Analysis with Calc

Question 1

What is the Domain and Range of a function?

Answer 1

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.

Question 2

What is the Vertical Line test?

Answer 2

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.

Question 3

Increasing Function

Answer 3

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

Question 4

Decreasing Function

Answer 4

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

Question 5

Even Function

Answer 5

f(x) = f(-x)

Question 6

Odd Function

Answer 6

f(-x) = -f(x)

Question 7

How to work out the Horizontal asymptotes?

Answer 7

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

Question 8

How to work out the Vertical asymptotes?

Answer 8

Where the denominator is undefined

Question 9

What is the continuity test?

Answer 9

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)

Question 10

What is the Intermediate Value Theorem?

Answer 10

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]

Question 11

What is the Sandwich Theorem?

Answer 11

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

Question 12

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

Answer 12

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

provided the limit exists

Question 13

When is a function differentiable?

Answer 13

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

Question 14

Does a differentiable function have to be continuous?

Answer 14

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

Question 15

Product rule

Answer 15

uv’ + vu’

Question 16

Quotient Rule

Answer 16

vu’-uv’
v^2

Question 17

d/dx (sinx)

Answer 17

cosx

Question 18

d/dx (cosx)

Answer 18

-sinx

Question 19

d/dx (tanx)

Answer 19

sec^2x

Question 20

d/dx (secx)

Answer 20

secxtanx

Question 21

d/dx (cotx)

Answer 21

-csc^2x

Question 22

d/dx (cscx)

Answer 22

-cscxcotx

Question 23

Chain rule

Answer 23

dy/dx = dy/du x du/dx

Question 24

What is the equation for linearization?

Answer 24

L(x) = f(a) + f’(a)(x-a)

is the linearization of f at a

Question 25

When does a function have an absolute maximum?

Answer 25

f has an absolute maximum value at a point c if

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

Question 26

When does a function have an absolute minimum?

Answer 26

f has an absolute minimum value at a point c if

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

Question 27

Extreme Value Theorem

Answer 27

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.

Question 28

When does a function have a local maximum?

Answer 28

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

Question 29

When does a function have a local minimum?

Answer 29

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

Question 30

The First Derivative Theorem for Local Extreme Values

Answer 30

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

Question 31

What is a critical point of f?

Answer 31

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

Question 32

How to find the Absolute Extrema?

Answer 32

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

Question 33

Rolle’s Theorem

Answer 33

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

Question 34

The Mean Value Theorem

Answer 34

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

Question 35

First Derivative Test for Local Extrema

Answer 35

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

Question 36

When is a function concave up?

Answer 36

When f’ is increasing on I

Question 37

When is a function concave down?

Answer 37

When f’ is decreasing on I

Question 38

Second Derivative Test for Local Extrema

Answer 38

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.

Question 39

What is an antiderivative

Answer 39

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

Question 40

ant x^n

Answer 40

1 x^n+1 + C
n + 1

Question 41

ant sinkx

Answer 41

-1/k coskx + c

Question 42

ant coskx

Answer 42

1/k sinkx + c

Question 42

ant sec^2(x)

Answer 42

1/k tankx + c

Question 43

ant csc^2(x)

Answer 43

-1/k cotkx + c

Question 44

ant sec(kx)tan(kx)

Answer 44

1/k sec(kx) + c

Question 45

ant csc(kx) cot(kx)

Answer 45

-1/k csc(kx) + c

Question 46

What is an indefenite integral?

