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Republic of the Philippines
Department of Education
REGION III-CENTRAL LUZON
REGIONAL DIAGNOSTIC ASSESSMENT
BASIC CALCULUS
Name: _________________________________ Score: _________________
Date: ____________
Direction: Write the letter of the correct answer on the blank before each number.
For items 1 and 2. Given the function 𝑥2
− 5𝑥 + 6 and the table of values below
𝑥−
2.5 2.7 2.99 2.9999 𝑥+
3.3 3.1 3.01 3.0001
𝑓(𝑥) 𝑓(𝑥)
_____1. Which of the following statements is TRUE?
A. The 𝑓(𝑥) does not exist. C. The 𝑓(𝑥) is indeterminate.
B. The 𝑓(𝑥) is ∞. D. The 𝑓(𝑥) is zero.
_____2. What is the value of f(x) if 3.01?
A. 0.01 C. 0.010101
B. 0.0101 D. 0.01010101
_____3. Based on the graph below, what is the limit of 𝑓(𝑥) ?
A. 1 C. 5
B. 2 D. DNE
For items 4 and 5, evaluate the following limits.
_____4. (6𝑥3
+ 5𝑥2
+ 4𝑥 − 5)
A. −71 C. 41
B. −41 D. 71
_____5. 𝑙𝑜𝑔 𝑙𝑜𝑔 𝑥
A. 0 C. 10
B. 1 D. 100

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For items 6 and 7, consider the graph of the cosine function 𝑓(𝑥) =𝑐𝑜𝑠 𝑐𝑜𝑠 𝑥 below.
_____6. Determine 𝑐𝑜𝑠𝑐𝑜𝑠 𝑥
A. 𝐷𝑁𝐸 C. 0
B. −1 D. 1
_____7. Determine 𝑐𝑜𝑠𝑐𝑜𝑠 𝑥
A. 𝐷𝑁𝐸 C. 0
B. −1 D. 1
For items 8 and 9, consider the function 𝑓(𝑥) =
𝑥2
−49
𝑥−7
_____8. Determine (
𝑥2−49
𝑥−7
) ?
A. 𝐷𝑁𝐸 C. 0
B. Indeterminate D. 14
_____9. Which of the following is the correct conclusion regarding the continuity of the
given at 𝑥 = 7?
A. The function is discontinuous at 𝑥 = 7 since the limit of the given function does
not exist.
B. The function is discontinuous at 𝑥 = 7 since one or more of the conditions were
not satisfied.
C. The function is continuous at 𝑥 = 7 the limit of the given function exists.
D. The function is continuous at 𝑥 = 7 all conditions were satisfied.
For items 10 and 11, consider the problem: A rectangular garden is to be closed with 980
meters of fencing material. If we let 𝑥 be the length of the field, express its area in square
meters in terms of 𝑥. Identify the domain of the function representing the area and show
that it is continuous on its domain.
_____10. Which of the following is the equation representing the area of the rectangular
garden?
A. 𝐴 = (𝑥2
+ 490) 𝑚2
C. 𝐴 = (490− 𝑥2
) 𝑚2
B. 𝐴 = (490 + 𝑥2
) 𝑚2
D. 𝐴 = (𝑥2
− 490) 𝑚2
_____11. What conclusion can be made regarding the problem?
A. Since the function representing the area is a polynomial function, it is continuous
on its domain [0,490]
B. Since the function representing the area is a polynomial function, it is
discontinuous on its domain [0,490]
C. Since the function representing the area is a rational function, it is continuous on
its domain [0, 490]
D. Since the function representing the area is a rational function, it is continuous on
its domain [0, 490]
For items 12 to 14, consider 𝑦 = 5 − 9𝑥 − 4𝑥2
and the point (−2,4).

