Answer with short solution

13. What is f 208 cost dil A." C. D. InD 14. Which of the following has e""-as an antiderivative? C. drew-1 D. xet-1 15. What is the equation of the curve that passes through (1,3), if its slope is given by ! = 3x' at any point x? Ayer+2 C. y = 6x + 2 B. yer - 2 D. y = 6x - 2 For numbers 16 to 18: The population of Guinea pigs in a certain place increases each month exponentially. After a month. 100 Guinea pigs were counted. After 3 months, there are 500 Guinea pigs already. 16. How many are there originally? A. Approximately 5 Guinea pigs B. Approximately 25 Guinea pigs PAGE C. Approximately 45 Guinea pigs D. Approximately 50 Guinea pigs 120 17. When will the population of the Guinea pigs be tripled? A. After In 3 LOVE months B. After- In V'S In 3 - month C. After In 3 In 5 month D. After months 18. When will the population reach 10,000? A. After ; In v5 In (100 V/5) `month B. After In (100,/5) In v5 months C. After In v5 In 100 month D. After- In 100 In V6 monthSECOND GRADING, SECOND SEMESTER NAME: GRADE & SECTION: PAGE 118 General Instructions: This is a written work worth 30 points from module 6. Cut this part (broken lines) and staple them properly and submit them in school or (this is encouraged) take clear pictures or scan your answer sheets and upload them to GENYO. Don't Forget to write your name grade and section on the upper left corner of the next pages. ALL final answers should be written in blue or black ink only. Strictly NO ERASURE of any kind or change in answer, NO superimposition on final answers. L Read carefully the statements below. Write TRUE if the statement is always correct, otherwise, FALSE. Use the space provided before each item. (1 point each) 1. There is no function / (x) such that J f (x) = /(x) + C. 2. If f (x) and g(x) are continuous functions, then 3. If / (x) and g(x) are continuous functions, then [ V() + s(x]] dx = ] ,(x)dx + | f(x)dx. 4. If f(x) and g(x) are continuous functions, then ]2 dx = [!(x)ar 5. [=dx = Inlx| + C and Jadx = Inlx/| + C. 6. Jx" = + C. for any real number n. 7. If G (x) is an antiderivative of g(x) and F(x) = G(x) -5, then F(x) is also an antiderivative of g (x). 8. If the integrand is positive the antiderivative is also positive. . The antiderivative of a function is not unique. 10. If G (x) is an antiderivative of g(x) then y = G(x) Is a solution to the differential on 2 = (x). equation d 11. The differential equation = = xy - y + x is separable. 12. If a population grows exponentially, the doubling time is given by t = 12, where k is the growth constant. II. MULTIPLE CHOICE. Read and analyze the questions then write the letter that corresponds to your answer on the space provided before each number. If the correct answer is not among the choices, write E. Use uppercase letters. (1 point each) _1. If we are trying to evaluate the f esin* cos x dax , which substitution would be most helpful? A. u = Cosx C. u = esinx B. u = sinx D. u =" cos I2. Which of the following can be integrated using substitution? A x sinx dx C. [ x sin(x*) dx B. fx sinx dx D. [ x sin x dx 3. Which of the following is an antiderivative of Br"? A 2x' + X G. 2(x* - 1) B. 2x - x D. 2x(x] - 1) 4. Which of the following is not a separable differential equation? PACE A dx C. y=0 119 - 2 cos x D. 5. Ify = ax"+),n * -2, then which of the following is equal to f ydx? A +C -+C n+1 7142 B. -+C + C D. n+1 n+2 6. Which of the following is a particular solution to the differential equation " = = when x = e" and y = 107 C. y = In|8x] - 30 A y= - 30 D. y = 8In/x| - 30 B. y = 8In|x] + c 7. Using integration by substitution in J x5 (x6 - 2)3dx, what is du? A x'dx C. (x5 - 2)dr B. 6x-dx D. x5(x6 - 2)'dx B. If [ (cos x . B) dx = _ costs + C. what Is the B? A - sinx C. - CO5 X D. B. = CSCX SOCI 9. Which of the following is equal to J e* at? A. e C. ex + C B. es + C D. est + C 10. What should be the value of A so that f A - sin (6x]) dx = _ cost63-1 4 0 7 A. x C. 3x B. 2x D. 4x 11. What is A. =+C C. " + + 2x + C B. =+x+C D. X(2x7+32+6) . + C 6 12. In solving fax, which substitution should be used? A. HE= C. u = lor B. u = Inx D. u = In(Inx)