1. 2 nd Semester AP Calculus AB

2.  The next two days come with a hefty assignment.  On a separate sheet of paper --- you will be doing test corrections. I will give you the questions and answers, you will have to show STEP BY STEP how to get there.  No work necessary? No problem! Write a complete sentence explaining the logic.  Today you should get through AT LEAST all of the multiple choice questions.

3.  This will go into the Test category of your grade.  It is worth 15 points.  You are essentially earning a curve for your last test.  This will NOT be a completion grade. It will be accuracy.  You must show all your work for ALL of the problems.  Right or wrong  You may work together or alone but when the assignment is turned in, it will speak for itself  Don’t say that part of the work is shown on someone else’s paper….that will not get you any points!

7.  Use the table to create a trapezoidal approximation for [0,2] with four equal subintervals:  A) 8  B) 12  C) 16  D) 24  E) 32

8.  Use the table to create a trapezoidal approximation for [0,2] with four equal subintervals:  Equation for the area: ½ h(bı +b ₂)  (b-a)/n = ½ in this case.  A) 8  B) 12  C) 16  D) 24  E) 32

9.  Use the table to create a trapezoidal approximation for [0,2] with four equal subintervals:  Equation for the area: ½ h(bı +b ₂)  (b-a)/n = ½ in this case.  A) 8  B) 12  C) 16  D) 24  E) 32

10.  If the integral from a to b of f(x) is a+2b then the integral of f(x) + 5 is ??  A) a + 2b + 5  B) 5b – 5a  C) 7b – 4a  D) 7b – 5a  E) 7b – 6a

11.  If the integral from a to b of f(x) is a+2b then the integral of f(x) + 5 is ??  Can we integrate 5dx from a to b?  YES! The answer will have a and b in it but that is ok!!!  Can we add integrals?  A) a + 2b + 5  B) 5b – 5a  C) 7b – 4a  D) 7b – 5a  E) 7b – 6a

12.  If the integral from a to b of f(x) is a+2b then the integral of f(x) + 5 is ??  Can we integrate 5dx from a to b?  YES! The answer will have a and b in it but that is ok!!!  Can we add integrals?  A) a + 2b + 5  B) 5b – 5a  C) 7b – 4a  D) 7b – 5a  E) 7b – 6a

13.  When is the graph concave down?  A) x > 0  B) 1 < x < 4  C) -4 < x < -1  D) x < 0  E) -1 < x < 4

14.  When is the graph concave down?  How do we find concavity?  2 nd Derivative Test!  Sign Chart  A) x > 0  B) 1 < x < 4  C) -4 < x < -1  D) x < 0  E) -1 < x < 4

15.  When is the graph concave down?  How do we find concavity?  2 nd Derivative Test!  Sign Chart  A) x > 0  B) 1 < x < 4  C) -4 < x < -1  D) x < 0  E) -1 < x < 4

16.  If 2x² - y² = 4, what is the value of the second derivative at the point (2,2)?  A) -1  B) 1  C) 0  D) ¼  E) -¼

17.  If 2x² - y² = 4, what is the value of the second derivative at the point (2,2)?  Can we find the second derivative?  YOU BETCHA!  Do we plug in first?  NOOOOO!  Wait until the end!  A) -1  B) 1  C) 0  D) ¼  E) -¼

18.  If 2x² - y² = 4, what is the value of the second derivative at the point (2,2)?  Can we find the second derivative?  YOU BETCHA!  Do we plug in first?  NOOOOO!  Wait until the end!  A) -1  B) 1  C) 0  D) ¼  E) -¼

19.  The expression given is a Riemann Sum approximation for???

20.  Riemann sum with the first term equal to what?  The last term equal to what?

21.  The expression given is a Riemann Sum approximation for???  Riemann sum with the first term equal to what?  The last term equal to what?

