INTERNATIONAL UNIVERSITY – VIETNAM NATIONAL UNIVERSITY – HCMC
CALCULUS 1 – FINAL EXAMINATION – SAMPLE QUESTION PAPER
This paper shows SAMPLE exam questions. In actual exams, Part 1 will contain 20 multiple choice
questions (total 60 points), Part 2 will contain three or four questions (total 40 points).
INSTRUCTIONS: Each student is allowed one double-sided sheet of reference material (size A4 or sim-
ilar) marked with their name and ID. A calculator is allowed only for Part 2. All other documents and
electronic devices are forbidden.
PART 1: 20 MULTIPLE CHOICE QUESTIONS – 60 Minutes – 60 points – NO CALCULATOR
Indicate your choice of answer (A, B, C or D) by filling in the correct oval on the attached answer sheet.
If you select more than one answer, no marks will be awarded.
1. Let f (x) = x6 − 10x4. The inflection point(s) of the graph of f is (are): (A) (0, 0) (C) (0, 0) and (1, −9) √ √
(B) (−2, −96) and (2, −96)
(D) (−2 15/3, 12) and (2 15/3, 12)  2. The limit lim x ln 1 + 1 is x→ x ∞ (A) 0 (B) −1 (C) 1 (D) e
3. On a certain day in Ho Chi Minh city, the temperature T (in ◦C) could be modeled by the function:
T (t) = 0.0004t4 − 0.12t2 + 33, −12 ≤ t ≤ 12, where t is the time in hours and t = 0 at noon. What
is the maximum temperature during the day? (A) 35◦C (B) 24◦C (C) 32◦C (D) 33◦C Z 0 p dA 4. Let A (x) = 1 + t3dt. The derivative is x2 dx √ √ √ √ (A) −2x 1 + x3 (B) −x 1 + x6 (C) −2x 1 + x6 (D) − 1 + x6 π
5. The area under the graph of y = sin x over the interval [0, ] is 3 √ 1 1 (A) (B) 1 3 (C) 1 (D) − 2 − 2 2 Z 0 6. The value of te3t dt is −∞ 1 1 (A) (B) 0 (C) ∞ (D) − 9 9
7. Which integral represents the volume of the solid generated when the region under the curve y = sin x
on [0, π] is rotated about the x-axis? (A) R π R π R π R π π sin x dx (B) 2π sin x dx (C) 2π x sin x dx (D) π 0 0 0 0 sin2 x d x
CALCULUS 1 – FINAL EXAMINATION – SAMPLE QUESTION PAPER Student’s name: Student’s mark Student ID:
PART 2: FULL WRITTEN ANSWERS – 40 minutes – 40 points – CALCULATORS ALLOWED
You must explain your answer in detail. No credit will be given for the answer alone.
1. (10 points) The start of the rainy season causes a rapid increase in the mosquito population. In a
certain village, the mosquito population growth rate r (in mosquitos per week) is modelled by the
function r(t) = 600 + 120e0.8t where t is time in weeks. If the initial population is 1000 mosquitos,
find the population after 3 weeks.
2. (10 points) Find the area of the region enclosed by the curves y = x2 − 2x and y = x + 4.
Solutions – Cal 1 Final Exam Sample Paper Part 1 1. B 2. C 3. D 4. C 5. A 6. D 7. D Part 2
1. Let P be the mosquito population at time t. By the net change theorem, with r = dP, we have dt Z 3 P(3) = P(0)+
600+120e0.8tdt = 1000+ [600t +150e0.8t]30 ≈ 1000+1800 +1503 = 4303 (mosquitos) 0
2. The intersection points (x, y) of the two curves satisfy ( ( ( y = x2 − 2x x + 4 = x2 − 2x x = −1 or x = 4 ⇐⇒ ⇐⇒ y = x + 4 y = x + 4 y = x + 4.
Thus the two intersection points are (−1, 3) and (4, 8). The area of the region enclosed by the two curves is Z 4 Z 4 A = x + 4  − (x2 − 2x) dx = 3x + 4 − x2 dx −1 −1 4  3x2 x3  125 = + 4x  −  = . 2 3  6 −1