Analysis with Calc Flashcards
What is the Domain and Range of a function?
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.
What is the Vertical Line test?
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.
Increasing Function
If f(x2) > f(x1) where x2 > x1
Decreasing Function
If f(x2) < f(x1) where x2 > x1
Even Function
f(x) = f(-x)
Odd Function
f(-x) = -f(x)
How to work out the Horizontal asymptotes?
Work out the lim of the function as it tends to infinity and -infinity
How to work out the Vertical asymptotes?
Where the denominator is undefined
What is the continuity test?
A function f(x) is continuous at a point x=c iff it meets the following conditions:
- f(c) exists
- lim x->c f(x) exists
- lim x->c f(x) = f(c)
What is the Intermediate Value Theorem?
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]
What is the Sandwich Theorem?
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
What is the derivative of a function f at a point x0?
lim f(x0 + h) - f(x0)
h->0 h
provided the limit exists
When is a function differentiable?
When it has a derivative at each point of the interbal. and if the left and right hand exist at the endpoints.
Does a differentiable function have to be continuous?
Yes, If f has a derivative at x=c, then f is continuous at x=c
Product rule
uv’ + vu’
Quotient Rule
vu’-uv’
v^2
d/dx (sinx)
cosx
d/dx (cosx)
-sinx
d/dx (tanx)
sec^2x
d/dx (secx)
secxtanx
d/dx (cotx)
-csc^2x
d/dx (cscx)
-cscxcotx
Chain rule
dy/dx = dy/du x du/dx
What is the equation for linearization?
L(x) = f(a) + f’(a)(x-a)
is the linearization of f at a
When does a function have an absolute maximum?
f has an absolute maximum value at a point c if
f(x)<= f(c) for all x a closed interva
When does a function have an absolute minimum?
f has an absolute minimum value at a point c if
f(x)>= f(c) for all x in a closed interval
Extreme Value Theorem
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.
When does a function have a local maximum?
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
When does a function have a local minimum?
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
The First Derivative Theorem for Local Extreme Values
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
What is a critical point of f?
An interior point of the domain of a function ƒ where ƒ′ is zero
or undefined is a critical point of ƒ
How to find the Absolute Extrema?
- Find all the critical points of f on the interval
- Evaluate f at all critical points and endpoints
- Take the largest and smallest of these values
Rolle’s Theorem
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
The Mean Value Theorem
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
First Derivative Test for Local Extrema
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
When is a function concave up?
When f’ is increasing on I
When is a function concave down?
When f’ is decreasing on I
Second Derivative Test for Local Extrema
- If f’(c) = 0 and f’‘(c) < 0, then f has a local maximum at x = c.
- If f’(c) = 0 and f’‘(c) > 0, then f has a local minimum at x = c.
- 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.
What is an antiderivative
A function F is an antiderivative of f on an interval I if f’(x) = f(x) for all x in I
ant x^n
1 x^n+1 + C
n + 1
ant sinkx
-1/k coskx + c
ant coskx
1/k sinkx + c
ant sec^2(x)
1/k tankx + c
ant csc^2(x)
-1/k cotkx + c
ant sec(kx)tan(kx)
1/k sec(kx) + c
ant csc(kx) cot(kx)
-1/k csc(kx) + c
What is an indefenite integral?
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
sum of k^2
n(n+1)(2n+1)
6
sum of k^3
(n(n+1))^2
2^2
Mean Value theorem for defenite integrals
f(c) = 1/(b-a) integral f(x) dx
Fundamental Theorem of Calculus, Part 1
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)
Fundamental Theorem of Calculus, Part 2
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)
How to find the area between the graph of y=f(x) and the x-axis
- Subdivide [a,b] at the zeros of f
- Integrate f over each subinterval
- Add the absolute values of the integrals
The Derivative Rule for Inverses
(f^-1)’(b2) = 1/ f’(f^-1(b1))
Product Rule of Logs
lnbx = lnb + lnx
Quotient Rule of Logs
ln b/x = lnb - lnx
Reciprocal Rule of Logs
ln 1/x = -lnx
Power Rule of Logs
ln x^r = rlnx
Integral of 1/u
ln |u| + c
Integral of tan u
ln|sec u| + c
integral of sec u
ln|sec u + tan u| + c
integral of cot u
ln |sin u| + c
integral of csc u
-ln|csc u + cot u| + c
integral of e^u
e^u + c
The General Exponential Function a^x
For any numbers a > 0 and x, the exponential function with base a is
a^x = e^xlna
Logarithms of Base a
log a x is the inverse function of a^x
L’Hopital’s Rule
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
Intermediate Powers
If lim x-> a ln f(x) = L, then
lim f(x) = lim e^lnf(x) = e^L
x->a x->a
Caunchy’s Mean Value Theorem
f’(c) = f(b) - f(a)
g’(c) g(b) - g(a)
d/dx (arcsin u)
1 du
sqrt(1-u^2) dx
|u| < 1
d/dx (arctan u)
1 du
1+ u^2 dx
d/dx (arccos u)
-1 du
sqrt(1-u^2) dx
|u| < 1
d/dx (arccot u)
-1 du
1+u^2 dx
d/dx(arcsec u)
1 du
|u| sqrt(u^2 -1) dx
d/dx(arccsc u)
-1 du
|u| sqrt(u^2 -1) dx
Integral 1
sqrt(a^2 - u^2)
arcsin(u/a) + c
integral 1
a^2 + u^2
1/a arctan(u/a) + c
integral 1
u sqrt(u^2 - a^2)
1/a arcsec(u/a) + c
sinhx
e^x - e^-x
2
cosh x
e^x + e^-x
2
d/dx arcsinh u
1
sqrt(1 + u^2)
d/dx arccosh u
1
sqrt(u^2 - 1)
d/dx arctanh u
1
1-u^2
integral 1
sqrt(a^2 + u^2)
arcsinh(u/a) + c
integral 1
sqrt(u^2 - a^2)
arccosh(u/a) + c
integral 1
a^2 - u^2
1/a arctanh(u/a) +c
when u^2 < a^2
Integration by parts
integral u(x)v’(x) dx = u(x)v(x) - integral u’(x)v(x)dx
Sequence
A sequence is a list of numbers in a given order:
a1, a2, a3, . . . , an, . . . .
