BIG HUMONGOUS BC PRACTICE MEMORY BOOK TEST
Short Answer
1. In the limit comparison test you take the infinite limit of________________________
and then if the answer is ____________________ it means______________________.
.
2. If the terms of a series go to zero, then the series________________________________.
.
3. If the terms of an alternating series go to zero, then the series _______________________.
.
4. Give the formula for arc length in parametric (or vector) form.
.
5. Give the Trapezoidal Rule Formula for approximating area under the curve.
.
6. Give the formula for any Taylor Series.
.
7. Give the series for arctan x =
8. Give the series for 1/(1+x) =
9. What's the Lagrange Remainder formula?
.
10. Give the formula for
in terms of t
.
11. Give dy/dx in terms of r and
.
12. Define the range of a function.
13. Sketch y = sin x
.
14. The limit is the y-value ______________________________________ and not
necessarily the ___________________________________.
15.
x
lim
0
1 - cos x
x
16. What is the definition of the slope of the tangent line at x ?
.
17. If f(x) is continuous at x = c, then what do we know about the derivative of f at x = c ?
.
18. If f(x) has a sharp corner at x = c, what do we know about f ' (c) ?
.
19. If the derivative does not exist at x = c but it's limit is infinite, what occurs at x = c on the curve?
.
20. d/dx ( mx ) = _______________
21. d/dx ( x ) = ________________________
22. d/dx ( f(x) + g(x) ) = __________________________________
23. What is the Product Rule?
.
24. d/dx ( sin x ) = ____________________
25. d/dx ( tan x ) = ________________________________________
26. d/dx ( sec x ) = _______________________________________
27. d/dx (csc x) = ________________________________________________
28. What is the Chain Rule (Leibniz's notation)
.
29. Given dy/dx, what is the independent variable?
30. When is a particle slowing down?
.
31. If v(t) > 0 then the particle is ____________________________________________.
32. What is a normal line ?
.
33. Why is it important to use variable names for all variable quantities until after the derivative is taken in a
related rates problem?
.
34. In a related rates problem, the derivative of V is ______________________.
35. Name the three types of critical points.
(1)
(2)
(3)
36. What two properties must f '' (x) have in order for (c,f(c)) to be an inflection point?
.
37. What is the difference between a relative (or local) maximum and an absolute (or global) maximum?
.
38. Besides points where f ' (c) = 0 where else may extrema occur?
.
39. cost = C(x) = _____________________________+ _________________________* _____
40. In cost/profit problems, what does x stand for?
.
41. How do we verify that there is a vertical asymptote at x=a ?
.
42. What are the two requirement for f(x) in order for the Mean Value Theorem to apply ?
.
43. If f(x)
0, then an Lower Sum is drawn with ___________________________rectangles.
44.
= lim
n
____________________________________________
oo
45. What happens to the value of the integral when the limits of integration (a and b) are switched?
.
46.
=-
47. Under what circumstances will the integral equal the area under the curve?
.
48. What is the Mean Value Theorem for Integrals ?
.
49. Fill in the blanks:
average value of f ' (x) on [a,b] =
=
-------------------------(b-a)
--------------------------------(b-a)
50. What is the Mean Value Theorem for Integrals?
.
51. What is the correct integral for finding the are between f(x) and g(x) from x=a to x=b if they do NOT
intersect, g(x) > f(x) and you are NOT permitted to use absolute value ?
.
52. What is the correct integral for computing displacement ?
.
53. When using the shell method and rotating about the y-axis, what variable is used in the integral?
.
54. What is the basic integral for computing work ?
.
55. How do you compute average velocity?
.
56. Explain the difference in computing total distance traveled verses displacement.
.
57.
dx =
58. ln ab =
59. Explain how to take the derivative of a function written as a variable expression raised to a variable power.
.
60. e
=
61. e
=
62. e
=
63. d/dx (e
)=
64. d/dx (a
)=
65. If f ' (x) = k · f(x) then f(x) =
66.
lim ( 1 + c/x ) =
x oo
67. What is the equation for computing compound interest ?
.
68. Give the principal domain of sin x
69. Give the principal domain of cos x
70. Give the principal domain of tan x
71. d/dx ( tan x ) =
72.
1/
73.
1/
74.
dx =
dx =
dx =
75. Give two forms of the antiderivative of tan x
.
76. What identity is used if there is a sin x in the integrand ?
.
77. What identity is used if there is a cos x in the integrand ?
.
78. What identity is used if there is a sin
x in the integrand ?
.
79. If regular u-sub does not work, what substitution should be tried if a - b x appears in the integrand ?
.
80. If regular u-sub does not work, what substitution should be tried if a + b x appears in the integrand ?
.
81. If regular u-sub does not work, what substitution should be tried if a x - b appears in the integrand ?
.
