ENGINEERING CORRELATION COURSE 1
ENGINEERING MATHEMATICS
DIFFERENTIAL CALCULUS 2
ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
MEAN VALUE THEOREM
The mean value theorem states that if a function f(x) is
c o n t i n u o u s o n t h e c l o s e d i n te r v a l [a, b] , a n d
differentiable on the open inter val (a, b) , then there
exists at least one point c in (a, b) such that:
f′(c) =
f(b) − f(a)
b−a
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
ROLLE’S THEOREM
Rolle s theorem is a special case of the Mean value
theorem. It states that if a function f(x) is continuous on
the closed inter val [a, b] , differentiable on the open
inter val (a, b), and f(a) = f(b), then there exists at least
one point c in (a, b) such that:
f′(c) = 0
PREPARED BY: ENGR. JOEY A. DANDAN
’


ECE/EE REVIEW COORDINATOR
QUESTION#1
6
− 3 on the closed inter val
x
[1, 2] using mean value theorem.
Find c of the function f(x) =
A. 0
B.
2
C. 1
D. − 2 and
2
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#2
Find c of the function f(x) = 2x 3 + 4x 2 − 10x + 5 on the
closed interval [−5, − 1] using mean value theorem.
A. 1.94
B. 2.04
C. −3.27
D. −1.56
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#3
Suppose that we know that f(x) is continuous and differentiable
on [6, 15]. Let s also suppose that f(6) = − 2 and that we know
that f′(x) ≤ 10. What is the largest possible value for f(15)?
A. 72
B. 90
C. 92
D. 88
PREPARED BY: ENGR. JOEY A. DANDAN
’

ECE/EE REVIEW COORDINATOR
OPTIMIZATION
Optimization in differential calculus involves nding the
maximum of minimum values of a function. These values
can represent the highest or lowest points on a graph
and are often used to solve real-world problems like
maximizing pro t, minimizing cost, or optimizing
resource usage.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#4
The volume of a closed cylindrical tank is 11.3 m 3 . If the
total surface area is a minimum, what should be the base
radius?
A. 1.39 m
B. 2.04 m
C. 1.54 m
D. 1.22 m
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#5
A wire 24 in long is cut into t wo pieces, one part being bent into
a circle and the other into a square. If the sum of their areas is to
be minimized, what is the difference in the length of each piece?
A. 4.32 in
B. 3.25 in
C. 2.88 in
D. 1.09 in
PREPARED BY: ENGR. JOEY A. DANDAN
fi
ECE/EE REVIEW COORDINATOR
QUESTION#6
A rectangular eld is to be fenced into four equal parts. What is
the size of the largest eld that can be fenced this way with a
fencing length of 1500 ft if the division is to be parallel to one
side?
A. 56250 ft 2
B. 64000 ft 2
C. 35200 ft 2
D. 40500 ft 2
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#7
Find the volume of the largest cylinder that can be
inscribed in a right circular cone of radius 3 in and whose
height is 10 in.
A. 35.63 in 3
B. 41.89 in 3
C. 43.27 in 3
D. 50.25 in 3
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
POINT OF INFLECTION
A point of in ection is a point on the graph of a function
where the concavity changes. This means the curve
changes from being concave up to concave down, or vice
versa.
PREPARED BY: ENGR. JOEY A. DANDAN
fi
fi
fl
ECE/EE REVIEW COORDINATOR
QUESTION#8
Find the points of in ection for f(x) = x 3 − 3x 2 + 4.
A. (1, 2)
B. (2, 0)
C. (3, 4)
D. (0, 4)
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#9
Find the points of in ection for y = 5x 4 − x 5.
A. (0, 0)
B. (3, 162)
C. (4, 256)
D. (0, 0) and (3, 162)
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
IMPLICIT DIFFERENTIATION
Implicit differentiation is a technique used to nd the
derivative of a function when it is not explicitly solved
for one variable in terms of another. In other words, it s
used when the relationship bet ween the variables is
given implicitly, rather than as an explicit function like
y = f(x).
PREPARED BY: ENGR. JOEY A. DANDAN
fl
fl
ECE/EE REVIEW COORDINATOR
QUESTION#10
If x 3y − 2x 2 + y 4 = 8, nd
A.
x 3 + 4y 3
4x − 3x 2y
B.
x 3 − 4y 3
4x + 3x 2y
C.
4x − 3x 2y
x 3 + 4y 3
D.
4x + 3x 2y
x 3 − 4y 3
dy
.
dx
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#11
Find
dy
at point (0, − 2) of the equation x 3 − xy + y 2 = 4.
dx
A. 0.5
B. 2
C. 1
D. 0.25
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
RELATED RATES (TIME RATES)
Related rates problems involves nding the rate at which
one quantity changes with respect to time, given the
rate of change of another related quantity. These
problems typically involve t wo or more variables that
are functions of time.
