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Mathematics > Analysis with Calc > Flashcards

Analysis with Calc Flashcards

1
Q

What is the Domain and Range of a function?

A

The Domain is the set of all possible input x-values of the function

The Range is the set of all possible output y-values of the function.

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2
Q

What is the Vertical Line test?

A

A function f can only have one value f(x) for each x in its domain, so no vertical line can intersect the graph more than once.

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3
Q

Increasing Function

A

If f(x2) > f(x1) where x2 > x1

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4
Q

Decreasing Function

A

If f(x2) < f(x1) where x2 > x1

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5
Q

Even Function

A

f(x) = f(-x)

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6
Q

Odd Function

A

f(-x) = -f(x)

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7
Q

How to work out the Horizontal asymptotes?

A

Work out the lim of the function as it tends to infinity and -infinity

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8
Q

How to work out the Vertical asymptotes?

A

Where the denominator is undefined

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9
Q

What is the continuity test?

A

A function f(x) is continuous at a point x=c iff it meets the following conditions:

  1. f(c) exists
  2. lim x->c f(x) exists
  3. lim x->c f(x) = f(c)
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10
Q

What is the Intermediate Value Theorem?

A

If f is a continuous function on a closed interval [a,b], and if yo is any value between f(a) and f(b), then yo = f(c) for some c in [a,b]

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11
Q

What is the Sandwich Theorem?

A

Suppose g(x) <= f(x) <= h(x) for all x in some open interval containingc.

Suppose also that lim g(x) = lim h(x) = L

then lim f(x) = L

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12
Q

What is the derivative of a function f at a point x0?

A

lim f(x0 + h) - f(x0)
h->0 h

provided the limit exists

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13
Q

When is a function differentiable?

A

When it has a derivative at each point of the interbal. and if the left and right hand exist at the endpoints.

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14
Q

Does a differentiable function have to be continuous?

A

Yes, If f has a derivative at x=c, then f is continuous at x=c

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15
Q

Product rule

A

uv’ + vu’

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16
Q

Quotient Rule

A

vu’-uv’
v^2

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17
Q

d/dx (sinx)

A

cosx

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18
Q

d/dx (cosx)

A

-sinx

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19
Q

d/dx (tanx)

A

sec^2x

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20
Q

d/dx (secx)

A

secxtanx

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21
Q

d/dx (cotx)

A

-csc^2x

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22
Q

d/dx (cscx)

A

-cscxcotx

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23
Q

Chain rule

A

dy/dx = dy/du x du/dx

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24
Q

What is the equation for linearization?

A

L(x) = f(a) + f’(a)(x-a)

is the linearization of f at a

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25
Q

When does a function have an absolute maximum?

A

f has an absolute maximum value at a point c if

f(x)<= f(c) for all x a closed interva

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26
Q

When does a function have an absolute minimum?

A

f has an absolute minimum value at a point c if

f(x)>= f(c) for all x in a closed interval

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27
Q

Extreme Value Theorem

A

If f is continuous on a closed interval [a,b], then f attains both an absolute maximum value M and an absolute minimum value m.

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28
Q

When does a function have a local maximum?

A

A function ƒ has a local maximum value at a point c if ƒ(x) <= ƒ(c) for all x∊D lying in some open interval containing c

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29
Q

When does a function have a local minimum?

A

A function ƒ has a local minimum value at a point c within its
domain D if ƒ(x) >= ƒ(c) for all x∊D lying in some open interval containing c

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30
Q

The First Derivative Theorem for Local Extreme Values

A

If ƒ has a local maximum or minimum value at an interior point c of its domain,
and if ƒ′ is defined at c, then

f’(c) = 0

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31
Q

What is a critical point of f?

A

An interior point of the domain of a function ƒ where ƒ′ is zero
or undefined is a critical point of ƒ

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32
Q

How to find the Absolute Extrema?