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_____12. Which of the following represents the derivative of the function?
A.
𝑑𝑦
𝑑𝑥
= −4𝑥 C.
𝑑𝑦
𝑑𝑥
= −9 − 4𝑥
B.
𝑑𝑦
𝑑𝑥
= −8𝑥 D.
𝑑𝑦
𝑑𝑥
= −9 − 8𝑥
_____13. Find the slope of the line tangent to the graph of the given function at the given
point.
A. 𝑚 = −25 C. 𝑚 = 7
B. 𝑚 = −7 D. 𝑚 = 25
_____14. Determine the equation of the line tangent to the graph of the given function.
A. 𝑦 = 25𝑥 + 54 C. 𝑦 = −7𝑥 − 10
B. 𝑦 = 7𝑥 + 18 D. 𝑦 = −25𝑥 − 46
15. Compute the derivatives of 𝑦 =𝑐𝑜𝑠 𝑐𝑜𝑠 (2𝑥3
) − 5𝑐𝑜𝑡(4𝑥2
).
A. 𝑦′
= −6𝑥2
𝑠𝑖𝑛 𝑠𝑖𝑛 (2𝑥3) − 40𝑥𝑐𝑠𝑐2
(4𝑥2
) C. 𝑦′
= 𝑐𝑜𝑠(3𝑥2)− 𝑐𝑜𝑡(4𝑥2
)
B. 𝑦′
= 6𝑥2
𝑠𝑖𝑛 𝑠𝑖𝑛 (2𝑥3) + 40𝑥𝑐𝑠𝑐2
(4𝑥2
) D. 𝑦′
= −𝑐𝑜𝑠(3𝑥2) + 𝑐𝑜𝑡(4𝑥2
)
For items 16 to 18 consider the problem: Dino wants to construct a rectangular garden.
He wants it to be built next to his room. There are 90 meters of fencing materials available
for the three sides.
_____16. Formulate the optimization equation.
A. 𝐴 = 𝑥𝑦 C. 𝐴 = 2𝑥 + 𝑦
B. 𝐴 = 𝑥2
D. 𝐴 = 2𝑥 + 2𝑦
_____17. Which of the following represents the constraint equation?
A. 𝑃 = 𝑥𝑦 C. 𝑃 = 2𝑥 + 𝑦
B. 𝑃 = 𝑥2
D. 𝑃 = 2𝑥 + 2𝑦
_____18. What should the dimensions of the rectangular garden be to maximize the
area?
A. 22.5 𝑚 & 45 𝑚 C. 45 𝑚 & 45 𝑚
B. 22.5 𝑚 & 22.5 𝑚 D. 45 𝑚 & 90 𝑚
For items 19 and 20, consider the equation 3𝑥2
+ 𝑦2
= 81.
_____19. Which of the following must be used to obtain the derivative of the given
equation?
A. Chain Rule of Differentiation C. Implicit Differentiation
B. Explicit Differentiation D. Partial Differentiation
_____20. Determine 𝑦′
.
A. 𝑦′
= −
3𝑥
𝑦
C. 𝑦′
=
3𝑥
𝑦
B. 𝑦′
= −
𝑦
3𝑥
D. 𝑦′
=
𝑦
3𝑥
_____21. Given 𝑦 =𝑙𝑛 𝑙𝑛 (5𝑥2) determine 𝑦′.
A. 𝑦′
=
𝑥
2
C. 𝑦′
=
2
5𝑥2
𝑙𝑛𝑙𝑛 𝑒

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B. 𝑦′
=
2
𝑥
D. 𝑦′
=
𝑒
2𝑙𝑛𝑙𝑛 5𝑥2
For items 22 to 25, consider the problem: A spherical balloon is being inflated at the rate
of 300 𝑐𝑚3
/𝑚𝑖𝑛. At what rate is the radius increasing when the diameter is 12 𝑐𝑚?
_____22. Which of the following represents the radius of the spherical balloon?
A. 𝑟 = 6 𝑐𝑚 C. 𝑟 = 225 𝑐𝑚
B. 𝑟 = 12 𝑐𝑚 D. 𝑟 = 450 𝑐𝑚
_____23. Which rate of change is being asked?
A.
𝑑ℎ
𝑑𝑡
C.
𝑑𝑉
𝑑𝑡
B.
𝑑𝑟
𝑑𝑡
D.
𝑑𝑦
𝑑𝑥
_____24. Which of the following equation will show the relationship of all the variables
involved in the problem?
A. 𝑉 = 𝜋𝑟2
ℎ C.
𝑑𝑉
𝑑𝑡
=
4𝜋𝑟3
3
B. 𝑉 = 𝜋𝑟3ℎ D. 𝑉 =
𝜋𝑟2
ℎ
3
_____25. Which of the following represents the conclusion for the problem?
A. The rate at which the radius of the of the balloon increases is 0.66 𝑐𝑚/𝑚𝑖𝑛.
B. The rate at which the radius of the of the balloon increases is −0.66 𝑐𝑚/𝑚𝑖𝑛.
C. The rate at which the radius of the of the balloon increases is 0.17 𝑐𝑚3
/𝑚𝑖𝑛.
The rate at which the radius of the of the balloon decreases is −0.17 𝑐𝑚3
/𝑚𝑖𝑛
For items 26 to 28, evaluate the following indefinite integrals.
_______26. ∫ (7𝑥6
− 2𝑥4
+ 5𝑥2
− 6)𝑑𝑥
A. 7𝑥7 −
2𝑥5
5
+
5𝑥3
3
− 6𝑥 + 𝑐 C.
42𝑥5
5
−
8𝑥3
3
+ 10𝑥 + 𝑐
B. 𝑥7 −
2𝑥5
5
+
5𝑥3
3
− 6𝑥 + 𝑐 D. 42𝑥5 − 8𝑥3 + 10𝑥
_______27.∫ (7𝑠𝑒𝑐2
𝑥− 5𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥)𝑑𝑥
A. 7 𝑡𝑎𝑛 𝑡𝑎𝑛 𝑥 + 5 𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 + 𝑐 C. 14 𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 + 5
𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 + 𝑐
B. 7 𝑡𝑎𝑛 𝑡𝑎𝑛 𝑥 − 5 𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 + 𝑐 D. 14 𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 − 5
𝑠𝑒𝑐 𝑠𝑒𝑐 𝑥 + 𝑐
_______28. ∫ 53𝑥
𝑑𝑥
A. 3 ∙ 53𝑥
+ 𝑐 C.−
53𝑥
5𝑙𝑛𝑙𝑛 5
+ 𝑐
B.
1
3
53𝑥
+ 𝑐 D.
53𝑥
3𝑙𝑛𝑙𝑛 5
+ 𝑐
For items 29 to 31, evaluate the following integrals using Substitution Rule.