82.
dx =
83. What identity is used if there is a cos
x in the integrand ?
.
84. What identity is used if there is a tan x in the integrand ?
.
85. When do you use integration by parts ?
.
86. How do you find a y-intercept?
87. How do you find x-intecepts?
88. Sketch y = mx + b
.
89. Sketch y =
.
90. Sketch y =
.
91. Complete the identity:
tan x + 1 = _________________
92. Complete the identity:
cos 2x = __________________________________________
93. Complete the identity:
tan x/2 =
_____________________________________
94. Draw the unit circle and lable the x and y-coordinates at the following angles:
.
95. Draw the unit circle and lable the x and y-coordinates at the following angles:
.
96. Explain the standard way of dealing with absolute value problems.
.
97. Write the formula for the distance between two points on the plane.
.
98. Write the equation of a circle with center (h,k) and radius r.
.
99. What is the slope of a horizontal line?
100.
______________________________________
101.
f ' (g(x)) g ' (x) dx = _________________________________
102. What are the three steps in solving a separable differential equation ?
1.
2.
3.
103.
sec x dx = ______________________________________
104. a - b =
105. When do we start rounding?
BIG HUMONGOUS BC PRACTICE MEMORY BOOK TEST
Answer Section
SHORT ANSWER
1. the ratio a(n) / b(n), a positive real, both series do the same thing
2. might converge
3. converges
4.
5.
6.
7. ...uh...I forget...go look it up...or take antideriv of the series for 1/(1-(-x^2))...oh now I remember...it's like
sine only no factorials
8. 1-x+x^2-x^3+....
9.
10.
11.
I think!
12.
13.
14.
15.
16.
17.
18.
19.
20.
all the y-values
dr
the number we are getting close to, function value itself
0
lim f(x+h) - f(x)
or
lim f(x) - f(t)
h 0
h
t x
x-t
f ' (c) might exist as long as there is not a sharp corner.
It does not exist.
a vertical tangent line
m
21.
22.
23.
24.
nx
f ' (x) + g ' (x)
d/dx (f(x) · g(x)) = f(x) · g ' (x) + f ' (x) · g(x)
cos x
25. sec x
26. sec x tan x
27. - csc x cot x
28. dy du
dy
--- · --- = ---du dx
dx
29. x
30. When v(t) and a(t) have opposite signs.
31. moving right or up
32. perpendicular to the tangent line
33. Otherwise the derivatives will be zero (useless)
34. dV/dt
35. endpts, stationary pts, singular pts
36. zero (or undefined) and change signs at x=c
37. relative (local) is only higher than neighboring points wheras absolute (global) is the highest of all points
38. at endpoints OR wenever f ' (c) is undefined but f(c) is defined
39. fixed cost + price per item * x
40. the number of items bought or sold
41. Show that there is an infinite limit as x
a
42. f is continuous on [a,b] and differentiable on (a,b)
43. inscribed
44. any Rieman sum or
45. It becomes the opposite of the previous value.
46. True
47. When f(x)
48.
= f(c) * (b-a) for some c between a and b.
49. a,b, f ' (x) dx, f(b) - f(a)
50.
dt = f(c) [ b - a ] for some c between a and b
51.
dx
52.
53. x
54.
55.
56.
57.
58.
59.
60.
displacement / time
answers vary
ln |x| + C
ln a + ln b
take the ln of both sides first, bring down the power, use implicit differentiation
x
61. e · e
62. e / e
63. e
· f ' (x)
64. a
· f ' (x) · ln a
65. A e
66. e
67. A(t) = A (1 + r/n)
68. [-
/2 ,
69. [0,
]
70. [-
/2 ,
/2 ]
/2 ]
71.
dx
72. sin x + C
73. sec x + C
74. tan x + C
75. -ln |cos x| or ln |sec x| + C
76. sin x = 1 - cos x
77. cos x = 1 - sin x
78.
79.
80.
81.
82.
sin x = 1/2 - 1/2 cos 2x
bx = au
bx = au
ax = bu
ln | sec x + tan x | + C
83. cos x = 1/2 + 1/2 cos 2x
84.
85.
86.
87.
88.
89.
90.
tan x = sec x - 1
When there is a simple product and/or if u-sub fails
Set x=0
Set y or f(x) = 0
(drawing)
dr
dr
91. sec x
92. cos x - sin x
93. sin x / (1 + cos x)
94. (
,
, 1/2), (-
95. (
96. Split it into parts without absolute value.
, -1/2), (
,-1/2)
97.
98.
99.
100.
101.
102.
103.
(x-h) + (y-k) = r
zero
-cos x + C
f(g(x)) + C
1) separate the variables 2) intergrate both sides 3) solve for y
tan x + C
104. (a-b)(a +ab +b )
105. Not until the very end of the problem