A
  1. Find all the critical points of f on the interval
  2. Evaluate f at all critical points and endpoints
  3. Take the largest and smallest of these values
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33
Q

Rolle’s Theorem

A

Suppose that y =ƒ(x) is continuous over the closed interval [a, b] and differentiable at every point of its interior (a, b). If ƒ(a) =ƒ(b), then there is at least one number c in (a, b) at which ƒ′(c) =0

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34
Q

The Mean Value Theorem

A

Suppose y =ƒ(x) is continuous over a closed interval [a, b] and differentiable on the interval’s interior (a, b). Then there is at least one point c in (a, b) at which

f(b) - f(a) = f’(c)
b - a

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35
Q

First Derivative Test for Local Extrema

A

Suppose that c is a critical point of a continuous function ƒ

1.if ƒ′ changes from negative to positive at c, then ƒ has a local minimum at c;
2.if ƒ′ changes from positive to negative at c, then ƒ has a local maximum at c;
3.if ƒ′ does not change sign at c (that is, ƒ′ is positive on both sides of c or nega-tive on both sides), then ƒ has no local extremum at c

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36
Q

When is a function concave up?

A

When f’ is increasing on I

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37
Q

When is a function concave down?

A

When f’ is decreasing on I

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38
Q

Second Derivative Test for Local Extrema

A
  1. If f’(c) = 0 and f’‘(c) < 0, then f has a local maximum at x = c.
  2. If f’(c) = 0 and f’‘(c) > 0, then f has a local minimum at x = c.
  3. If f’(c) = 0 and f’‘(c) = 0, then the test fails. The function f may have a local maximum, a local minimum or neither.
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39
Q

What is an antiderivative

A

A function F is an antiderivative of f on an interval I if f’(x) = f(x) for all x in I

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40
Q

ant x^n

A

1 x^n+1 + C
n + 1

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41
Q

ant sinkx

A

-1/k coskx + c

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42
Q

ant coskx

A

1/k sinkx + c

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42
Q

ant sec^2(x)

A

1/k tankx + c

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43
Q

ant csc^2(x)

A

-1/k cotkx + c

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44
Q

ant sec(kx)tan(kx)

A

1/k sec(kx) + c

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45
Q

ant csc(kx) cot(kx)

A

-1/k csc(kx) + c

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46
Q

What is an indefenite integral?

A

The collection of all antiderivatives of ƒ is called the indefinite integral of ƒ with respect to x, and is denoted by

Integral f(x) dx

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47
Q

sum of k^2

A

n(n+1)(2n+1)
6

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48
Q

sum of k^3

A

(n(n+1))^2
2^2

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49
Q

Mean Value theorem for defenite integrals

A

f(c) = 1/(b-a) integral f(x) dx

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50
Q

Fundamental Theorem of Calculus, Part 1

A

If ƒ is continuous on [a, b], then F(x) = integral ƒ(t) dt is continuous on [a, b] and differentiable on (a, b) and its derivative is ƒ(x):

F’(x) = d/dx integral f(t)dt = f(x)

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51
Q

Fundamental Theorem of Calculus, Part 2

A

If f is continuous over [a.b] and F is any antiderivative of f on [a,b], then

integral f(x) dx = F(b) - F(a)

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52
Q

How to find the area between the graph of y=f(x) and the x-axis

A
  1. Subdivide [a,b] at the zeros of f
  2. Integrate f over each subinterval
  3. Add the absolute values of the integrals
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53
Q

The Derivative Rule for Inverses

A

(f^-1)’(b2) = 1/ f’(f^-1(b1))

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54
Q

Product Rule of Logs

A

lnbx = lnb + lnx

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55
Q

Quotient Rule of Logs

A

ln b/x = lnb - lnx

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56
Q

Reciprocal Rule of Logs

A

ln 1/x = -lnx

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57
Q

Power Rule of Logs

A

ln x^r = rlnx

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58
Q

Integral of 1/u

A

ln |u| + c

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59
Q

Integral of tan u

A

ln|sec u| + c

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60
Q

integral of sec u

A

ln|sec u + tan u| + c

61
Q

integral of cot u

A

ln |sin u| + c

62
Q

integral of csc u

A

-ln|csc u + cot u| + c

63
Q

integral of e^u

A

e^u + c

64
Q

The General Exponential Function a^x

A

For any numbers a > 0 and x, the exponential function with base a is

a^x = e^xlna

65
Q

Logarithms of Base a

A

log a x is the inverse function of a^x

66
Q

L’Hopital’s Rule

A

Suppose that ƒ(a) =g(a) =0, that ƒ and g are differentiable on an open interval I containing a, and that g′(x)≠0 on I if x≠a. Then

lim f(x) = lim f’(x)
z->a g(x) x->a g’(x)

assuming that the limit on the rigght side of this equation exists

67
Q

Intermediate Powers

A

If lim x-> a ln f(x) = L, then

lim f(x) = lim e^lnf(x) = e^L
x->a x->a

68
Q

Caunchy’s Mean Value Theorem

A

f’(c) = f(b) - f(a)
g’(c) g(b) - g(a)

