CALC 2 FINAL Flashcards
(112 cards)
1
Q
∫ k dx =
A
kx + C
2
Q
∫ x^n dx =
A
x^(n+1)/(n+1) + C (n ≠ -1)
3
Q
∫ 1/x dx =
A
ln|x| + C
4
Q
∫ e^x dx =
A
e^x + C
5
Q
∫ b^x dx =
A
b^x / ln(b) + C
6
Q
∫ sin(x) dx =
A
-cos(x) + C
7
Q
∫ cos(x) dx =
A
sin(x) + C
8
Q
∫ sec^2(x) dx =
A
tan(x) + C
9
Q
∫ csc^2(x) dx =
A
-cot(x) + C
10
Q
∫ sec(x)tan(x) dx =
A
sec(x) + C
11
Q
∫ csc(x)cot(x) dx =
A
-csc(x) + C
12
Q
∫ tan(x) dx =
A
ln|sec(x)| + C
13
Q
∫ cot(x) dx =
A
ln|sin(x)| + C
14
Q
∫ 1/(x^2 + a^2) dx =
A
(1/a) tan^(-1)(x/a) + C
15
Q
∫ 1/√(a^2 - x^2) dx =
A
sin^(-1)(x/a) + C, a > 0
16
Q
∫ sinh(x) dx =
A
cosh(x) + C
17
Q
∫ cosh(x) dx =
A
sinh(x) + C
18
Q
What is the general formula for exponential change?
A
y(t) = Ce^(kt)
19
Q
What differential equation models exponential growth or decay?
A
dy/dt = ky
20
Q
What does k > 0 mean in exponential change?
A
Exponential growth (increasing over time).
21
Q
What does k < 0 mean in exponential change?
A
Exponential decay (decreasing over time).
22
Q
What is the special formula for continuous compound interest?
A
P(t) = Pe^{rt}, where P = principal, r = interest rate, t = time.
23
Q
What is Newton’s Law of Cooling (differential form)?
A
dT/dt = k(T - Ts), where Ts = surrounding temperature.
24
Q
What is the solution to Newton’s Law of Cooling?
A
T(t) = Ts + (T0 - Ts)e^{kt}, where T0 = initial object temperature.
25
What is the definition of sinh(x)?
sinh(x) = (e^x - e^{-x})/2
26
What is the definition of cosh(x)?
cosh(x) = (e^x + e^{-x})/2
27
What is the derivative of sinh(x)?
d/dx[sinh(x)] = cosh(x)
28
What is the derivative of cosh(x)?
d/dx[cosh(x)] = sinh(x)
29
Do hyperbolic functions have negative signs in their derivatives?
No! Unlike regular trig functions, hyperbolic functions do not introduce negatives when differentiating.
30
∫ secx dx =
ln|sec x + tan x| + C
31
∫ cscx dx =
-ln|csc(x) + cot(x)| + C
32
33
What is a separable differential equation?
An equation where variables can be separated: dy/dx = f(x)g(y) becomes (1/g(y)) dy = f(x) dx
34
How do you solve a separable differential equation?
Separate variables, integrate both sides, solve for y (if possible), and apply initial conditions if given.
35
What is the general solution to dy/dx = ky?
y = Ce^(kt)
36
What step must not be forgotten when integrating both sides?
Include the constant of integration (usually on the right side).
37
After separating and integrating, what do you do if an initial condition is given?
Plug in the initial values to solve for C, the constant of integration.
38
Solve: du/dt = (2t + sec^2 t)/(2u)
u(t) = -√(t^2 + tan t + 25), using initial condition u(0) = -5
39
Solve: dA/dr = Ab^2 cos(br), A(0) = b^3
A(r) = b^3 e^{b sin(br)}
40
Exponential Change Formula
y(t) = Ce^(kt) — models continuous exponential change.
41
Compound Interest Formula
P(t) = Pe^(rt) — exponential growth with interest rate r.
42
Definition of e via Integral
e = ∫₁^e (1/x) dx = 1.
43
Derivative of sinh(x)
cosh(x)
44
Derivative of cosh(x)
sinh(x)
45
sinh(x) definition
(e^x - e^(-x))/2
46
Fundamental Theorem of Calculus
∫_a^b f(x) dx = F(b) - F(a), where F'(x) = f(x)
47
Substitution Rule
Let u = g(x), then ∫f(g(x))g'(x)dx = ∫f(u)du
48
Integration by Parts
∫u dv = uv - ∫v du
49
Trig Integral Strategy (odd/even)
Odd power of sine/cosine: split and use identities.
50
Trig Substitution (√a² - x²)
Use x = a sin(θ)
51
Partial Fractions
Break rational function into simpler fractions to integrate.
52
When to Use Long Division
Use when degree of numerator ≥ degree of denominator.
53
Improper Integral Test (∞ bounds)
Take limit as bound → ∞ and evaluate if it converges.
54
Improper Integral: Discontinuity
Check for vertical asymptotes and use limits.
55
Comparison Theorem
0 ≤ f(x) ≤ g(x). If ∫g converges, so does ∫f.
56
Limit Comparison Test
If lim x→∞ f(x)/g(x) = L (0 < L < ∞), then f and g both converge or diverge.
57
Definition of Convergence
Sequence {a_n} converges if lim(n→∞) a_n = L.
58
Definition of Divergence
A sequence diverges if it does not settle to a finite limit.
59
Monotonic Sequence Theorem
If a sequence is bounded and monotonic, it converges.
60
Squeeze Theorem
If a_n is squeezed between two sequences that converge to L, then a_n → L.