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_______29.∫ (2𝑥3
− 5)
1
2 ∙ 2𝑥2
𝑑𝑥
A.
2√(2𝑥3
−5)3
9
+ 𝑐 C.
3(2𝑥3
−5)
3
2
2
+ 𝑐
B.
√(2𝑥3
−5)3
3
+ 𝑐 D.
(2𝑥3
−5)
3
2
9
+ 𝑐
_______30.∫
5
(4𝑥 + 3)4
𝑑𝑥
A.
−5
12(4𝑥+3)3 + 𝑐 C.
(4𝑥+3)5
5
+ 𝑐
B.
−5
12(4𝑥+3)3 + 𝑐 D.
(4𝑥+3)5
12
+ 𝑐
_______31.∫ (4𝑥2
− 6𝑥 − 7)5(4𝑥− 3)𝑑𝑥
A.
(4𝑥−3)6
6
+ 𝑐 C.
(4𝑥−3)6
12
+ 𝑐
B.
(4𝑥2
−6𝑥−7)
6
6
+ 𝑐 D.
(4𝑥2
−6𝑥−7)
6
12
+ 𝑐
_____32. Consider the differential equation
𝑑𝑦
𝑑𝑥
= (𝑥 − 5)(2𝑥 − 3), which of the following
represents the particular solution of given if 𝑦 = 5 𝑎𝑛𝑑 𝑥 = 0?
A. 𝑦 =
2𝑥3
3
−
13𝑥2
2
+ 15𝑥 + 5 C. 𝑦 =
2𝑥3
3
−
13𝑥2
2
− 15𝑥 − 5
B. 𝑦 =
2𝑥3
3
+
13𝑥2
2
+ 15𝑥 + 15 D. 𝑦 =
2𝑥3
3
+
13𝑥2
2
− 15𝑥 − 15
For items 33 to 35, consider the problem: Certain bacteria cells are being observed in
an experiment. The population doubles every 5 hours. How many bacteria cells will
there be after 12 hours if at the beginning there were 900?
______33. Which of the following represents 𝑦0 ?
A. 5 C. 900
B. 12 D. 1800
_____34. Determine the value of k.
A. 0.13862 C. −0.13862
B. 0.13863 D. −0.13863
_____35. Which the following is the conclusion for the problem?
A. There are approximately 6428 bacteria cells after 12 hours.
B. There are approximately 6427 bacteria cells after 12 hours.
C. There are approximately 4751 bacteria cells after 12 hours.
D. There are approximately 4750 bacteria cells after 12 hours.
For items 36 to 38, consider the problem: Substance H has a half-life of 80 years. If in
2022, 150 g of substance H was at hand, how much will be at hand in 2077?
_____36. Which of the following is true regarding the given on the problem?
A. 𝑦0 = 75 𝑔
B. 𝑦 = 75 𝑔 when 𝑡 = 160 𝑦𝑒𝑎𝑟𝑠

