The Rules. The Math Game. More Rules. Teams. 1. Slope of tangent line. Are you ready??? 10/24/17

Transcription

1 The Rules The Math Game Four Teams 1-2 players per team at the buzzers each =me. First to buzz in gets to answer Q. Correct answer: 1 point for your team, and con=nue playing Incorrect answer/ no answer/ missed buzzing in: change with replacement team members. More Rules Teams Keep moving Each team member only plays once un=l all team members had a chance. Audience par=cipa=on: by clickers No hints from the audience! Encouragement yes!! 1. Slope of tangent line Are you ready??? 1

2 2. Tangent line equa=on Which of the following is the equa=on of the tangent line to f(x) at the point x 0 3. Above or below In the following graph, the tangent line is above the graph of the func=on. I could determine this based on the fact that A. f(x) > 0 B. f (x) > 0 C. f (x) < 0 D. f (x) > 0 E. f (x) < 0 (4) The func=on y = f(x) = x 3 a x (a>0) looks like: 5. Over or under? A linear approxima=on of f(1.1) to the func=on based at the point x 0 =1 would be an A. under-es=mate B. over-es=mate C. given by D. Both A and C E. Both B and C 6. Which graph corresponds to the func=on A. B. ßf (x) 7. What does f(x) look like? The original func=on would look like this: A. B. C. D. C. D. E. 2

3 8. MC 1 When x=100 the func=on Is closest to (A) 3 (B) 0 (C) 1/50000 (D) π (E) Which is which? The graph of the func=on Looks like which of the following? 10. Even or Odd? 11. Velocity of falling object If the height of a falling ball is h(t) = 100 -½ g t 2 then A. Its instantaneous velocity is constant B. Its average velocity is constant C. Its accelera=on is constant D. It will hit the ground at t=10 E. It fell from height h=10 Evaluate the limit Evaluate the limit lim xà0 100 x 3 3 x lim xà0 100 x 3 3 x : func=on is con=nuous so we an plug in x=0 and we get 10 3

4 Find the deriva=ve Compute the deriva=ve and write in simplest form (assume k>0) y(x)= y (x) = Find the deriva=ve Compute the deriva=ve and write in simplest form y(x)= y (x) = Find the deriva=ve Compute the 2 nd deriva=ve and write in simplest form y (x)= y (x) = 4

5 Recap of what we ve found The func=on: y(x)= Its 1 st deriva=ve: y (x)= Its 2 nd deriva=ve: y (x) = Where are the cri=cal points of this func=on? The func=on: y(x)= Its 1 st deriva=ve: y (x)= Its 2 nd deriva=ve: y (x) = So the cri=cal point(s) are at: 12. What kind of cri=cal point? The func=on: y(x)= Its 1 st deriva=ve: y (x)= Its 2 nd deriva=ve: y (x) = So the cri=cal point(s) are : (A) Local MAX (B) Local MIN (C) Neither Where are the inflec=on points of this func=on? The func=on: y(x)= Its 1 st deriva=ve: y (x)= Its 2 nd deriva=ve: y (x) = So the inflec=on point(s) are at: The func=on: Cri=cal points: x=0 (13) The graph of the func=on looks like Inflec=on points: 5

6 14. Distance, velocity, accelera=on Cri=cal points Where does this func=on have cri=cal points? Inflec=on points Where does this func=on have inflec=on points? Func=on: f(x)= 1 st deriva=ve: f (x) = = So CP at x=0, 4/3 2 nd deriva=ve: f (x) = = Changes sign only at x=1, NOT at x=0; IP at x=1 15. Evaluate the limit 16. Straight line lim xà0 A. 0 B. 1 C. ½ D. DNE x/x The equa=on of this line is (A) m=1, b=1 (B) m=-1, b=2 (C) m=-2, b=1 (D) m=-2, b=2 (E) m=2, b=2 where 6

7 (17) Deriva=ve 18. Which of these is problems is not suitable for Newton s method? (A) Solve f(x)=0 for x. (B) Find the zeros of a func=on f(x). (C) Find where the graph of f(x) crosses the x axis. (D) Approximate the value of a func=on close to a known point x 0. (E) Find roots of an equa=on g(x)=c. 19. T or F? 20. T or F? (21) A B 22. When x=1000, the func=on is closest to C D 7

8 (23) Changing sign Which of the following func=ons does not change sign at x = 1? (24) Which of the following graphs looks like the func=on 25. MC Which graph shows both func=ons Let y = f(x) be a smooth func=on (deriva=ves of all orders exist) at x 0. Which of the following statements is correct? Where do the func=ons intersect? Where do the func=ons intersect? SOLUTION: At x = 0, a Only if BOTH n, m even then at x = 0, ±a 8

9 27.Evaluate the limit SOLUTION Factor top and botom: (x-3)(x-1)/(x-3)(x+3) Cancel common factor: (x-1)/(x+3) Sub in x=3: 2/6 = 1/3 28. Rela=onship How are the labeled quan==es related? A. D 2 =2 x 2 +y 2 B. D 2 = x 2 -y 2 C. D 2 = x 2 +y 2 D. D 2 = 2(x 2 +y 2 ) E. D 2 = 2 ( x 2 +y 2 ) 29. Which func=on could represent a type II predator response? A. B. C. D. E. Either B or D Swimming tuna Compute the average velocity of the tuna over 20 hours of swimming based on the following data 30. Evaluate the limit A. 0 B. 1 C. 2 D. ½ E. DNE 9

10 Net rate of nutrient increase in a spherical cell is For what radius is N(r) largest? Cri=cal points: Check for max: so r=0 is a local min, and the other CP is a local max. 31. Rela=onship 32.How is x 1 related to x 0? How are the labeled quan==es related? A. L 2 = (h/2) + 2r B. L 2 = (h) 2 + 2r 2 C. L = (h/2) + (2r) D. L 2 = (h/2) 2 - (2r) 2 E. L 2 = (h/2) 2 + (2r) Even or Odd? Thanks for your par=cipa=on!! Hopefully you had some fun, and we all agree this is MUCH more exci=ng than that other show on TV! 10

11 Answers 1D 2C 3E 4D 5A 6B 7A 8C 9E 10B 11C 12B 13A 14A 15B 16D 17B 18D 19A 20B 21A 22B 23D 24C 25B 26B 27C 28A 29B 30D 31E 32E 33D 11