Infinite Sequence
An infinite sequence of numbers is a function whose domain is the set of positive integers
Convergence, Divergence and Limit of a Sequence
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
Divergence to Infinity
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
Sandwich Theorem for Sequences
For sequences {an}, {bn}, {cn}. If an<=bn<=cn beyond some index N, and if lim an = lim cn = L then lim bn =L
The Continuous Function Theorem for Sequences
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).
Connection between the Limit of a Sequence and of a Function
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
lim ln(n)/n
0
lim nthroot(n)
1
lim x^(1/n)
1 when x>0
lim x^n
0 when |x| < 1
lim (1 + x/n)^n
e^x
lim x^n/n!
0
The Monotonic Sequence Theorem
If sequence {an} is both bounded and monotonic, then the sequence converges.
Infinite Series
Given a sequence of numbers {an} and expression of the form
a1+a2+a3+…+an+…
is an infinite series.
Nth term of a Series
the number an is the nth term of a series
Sequence of Partial sums
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
Convergence and Divergence of the Sequence of Partial sums
If {sn} converges to L, we say that the series converges and its sum is L
sum to infinity (an) = L
Otherwise it Diverges
The anth term of a Convergent Series
If sum to infinity (an0 converges then an –> 0
Nth Term Test for Divergence
sum to infinity (an) diverges if lim(an) fails to exist or is not 0
The Integral Test
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
Integral test and P-series
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
Absolute Convergence
A series sum(an) converges absolutely if the series of absolute values sum(|an|), converges
Absolute Convergence Test
if sum to infinity (|an|) converges then sum to infinity (an) converges
Conditional Convergence
A series that converges but does not converge absolutely converges conditionally
Alternating Series Test
The Ratio Test
Power Series about x = 0
Where the coefficients c0,c1,…,cn are constants
Power Series about x = a
Convergence Theorem for Power Series
Converge of a Power Series Possibilities
Here R is called the radius of convergence and the interval of radius R centred at x = a is called the interval of convergence
Taylor Series
Maclaurin Series
Taylor Polynomial of Order n
Taylor’s Formula
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
The Remainder of order n
The Remainder Estimation Theorem
Level Curve
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
Limit of a Function of Two Variables
Continuous Function of Two Variables
Polar coordinates
Two-Path Test for Nonexistence of a Limit
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.
Mixed Derivative Theorem
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).
Differential Approximation formula for a single variable
∆y = f′(x0)∆x + e∆x
in which e → 0 as ∆x → 0
Differential Approximation formula for 2 variables
Chain Rule for Functions of Two Variables
Chain Rule for Functions of Three Variables
Formula for Implicit Differentiation with Two Variables
Directional Derivatives
Gradient Vector
The Directional Derivative Formula (Dot Product)
Equation of a tangent to a level curve
Linearization for a function of 2 variables
Standard Linear Approximation
Total differential of a function f
Local Maximum and Local Minimum of a Plane
First Derivative Test for Local Extreme Values In 3 Dimension
Critical Points in 3 Dimension
Saddle point in 3 Dimension
Second Derivative Test for Local Extremes in 3 Dimension
Hessian Matrix
Volume of a Solid
Fubini’s Theorem (First Form)
Fubini’s Theorem (Stronger Form)
How do you calculate the limits of integration?
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
How do you calculate the average value of a function f(x,y) over the region R?
Jacobian matrix
Jacobian determinant
Transformation formula for double integrals
Average value of a function
f(x, y, z) over a volume D
Jacobian Matrix in 3D
Jacobian determinant 3D
Integral under change of variables 3D