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C. 𝑦 = 150 𝑔 when 𝑡 = 40 𝑦𝑒𝑎𝑟𝑠
D. 𝑡 = 55 𝑦𝑒𝑎𝑟𝑠 will be used in computing the unknown.
_____37. Compute the value of k.
A. −0.00866 C. 0.00603
B. −0.00603 D. 0.00866
_____38. What can be inferred regarding the problem?
A. There is still 118.08 g of Substance H in 2077.
B. There is still 106.39 g of Substance H in 2077.
C. There is still 97.75 g of Substance H in 2077
D. There is still 93.16 g of Substance H in 2077
For items 39 – 43, evaluate the following definite integrals.
_______39.∫
5
5
(10𝑥10
− 9𝑥9
+ 8𝑥8
− 7𝑥7
+ 6𝑥6
− 5𝑥5
+ 4𝑥4
− 3𝑥3
+ 2𝑥2
− 𝑥)𝑑𝑥
A. indeterminate C. 0
B. undefined D. ∞
_______40.∫
1
0
(2𝑥2
− 3√𝑥)𝑑𝑥
A. −
4
3
C.
1
6
B. −
1
3
D.
5
6
_______41. ∫
2
−1
(9𝑥 − 5)3
∙ 3𝑑𝑥
A.
−28561
12
C.
3285
4
B.
−3285
4
D.
28561
12
_______42.∫
4
0
(4𝑥2
− 14𝑥 − 10)3
(4𝑥 − 7)𝑑𝑥
A. 1248 C. −1246
B. 1246 D. −1248
______43.∫
5
5
√5𝑥3 + 2 ∙ 15𝑥2
𝑑𝑥
A. DNE C. infinity
B. imaginary D. zero
For items 44 to 46, consider the problem: Find the area of the plane region bounded by
𝑦 = 2𝑥 + 5, 𝑥 = 1,𝑥 = 4 and the x-axis
_____44. The following are points that can be found on the graph EXCEPT _____.
A. (1,7) C. (3, 12)
B. (2,9) D. (4, 13)
_____45. Determine the values of the upper and lower limit.
A. 𝑎 = −1 & 𝑏 = 2 C. 𝑎 = 1 & 𝑏 = 4
B. 𝑎 = 0 & 𝑏 = 3 D. 𝑎 = 2 & 𝑏 = 5
_____46. What can be concluded about the problem?

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A. The area of the plane region bounded by 𝑦 = 2𝑥 + 5,𝑥 = 1, 𝑥 = 4 and the x-
axis is 28 square units.
B. The area of the plane region bounded by 𝑦 = 2𝑥 + 5,𝑥 = 1, 𝑥 = 4 and the x-
axis is 30 square units.
C. The area of the plane region bounded by 𝑦 = 2𝑥 + 5,𝑥 = 1, 𝑥 = 4 and the x-
axis is 32 square units.
D. The area of the plane region bounded by 𝑦 = 2𝑥 + 5,𝑥 = 1, 𝑥 = 4 and the x-
axis is 34 square units.
For items 47 to 50, consider the problem: Find the area of the plane region bounded by
𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 + 3.
_____47. Which of the following represents the points of intersection?
A. (4,7),(−3, 0) C. (4, −7),(−3, 0)
B. (4,7),(3, 0) D. (−4,7), (3,0)
_____48. Determine the values of the upper and lower limit.
A. 𝑎 = −3 & 𝑏 = 4 C. 𝑎 = 1 & 𝑏 = −3
B. 𝑎 = 0 & 𝑏 = 3 D. 𝑎 = 2 & 𝑏 = 7
______49. Which of the following is the upper function 𝑓(𝑥)?
A. 𝑦 = 𝑥2
+ 𝑥 + 12 C. 𝑦 = 𝑥 + 12
B. 𝑦 = 𝑥2
− 9 D. 𝑦 = 𝑥 + 3
_____50. What conclusion can be drawn on the problem?
A. The area of the plane region bounded by 𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 + 3 is 54.14
square units.
B. The area of the plane region bounded by 𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 + 3 is 55.15
square units.
C. The area of the plane region bounded by 𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 + 3 is 56.16
square units.
D. The area of the plane region bounded by 𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 + 3 is 57.17
square units.
For items 47 to 50, consider the problem: Find the area of the plane region bounded by
𝑦 = 𝑥2
− 9 and 𝑦 = 𝑥 +

8. Address: Matalino St. D.M. Government Center, Maimpis,City of San Fernando (P)
Telephone Number: (045) 598-8580 to 89; Email Address: region3@deped.gov.ph
Republic of the Philippines
Department of Education
REGION III-CENTRAL LUZON
DIAGNOSTIC TEST IN BASIC CALCULUS
KEY TO CORRECTION
1 D 26 B
2 B 27 B
3 C 28 D
4 B 29 A
5 B 30 B
6 B 31 D
7 C 32 A
8 D 33 C
9 B 34 B
10 C 35 C
11 A 36 D
12 D 37 A
13 C 38 D
14 B 39 C
15 B 40 A
16 A 41 B
17 C 42 D
18 A 43 D
19 C 44 C
20 A 45 C
21 B 46 B
22 A 47 A
23 B 48 A
24 C 49 B
25 A 50 D