
1
5. Evaluate lim [ex + x] x .
(4 points)
(4 points)
(4 points)
(4 points)
(5 points)
(5 points)
x→∞
6. MCT Corp. manufactures ready-to-drink (RTD) coffee in can. The daily revenue of selling x cans of the said
product is given by R(x) = − 13 x3 + 12 x2 + 90x. How many cans of RTD coffee must be sold daily to maximize
their revenue?
(5 points)
7. Keren’s puppy climbed at the edge of a top of a building 4 meters from the ground. If Keren is moving at a
rate of 0.5 m/s away from the building to find his puppy, at what rate is the angle θ changing when Keren is
5 meters away from the building?
(6 points)
End of exam
Total: 65 points