Answer 46

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

Question 47

sum of k^2

Answer 47

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

Question 48

sum of k^3

Answer 48

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

Question 49

Mean Value theorem for defenite integrals

Answer 49

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

Question 50

Fundamental Theorem of Calculus, Part 1

Answer 50

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)

Question 51

Fundamental Theorem of Calculus, Part 2

Answer 51

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)

Question 52

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

Answer 52

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

Question 53

The Derivative Rule for Inverses

Answer 53

(f^-1)’(b2) = 1/ f’(f^-1(b1))

Question 54

Product Rule of Logs

Answer 54

lnbx = lnb + lnx

Question 55

Quotient Rule of Logs

Answer 55

ln b/x = lnb - lnx

Question 56

Reciprocal Rule of Logs

Answer 56

ln 1/x = -lnx

Question 57

Power Rule of Logs

Answer 57

ln x^r = rlnx

Question 58

Integral of 1/u

Answer 58

ln |u| + c

Question 59

Integral of tan u

Answer 59

ln|sec u| + c

Question 60

integral of sec u

Answer 60

ln|sec u + tan u| + c

Question 61

integral of cot u

Answer 61

ln |sin u| + c

Question 62

integral of csc u

Answer 62

-ln|csc u + cot u| + c

Question 63

integral of e^u

Answer 63

e^u + c

Question 64

The General Exponential Function a^x

Answer 64

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

a^x = e^xlna

Question 65

Logarithms of Base a

Answer 65

log a x is the inverse function of a^x

Question 66

L’Hopital’s Rule

Answer 66

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

Question 67

Intermediate Powers

Answer 67

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

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

Question 68

Caunchy’s Mean Value Theorem

Answer 68

f’(c) = f(b) - f(a)
g’(c) g(b) - g(a)

Question 69

d/dx (arcsin u)

Answer 69

1 du
sqrt(1-u^2) dx

|u| < 1

Question 70

d/dx (arctan u)

Answer 70

1 du
1+ u^2 dx

Question 71

d/dx (arccos u)

Answer 71

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

|u| < 1

Question 72

d/dx (arccot u)

Answer 72

-1 du
1+u^2 dx

Question 73

d/dx(arcsec u)

Answer 73

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

Question 74

d/dx(arccsc u)

Answer 74

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

Question 75

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

Answer 75

arcsin(u/a) + c

Question 76

integral 1
a^2 + u^2

Answer 76

1/a arctan(u/a) + c

Question 77

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

Answer 77

1/a arcsec(u/a) + c

Question 78

sinhx

Answer 78

e^x - e^-x
2

Question 79

cosh x

Answer 79

e^x + e^-x
2

Question 80

d/dx arcsinh u

Answer 80

1
sqrt(1 + u^2)

Question 81

d/dx arccosh u

Answer 81

1
sqrt(u^2 - 1)

Question 82

d/dx arctanh u

Answer 82

1
1-u^2

Question 83

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

Answer 83

arcsinh(u/a) + c

Question 84

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

Answer 84

arccosh(u/a) + c

Question 85

integral 1
a^2 - u^2

Answer 85

1/a arctanh(u/a) +c

when u^2 < a^2

Question 86

Integration by parts

Answer 86

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

Question 87

Sequence

Answer 87

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

Question 88

Infinite Sequence

Answer 88

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

Question 89

Convergence, Divergence and Limit of a Sequence

Answer 89

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

Question 90

Divergence to Infinity

Answer 90

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

Question 91

Sandwich Theorem for Sequences

Answer 91

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

Question 92

The Continuous Function Theorem for Sequences

Answer 92

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).

Question 93

Connection between the Limit of a Sequence and of a Function

Answer 93

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

Question 94

lim ln(n)/n

Answer 94

0

Question 95

lim nthroot(n)

Answer 95

1

Question 96

lim x^(1/n)

Answer 96

1 when x>0

Question 97

lim x^n

Answer 97

0 when |x| < 1

Question 98

lim (1 + x/n)^n

Answer 98

e^x

Question 99

lim x^n/n!

Answer 99

0

Question 100

The Monotonic Sequence Theorem

Answer 100

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

Question 101

Infinite Series

Answer 101

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

a1+a2+a3+…+an+…

is an infinite series.