69
Q

d/dx (arcsin u)

A

1 du
sqrt(1-u^2) dx

|u| < 1

70
Q

d/dx (arctan u)

A

1 du
1+ u^2 dx

71
Q

d/dx (arccos u)

A

-1 du
sqrt(1-u^2) dx

|u| < 1

72
Q

d/dx (arccot u)

A

-1 du
1+u^2 dx

73
Q

d/dx(arcsec u)

A

1 du
|u| sqrt(u^2 -1) dx

74
Q

d/dx(arccsc u)

A

-1 du
|u| sqrt(u^2 -1) dx

75
Q

Integral 1
sqrt(a^2 - u^2)

A

arcsin(u/a) + c

76
Q

integral 1
a^2 + u^2

A

1/a arctan(u/a) + c

77
Q

integral 1
u sqrt(u^2 - a^2)

A

1/a arcsec(u/a) + c

78
Q

sinhx

A

e^x - e^-x
2

79
Q

cosh x

A

e^x + e^-x
2

80
Q

d/dx arcsinh u

A

1
sqrt(1 + u^2)

81
Q

d/dx arccosh u

A

1
sqrt(u^2 - 1)

82
Q

d/dx arctanh u

A

1
1-u^2

83
Q

integral 1
sqrt(a^2 + u^2)

A

arcsinh(u/a) + c

84
Q

integral 1
sqrt(u^2 - a^2)

A

arccosh(u/a) + c

85
Q

integral 1
a^2 - u^2

A

1/a arctanh(u/a) +c

when u^2 < a^2

86
Q

Integration by parts

A

integral u(x)v’(x) dx = u(x)v(x) - integral u’(x)v(x)dx

87
Q

Sequence

A

A sequence is a list of numbers in a given order:
a1, a2, a3, . . . , an, . . . .

88
Q

Infinite Sequence

A

An infinite sequence of numbers is a function whose domain is the set of positive integers

89
Q

Convergence, Divergence and Limit of a Sequence

A

The sequence {an} converges to the number L if for every positive number e there exists an integer N such that |an - L | < e when n>N

The sequence diverges if there is no such number L

L is the limit of the sequence

90
Q

Divergence to Infinity

A

A sequence diverges to infinity if for ever number M there is a integer N such that for all n larger than N, an > M

A sequence diverges to negative infinity if for all n > N we have an<m for every number m

91
Q

Sandwich Theorem for Sequences

A

For sequences {an}, {bn}, {cn}. If an<=bn<=cn beyond some index N, and if lim an = lim cn = L then lim bn =L

92
Q

The Continuous Function Theorem for Sequences

A

For sequence {an} if an –>L and if f is a continuous function at L and defined at all an then f(an) –>f(L).

93
Q

Connection between the Limit of a Sequence and of a Function

A

Let f(x) be defined for all x>= n0
and the sequence {an} such that an=f(n) for n>=n0. Then

lim f(x) = L implies lim an =L and x–> infinity

94
Q

lim ln(n)/n

A

0

95
Q

lim nthroot(n)

A

1

96
Q

lim x^(1/n)

A

1 when x>0

97
Q

lim x^n

A

0 when |x| < 1

98
Q

lim (1 + x/n)^n

A

e^x

99
Q

lim x^n/n!

A

0

100
Q

The Monotonic Sequence Theorem

A

If sequence {an} is both bounded and monotonic, then the sequence converges.

101
Q

Infinite Series

A

Given a sequence of numbers {an} and expression of the form

a1+a2+a3+…+an+…

is an infinite series.