61
cos(n)
Oscillates, does not settle → diverges.
62
cos(nπ)
Alternates between ±1 → diverges.
63
(1 + 2/n)^n
Exponential limit → converges to e^2.
64
ln(n+1) - ln(n)
Becomes ln(1 + 1/n) → converges to 0.
65
(-1)^n / n
Alternating + decreasing → converges to 0.
66
Arithmetic Sequence
a_n = a + (n−1)d. Diverges unless d = 0.
67
Factorial Growth Rule
n! grows faster than exponential or polynomial.
68
a_n = (–2)^n / (n+1)!
Converges — factorial dominates exponential.
69
Power Series Convergence
Use ratio/root test and find interval of convergence.
70
Squeeze Theorem with cos(n)/n
Converges to 0 since |cos(n)| ≤ 1 and 1/n → 0.
71
What is the formula for the sum of a geometric series starting at n = 1?
S = a / (1 - r), where |r| < 1
72
Does the geometric series ∑ 12(0.73)^(n-1) converge?
Yes, it converges because |0.73| < 1
73
What is the sum of ∑ 12(0.73)^(n-1)?
12 / (1 - 0.73) = 400/9 ≈ 44.44
74
Does ∑ arctan(n) converge or diverge?
Diverges (limit of terms is not 0)
75
What test shows that ∑ arctan(n) diverges?
Divergence Test (limit ≠ 0)
76
What is the sum of ∑ (1/e^n + 1/[n(n+1)])?
1 + 1/(e - 1)
77
What is the sum of ∑ (1/n^2 - 1/(n+1)^2)?
1/4 (telescoping series)
78
Does ∑ (-3)^(n-1)/4^n converge?
Yes, it's geometric with |r| < 1
79
What is the sum of ∑ (-3)^(n-1)/4^n?
36898
80
What is the sum of ∑ (1/n^2 - 1/(n+1)^2 - (-1)^n/3^n)?
36897
81
What does the divergence test state?
If lim a_n ≠ 0, then ∑ a_n diverges
82
Does ∑ (-1)^n / 3^n diverge?
No, it converges since the limit is 0 and it's geometric
83
Integral Test - Conditions
f(x) must be continuous, positive, and decreasing for x ≥ N.
84
Integral Test - Result
If ∫ from N to ∞ of f(x) dx converges, then ∑ a_n converges. If it diverges, so does ∑ a_n.
85
Integral Test - Use case
Used when a_n = f(n), and integrating f(x) is easier than summing a_n.
86
Integral Test - Example
∑ 1/n^p: converges if p > 1, diverges if p ≤ 1
87
Direct Comparison Test - Setup
Compare a_n with a known series b_n where 0 ≤ a_n ≤ b_n.
88
Direct Comparison Test - Convergence
If b_n converges and a_n ≤ b_n, then a_n converges.
89
Direct Comparison Test - Divergence
If a_n ≥ b_n and b_n diverges, then a_n diverges.
90
Direct Comparison Test - Example
∑ 1/(n^2 + 1) compared to ∑ 1/n^2 (converges).
91
Limit Comparison Test - Formula
lim (a_n / b_n) = c, where c > 0 and finite.
92
Limit Comparison Test - Result
If ∑ b_n converges/diverges, then ∑ a_n does the same.
93
Limit Comparison Test - Use
Used when direct comparison is hard but behavior is similar.
94
Limit Comparison Test - Example
∑ (n+1)/(n^3+5) with b_n = 1/n^2.
95
Alternating Series Test - Conditions
a_n positive, decreasing, and lim a_n → 0.
96
Alternating Series Test - Result
If conditions met, ∑ (-1)^n a_n converges.
97
Alternating Series Test - Error Bound
Error < first omitted term (|a_{n+1}|).
98
Alternating Series Test - Example
∑ (-1)^n / n converges (harmonic alt series).
99
Absolute Convergence
If ∑ |a_n| converges, then ∑ a_n converges absolutely.
100
Conditional Convergence
∑ a_n converges, but ∑ |a_n| diverges.
101
Test for Absolute Convergence
Check if ∑ |a_n| converges (e.g., via comparison or integral test).
102
Example of Conditional Convergence
∑ (-1)^n / n converges conditionally.
103
What is the Ratio Test for series convergence?
Given a_n, compute L = lim (n→∞) |a_{n+1}/a_n|. If L < 1, the series converges absolutely. If L > 1 or diverges, the series diverges. If L = 1, the test is inconclusive.
104
When is the Ratio Test especially useful?
When terms involve factorials or exponential expressions like n!, a^n, etc.
105
What is the Root Test for series convergence?
Given a_n, compute L = lim (n→∞) |a_n|^(1/n). If L < 1, the series converges absolutely. If L > 1 or diverges, the series diverges. If L = 1, the test is inconclusive.
106
When is the Root Test especially useful?
When the nth term contains expressions raised to the nth power, such as (b_n)^n.
107
Which test should you try first when a_n does not approach 0?
The Divergence Test.
108
Which test is best when a_n is geometric or similar to a geometric sequence?
The Geometric Series Test.
109
Which test works well for series with alternating signs?
The Alternating Series Test (AST).
110
When should you try the Integral Test?
When a_n = f(n) for a positive decreasing function f(x), and ∫f(x) is easy to evaluate.
111
When should you try the Comparison or Limit Comparison Test?
When a_n is a rational function or involves roots/polynomials/logs.
112
When is the Ratio Test preferable over the Root Test?
When terms have factorials or products. The Root Test is better when each term is raised to the nth power.