22.  A) 2  B) 0  C) 1/e²  D) 2e  E) e²

23.  A) 2  B) 0  C) 1/e²  D) 2e  E) e²

24.  A) 2  B) 0  C) 1/e²  D) 2e  E) e²

25.  A) 0  B) 1  C) sin(x)  D) cos(x)  E) DNE

26.  A) 0  B) 1  C) sin(x)  D) cos(x)  E) DNE

27.  A) 0  B) 1  C) sin(x)  D) cos(x)  E) DNE

31.  If y = xy + x² + 1 then when x = -1, the derivative is?  A) ½  B) -½  C) -1  D) -2  E) None of these

32.  If y = xy + x² + 1 then when x = -1, the derivative is?  Xs and Ys together? OH MY!  Need that implicit work…mmhmm.  Got some product rule in there for you too…  A) ½  B) -½  C) -1  D) -2  E) None of these

33.  If y = xy + x² + 1 then when x = -1, the derivative is?  Xs and Ys together? OH MY!  Need that implicit work…mmhmm.  Got some product rule in there for you too…  A) ½  B) -½  C) -1  D) -2  E) None of these

34.  A) I only  B) II only  C) I and II only  D) I and III only  E) II and III only

35.  A) I only  B) II only  C) I and II only  D) I and III only  E) II and III only

36.  A) I only  B) II only  C) I and II only  D) I and III only  E) II and III only

37.  A) 0  B) 1/2500  C) 1  D) 4  E) DNE

38.  A) 0  B) 1/2500  C) 1  D) 4  E) DNE

39.  A) 0  B) 1/2500  C) 1  D) 4  E) DNE

40.  Use the graph and the fact that the integral from 1 to 3 of f(x) is 2.3 to find F(3) – F(0)  A) 0.3  B) 1.3  C) 3.3  D) 4.3  E) 5.3

41.  Use the graph and the fact that the integral from 1 to 3 of f(x) is 2.3 to find F(3) – F(0)  ARG Definite integrals…they only told me F(3) – F(1)…How could I possibly figure the rest out?  A) 0.3  B) 1.3  C) 3.3  D) 4.3  E) 5.3

42.  Use the graph and the fact that the integral from 1 to 3 of f(x) is 2.3 to find F(3) – F(0)  ARG Definite integrals…they only told me F(3) – F(1)…How could I possibly figure the rest out?  A) 0.3  B) 1.3  C) 3.3  D) 4.3  E) 5.3

43.  Use the derivative graph (given) to find the parent graph  A) line with positive slope  B) line with negative slope  C)curve with zeros at -2 and 2  D) curve with max at - 2 and min at 2  E) curve with min at - 2 and max at 2

44.  Use the derivative graph (given) to find the parent graph  HOW DO I EVEN SORT OF DO THIS?  What does it mean when the derivative is equal to zero?  Happening at x = -2 and x = 2.  A) line with positive slope  B) line with negative slope  C)curve with zeros at -2 and 2  D) curve with max at - 2 and min at 2  E) curve with min at - 2 and max at 2

45.  Use the derivative graph (given) to find the parent graph  HOW DO I EVEN SORT OF DO THIS?  What does it mean when the derivative is equal to zero?  Happening at x = -2 and x = 2.  A) line with positive slope  B) line with negative slope  C)curve with zeros at -2 and 2  D) curve with max at - 2 and min at 2  E) curve with min at - 2 and max at 2

46.  At time > 0, a(t) = t + sin(t). v(0) = -2. For what value will v(t) = 0?  A) 1.02  B) 1.48  C) 1.85  D) 2.81  E) 3.14

47.  At time > 0, a(t) = t + sin(t). v(0) = -2. For what value will v(t) = 0?  v’(t) = a(t)  Can we anti-derive this?  Yes!  What about “c”?  We have a condition!  A) 1.02  B) 1.48  C) 1.85  D) 2.81  E) 3.14

48.  At time > 0, a(t) = t + sin(t). v(0) = -2. For what value will v(t) = 0?  v’(t) = a(t)  Can we anti-derive this?  Yes!  What about “c”?  We have a condition!  A) 1.02  B) 1.48  C) 1.85  D) 2.81  E) 3.14

49.  A) -.46  B).2  C).91  D).95  E) 3.73

50.  A) -.46  B).2  C).91  D).95  E) 3.73

51.  A) -.46  B).2  C).91  D).95  E) 3.73

52.  A) -6  B) -3  C) 3  D) 6  E) 8

53.  A) -6  B) -3  C) 3  D) 6  E) 8

54.  A) -6  B) -3  C) 3  D) 6  E) 8

58.  A)f’(e) where f(x) = ln x  B)f’(e) where f(x) = (ln(x))/x  C)f’(1) where f(x) =ln(x)  D)f’(1) where f(x) = ln(x+e)  E) f’(0) where f(x) = ln(x)