Question 102

Nth term of a Series

Answer 102

the number an is the nth term of a series

Question 103

Sequence of Partial sums

Answer 103

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

Question 104

Convergence and Divergence of the Sequence of Partial sums

Answer 104

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

sum to infinity (an) = L

Otherwise it Diverges

Question 105

The anth term of a Convergent Series

Answer 105

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

Question 106

Nth Term Test for Divergence

Answer 106

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

Question 107

The Integral Test

Answer 107

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

Question 108

Integral test and P-series

Answer 108

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

Question 109

Absolute Convergence

Answer 109

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

Question 110

Absolute Convergence Test

Answer 110

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

Question 111

Conditional Convergence

Answer 111

A series that converges but does not converge absolutely converges conditionally

Question 112

Alternating Series Test

Answer 112

Question 113

The Ratio Test

Answer 113

Question 114

Power Series about x = 0

Answer 114

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

Question 115

Power Series about x = a

Answer 115

Question 116

Convergence Theorem for Power Series

Answer 116

Question 117

Converge of a Power Series Possibilities

Answer 117

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

Question 118

Taylor Series

Answer 118

Question 119

Maclaurin Series

Answer 119

Question 120

Taylor Polynomial of Order n

Answer 120

Question 121

Taylor’s Formula

Answer 121

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

Question 122

The Remainder of order n

Answer 122

Question 123

The Remainder Estimation Theorem

Answer 123

Question 124

Level Curve

Answer 124

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

Question 125

Limit of a Function of Two Variables

Answer 125

Question 126

Continuous Function of Two Variables

Answer 126

Question 127

Polar coordinates

Answer 127

Question 128

Two-Path Test for Nonexistence of a Limit

Answer 128

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.

Question 129

Mixed Derivative Theorem

Answer 129

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).

Question 130

Differential Approximation formula for a single variable

Answer 130

∆y = f′(x0)∆x + e∆x

in which e → 0 as ∆x → 0

Question 131

Differential Approximation formula for 2 variables

Answer 131

Question 132

Chain Rule for Functions of Two Variables

Answer 132

Question 133

Chain Rule for Functions of Three Variables

Answer 133

Question 134

Formula for Implicit Differentiation with Two Variables

Answer 134

Question 135

Directional Derivatives

Answer 135

Question 136

Gradient Vector

Answer 136

Question 137

The Directional Derivative Formula (Dot Product)

Answer 137

Question 138

Equation of a tangent to a level curve

Answer 138

Question 139

Linearization for a function of 2 variables

Answer 139

Question 140

Standard Linear Approximation

Answer 140

Question 141

Total differential of a function f

Answer 141

Question 142

Local Maximum and Local Minimum of a Plane

Answer 142

Question 143

First Derivative Test for Local Extreme Values In 3 Dimension

Answer 143

Question 144

Critical Points in 3 Dimension

Answer 144

Question 145

Saddle point in 3 Dimension

Answer 145

Question 146

Second Derivative Test for Local Extremes in 3 Dimension

Answer 146

Question 147

Hessian Matrix

Answer 147

Question 148

Volume of a Solid

Answer 148

Question 149

Fubini’s Theorem (First Form)

Answer 149

Question 150

Fubini’s Theorem (Stronger Form)

Answer 150

Question 151

How do you calculate the limits of integration?

Answer 151

a) Sketch the region of integration

b) Find the y-limits of integration

c) Find the x-limits of integration

d) Reverse the order of integration if necessary

Question 152

How do you calculate the average value of a function f(x,y) over the region R?

Answer 152

Question 153

Jacobian matrix

Answer 153

Question 154

Jacobian determinant

Answer 154

Question 155

Transformation formula for double integrals

Answer 155

Question 156

Average value of a function
f(x, y, z) over a volume D

Answer 156

Question 157

Jacobian Matrix in 3D

Answer 157

Question 158

Jacobian determinant 3D

Answer 158

Question 159

Integral under change of variables 3D

Answer 159