Gateway Calculus Bowl 2014

1. Gateway Calculus Bowl 2014 Practice Problems Colorado Youth Education Connection, 2013

2. Problem 1

3. Problem 2

4. Problem 3 If what is the constant A for which y satisfies the equation Ay’’+y= 0 ? • 0 (d) -1/9 • 3 (e) 1/9 • 1/3

5. Problem 4

6. Problem 5

7. Problem 6 • A function f has a point of inflection when (best answer): • f’ = 0 • f’ < 0 and f’’ > 0 • f’ = 0 and f’’ < 0 • f’’ = 0 • the curvature changes sign at this point

8. Problem 7

9. Problem 8 The line segment drawn indicates an inflection point of the curve when It is blue It is green It is red It is blue and horizontal Never, because this curve has no inflection points

10. Problem 9

11. Problem 10

12. Problem 11 y 2 1 x -1 1 2 -1

13. Problem 12

14. Problem 13

15. Problem14 “calculus” is: (a) A Latin word meaning a small stone used for counting. (b) A branch of mathematics including the study of limits, derivatives, integrals, sequences and series. (c) Any method or system of calculation based on the symbolic manipulation of expressions. (d) A form of hardened dental plaque. (e) (b) and (c) (f) All of the above

16. Problem 15

17. Problem 16

18. Problem 17 y y=f(x) x=t R(t) y=g(x)

19. Problem 18

20. Problem 19

21. Problem 20 y x

22. Problem 21 ?

23. Problem 22

24. Problem 23

25. Problem 24 • The statement of the Mean Value Theorem (MVT) is • a) The statement that all functions have a mean value. • b) The statement that mean value can be computed via definite integrals. • c) The statement that continuous functions over an interval [a,b] take on their mean value somewhere in that interval. • d) The statement that for functions with a continuous derivative over an interval [a,b] , at at least one point in that interval the derivative takes on the value of the average slope. • e) A false conjecture first published by Parameshvara (1370-1460).

26. Problem 25

27. Problem 26 A particle moves along the x-axis with position at time t given by for . Find the time at which the particle is furthest to the right. • 0 (d) 3π/2 • π (e) 3π/4 • 2π

28. Problem 27

29. Problem 28

30. Problem 29

31. Problem 30 • Which of the following is a true story about a famous woman mathematician? Hypatia was the first recognized women mathematician. She was born in Alexandria, Egypt, around 350AD and was a recognized scholar. Besides being a mathematician, she was an astute astronomer and a philosopher. An angry Christian mob killed her. * She wrote a commentary on the 13th volume of the famous Greek mathematics text book, 'Arithmetica'. * She edited Ptolemy's famous version of the 'Almagest'. * She edited her father's commentary on 'Euclid's Elements'. Maria Gaetana Agnesi was a child prodigy, an Italian linguist and a math wizard. Born in a wealthy family in 1718, she was the 21st child of her parents. She was fluent in 6 languages. Some of her contributions are as follows: * She wrote the first book introducing integral and differential calculus. * She determined the equation of a peculiar curve, which came to be known as the 'Witch of Agnesi'. Marie Sklodowska Curie was born in 1868 in Poland, eventually becoming a famous biologist and mathematician in Germany. She was the first woman professor at the University of Stutgart and developed the mathematical theory behind radiocarbon dating. • Hypatia • Hypatia and Maria Agnesi • Maria Agnesi and Marie Curie • Hypatia and Marie Curie • All of the above

32. Problem 31

33. Problem 32

34. Problem 33 y d y=f(x) c x O a b

35. Problem 34 y

36. Problem 35

37. Problem 36

38. Problem 37

39. Problem 38