59.  A)f’(e) where f(x) = ln x  B)f’(e) where f(x) = (ln(x))/x  C)f’(1) where f(x) =ln(x)  D)f’(1) where f(x) = ln(x+e)  E) f’(0) where f(x) = ln(x)

60.  A)f’(e) where f(x) = ln x  B)f’(e) where f(x) = (ln(x))/x  C)f’(1) where f(x) =ln(x)  D)f’(1) where f(x) = ln(x+e)  E) f’(0) where f(x) = ln(x)

61.  s(t) = 2t³ - 24t² +90t +7 when t > 0. For what values of t is the speed increasing?  A) 3 < t < 4 only  B) t > 4 only  C) t > 5 only  D) 0 5  E) 3 5

62.  s(t) = 2t³ - 24t² +90t +7 when t > 0. For what values of t is the speed increasing?  Speed is increasing? Seems like we took notes over that this week just in case we needed the reminder!  A) 3 < t < 4 only  B) t > 4 only  C) t > 5 only  D) 0 5  E) 3 5

63.  s(t) = 2t³ - 24t² +90t +7 when t > 0. For what values of t is the speed increasing?  Speed is increasing? Seems like we took notes over that this week just in case we needed the reminder!  A) 3 < t < 4 only  B) t > 4 only  C) t > 5 only  D) 0 5  E) 3 5

64.  A) 8.5  B) 8.7  C) 22  D) 33  E) 66

65.  A) 8.5  B) 8.7  C) 22  D) 33  E) 66

66.  A) 8.5  B) 8.7  C) 22  D) 33  E) 66

69. ***Yes, there was a typo on this question. As you may have noticed, all your correct answers for the MC section are rounded up. ***

70.  A) I only  B) II only  C) I and II only  D) I and III only  E) I, II, and III

71.  A) I only  B) II only  C) I and II only  D) I and III only  E) I, II, and III

72.  A) I only  B) II only  C) I and II only  D) I and III only  E) I, II, and III

73.  How many gallons of water enter the tank during [0,7]?  Round your answer to the nearest gallon.  We should be thinking: How can I take every single gallon that enters into account?  If only there was a way to use the rate equation and the area under its curve….

74.  For [0,7], find the time intervals during which the amount of water in the tank is decreasing?  If we know the rate at which it comes in and the rate at which is leaves…how could we know when the amount itself is decreasing?  How many decimal places should you be concerned with?

75.  For [0,7] at what time t is the amount of water in the tank the greatest? To the nearest gallon, compute the amount of water at this time. Justify.  Closed interval…do the endpoints matter?  How could we find a max?  Do we have a test for that?

76.  Explain why there must be a value r for 1 < r < 3 such that h(r) = -5.  THIS LOOKS LIKE ONE OF OUR THEOREMS!!!  Does the function meet the conditions?  Can we use it?  How does it work?

77.  Explain why there must be a value c for 1 < c < 3 such that h’(c) = -5  THIS LOOKS LIKE ONE OF OUR THEOREMS!!!  Does the function meet the conditions?  Can we use it?  How does it work?

78.  Find w’(3).  Hmmm…This looks like the Fundamental Theorem of Calculus in disguise….  And there is a function in one of the endpoints….I wonder if that changes anything?

79.  Find the equation to the tangent line of y when y is the inverse of g.  What do we need for a line?  Slope, and a point.  Slope of the tangent line? That sounds familiar….  It should. We spent a whole semester on it!  But did we talk about inverses?  YES!

80.  Find f’(x) and f’’(x)  The hardest part here was not getting intimidated by the “k” that they threw in the mix.

81.  Critical point at x = 1, then find k. Then determine if it is a min, max, or neither. Justify.  How do we find critical points?  How could we solve for k?  Do we have a function and a value so that the only unknown is k?  How do we know max or min?

82.  For a certain value of the constant k, the graph has a point of inflection on the x- axis. Find this value of k.  By setting the second derivative equal to zero and the function itself equal to zero (on the x-axis) You can find two different equations that are equal to k.  The rest is up to you…