102
Q

Nth term of a Series

A

the number an is the nth term of a series

103
Q

Sequence of Partial sums

A

The sequence of partial sums {sn} defined by
s1 = a1
s2 = a1 + a2
…
sn = a1 + a2 +…+ an +…+ sum ak

where sn is the nth partial sum

104
Q

Convergence and Divergence of the Sequence of Partial sums

A

If {sn} converges to L, we say that the series converges and its sum is L

sum to infinity (an) = L

Otherwise it Diverges

105
Q

The anth term of a Convergent Series

A

If sum to infinity (an0 converges then an –> 0

106
Q

Nth Term Test for Divergence

A

sum to infinity (an) diverges if lim(an) fails to exist or is not 0

107
Q

The Integral Test

A

For the sequence {an} of positive terms. Suppose that an=f(n) where f is a continuous, positive, decreasing function of c for all x>=N

Then the series sum to infinity(an) and the integral of f(x) both converge or both diverge

108
Q

Integral test and P-series

A

The integral test can be used to show that the p-series sum to infinity (1/(n^p)) converges if p>1 and diverges if p<= 1

109
Q

Absolute Convergence

A

A series sum(an) converges absolutely if the series of absolute values sum(|an|), converges

110
Q

Absolute Convergence Test

A

if sum to infinity (|an|) converges then sum to infinity (an) converges

111
Q

Conditional Convergence

A

A series that converges but does not converge absolutely converges conditionally

112
Q

Alternating Series Test

A
113
Q

The Ratio Test

A
114
Q

Power Series about x = 0

A

Where the coefficients c0,c1,…,cn are constants

115
Q

Power Series about x = a

A
116
Q

Convergence Theorem for Power Series

A
117
Q

Converge of a Power Series Possibilities

A

Here R is called the radius of convergence and the interval of radius R centred at x = a is called the interval of convergence

118
Q

Taylor Series

A
119
Q

Maclaurin Series

A
120
Q

Taylor Polynomial of Order n

A
121
Q

Taylor’s Formula

A

The quantity Rn(x) in this formula is called the remainder of order n or the error term for the approximation of f by Pn(x) over I

122
Q

The Remainder of order n

A
123
Q

The Remainder Estimation Theorem

A
124
Q

Level Curve

A

The set of points in the domain where a function f(x, y) has a constant value, f(x, y) = c,
is called a level curve of f

125
Q

Limit of a Function of Two Variables

A
126
Q

Continuous Function of Two Variables

A
127
Q

Polar coordinates

A
128
Q

Two-Path Test for Nonexistence of a Limit

A

It states that if a
function f(x, y) has different limits along two different paths as (x, y) → (x0, y0), then lim (x,y)→(x0,y0)
f(x, y) does not exist.

129
Q

Mixed Derivative Theorem

A

If f(x, y) and its partial derivatives fx, fy, fxy and fyx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b) then

fxy(a, b) = fyx(a, b).

130
Q

Differential Approximation formula for a single variable

A

∆y = f′(x0)∆x + e∆x

in which e → 0 as ∆x → 0

131
Q

Differential Approximation formula for 2 variables

A
132
Q

Chain Rule for Functions of Two Variables

A
133
Q

Chain Rule for Functions of Three Variables

A
134
Q

Formula for Implicit Differentiation with Two Variables

A
135
Q

Directional Derivatives

A
136
Q

Gradient Vector

A
137
Q

The Directional Derivative Formula (Dot Product)

A
138
Q

Equation of a tangent to a level curve

A
139
Q

Linearization for a function of 2 variables

A
140
Q

Standard Linear Approximation

A
141
Q

Total differential of a function f

A
142
Q

Local Maximum and Local Minimum of a Plane

A
143
Q

First Derivative Test for Local Extreme Values In 3 Dimension

A
144
Q

Critical Points in 3 Dimension

A
145
Q

Saddle point in 3 Dimension

A
146
Q

Second Derivative Test for Local Extremes in 3 Dimension

A
147
Q

Hessian Matrix

A
148
Q

Volume of a Solid

A
149
Q

Fubini’s Theorem (First Form)

A
150
Q

Fubini’s Theorem (Stronger Form)

A

Mathematics (5 decks)

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