PREPARED BY: ENGR. JOEY A. DANDAN
fi
ECE/EE REVIEW COORDINATOR
QUESTION#12
Find the slope of the line whose parametric equations are
x = 2 + t and y = 5 − 3t.
A. −2
B. 3
C. 2
D. −3
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#13
Find the slope of the line whose parametric equations are
x = 2 − t and y = 2t + 1.
A. −1
B. −2
C. 2
D. 0.5
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#14
A balloon is being in ated so that its radius increases at a
rate of 2 cm/s . How fast is the volume of the balloon
increasing when the radius is 5 cm?
A. 180π cm 3 /s
B. 190π cm 3 /s
C. 200π cm 3 /s
D. 210π cm 3 /s
PREPARED BY: ENGR. JOEY A. DANDAN
fl
ECE/EE REVIEW COORDINATOR
QUESTION#15
A balloon is being in ated at a rate of 3 in 3 /s . How fast is
the radius of the balloon increasing when the radius is 5 in?
A. 0.0095 in/s
B. 0.0120 in/s
C. 0.1667 in/s
D. 3 in/s
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#16
A 3-meter long steel pipe has its upper end leaning against a
vertical wall and lower end on a level ground. The lower end moves
away at a constant rate of 2 cm/s . How fast is the upper end
moving down, in cm/s, when the lower end is 2 m from the wall?
A. 1.25
B. 1.79
C. 1.37
D. 1.50
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#17
The height of a right circular cylinder is 50 in and decreases at
the rate of 4 in/s , while the radius of the base is 20 in and
increases at the rate of 1 in/s . At what rate is the volume
changing initially?
A. 1306 in 3 /s
B. 1143 in 3 /s
C. 985 in 3 /s
D. 1257 in 3 /s
PREPARED BY: ENGR. JOEY A. DANDAN
fl
ECE/EE REVIEW COORDINATOR
QUESTION#18
If s = t 2 − t 3, where s is the displacement in meters, and t is
the time in seconds, nd the velocity when the rate of
change of velocity is zero.
A. 0.750 m/s
B. 0.667 m/s
C. 0.500 m/s
D. 0.333 m/s
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#19
A tank of water in the shape of inverted cone is being lled with
water at a rate of 12 m 3 /s. The radius of the tank is 26 m and the
height of the tank is 8 m. At what rate is the depth of the water in
the tank changing when the radius of the top of the water is 10 m?
A. 0.038 m/s
B. 0.047 m/s
C. 0.056 m/s
D. 0.065 m/s
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#20
Water is pouring into a swimming pool. After t hours, there
are t + t gallons of water in the pool. At what rate in
gallons per hour is the water pouring in the pool when t = 9?
A. 5/4
B. 6/5
C. 7/6
D. 8/7
PREPARED BY: ENGR. JOEY A. DANDAN
fi
ECE/EE REVIEW COORDINATOR
PARTIAL DERIVATIVES
Partial derivatives are used when dealing with functions
of multiple variables. They measure how the function
changes as one speci c variable changes, while keeping
all other variables constant.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#21
Find the rst partial derivative with respect to x of the given function:
2xy + 3y
z=
x
A.
3y
x2
B.
−3y
x2
C.
−3
x
D.
3
x
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#22
If f(x, y, z) = 2x 5 + 3y 5 − 8x 2y 2z 2, evaluate
∂f
.
∂x
A. 10x 4 − 16xyz
B. 10x 4 − 16xy 2z 2
C. 10x 4 − 16x 2y 2z 2
D. 10x 4 − 64xyz
PREPARED BY: ENGR. JOEY A. DANDAN
fi
fi
ECE/EE REVIEW COORDINATOR
QUESTION#23
If f(x, y) = x 3y 4 + x 2y, evaluate
∂2f
.
∂x 2
A. 12x 3y 2
B. 12x 2y 3 + 2x
C. 6xy 4 + 2y
D. 2x 4y + 6x
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#24
If f(x, y) = x 3y 4 + x 2y, evaluate
∂2f
.
∂x∂y
A. 12x 3y 2
B. 12x 2y 3 + 2x
C. 6xy 4 + 2y
D. 2x 4y + 6x
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#25
1
Divide 94 into 3 parts so that the sum of product of one pair plus
2
1
1
of another pair plus product of remaining pair will be maximum.
3
4
A. 12, 32, 50
B. 18, 36, 40
C. 16, 36, 42
D. 12, 40, 42
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR