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Title: Calculus Notes
Description: 50 pages worth of calculus notes that will help you in class.
Description: 50 pages worth of calculus notes that will help you in class.
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Calculus Notes
Calculus
Leong Y
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1 Functions and their graphs
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2 Transformations and combinations of graphs
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3 Trigonometric functions
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1 Limit of a function and limit laws
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2 Precise definition of a limit
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3 One-sided limits
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4 Continuity
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5 Limits involving infinity and asymptotes
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1 Tangent lines and the derivative at a point
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4 Applications of differentiation
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4 Concavity and curve sketching
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5 Applied optimization
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6 Newton’s method
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7 Antiderivatives
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1 Areas and estimating with finite sums
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2 Sigma notation and limits of finite sums
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3 Definite integrals
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4 The fundamental theorem of calculus
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5 Indefinite integrals and the substitution method
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6 Definite integral substitutions and the area between curves
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6 Applications of definite integration
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7 Transcendental functions
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5 Indeterminate forms and L’Hˆopital’s rule
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6 Inverse trigonometric functions
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7 Hyperbolic functions
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8 Relative rates of growth
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1 Using basic integration formulae
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2 Integration by parts
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3 Trigonometric integrals
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4 Trigonometric substitutions
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5 Integration of rational functions by partial functions
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1 Solutions, slope fields and Euler’s method
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2 First-order linear equations
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3 Applications
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4 Graphical solutions of autonomous equations
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1 Sequences
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2 Infinite series
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3 The integral test
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4 Comparison tests
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5 Absolute convergence, ratio and root tests
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6 Alternating series and conditional convergence
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11 Parametric equations and polar coordinates
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These
are intended to summarise the core concepts as simply and compactly as possible
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3
1
1
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D is called the domain and the set of possible f (x) is the range
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A function which outputs real numbers is real-valued
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The vertical line
test says that a graph is of a function if no vertical line intersects the graph
more than once
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Similarly, it is said to be decreasing if f (x1 ) > f (x2 )
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It is said to be odd if
f (−x) = −f (x)
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Odd functions have symmetry about the origin
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Rational functions are quotients of two polynomials
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Non-algebraic functions are transcendental
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2
Transformations and combinations of graphs
Algebraic operations can be used to form new functions f + g, f − g, f g and
f /g from f and g
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(For f /g, g(x) ̸= 0
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Note that f ◦g ̸= g ◦f
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For a graph of a function y = f (x), a vertical shift of k units is given by
y = f (x) + k
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A
vertical scaling by k is given by y = kf (x)
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Scaling by −1 represents a reflection in either axis
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3
Trigonometric functions
The 6 trigonometric functions are defined using the circle x2 + y 2 = r2
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Trigonometric identities are useful in calculus
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The law of cosines relates the sides of a triangle, a, b and c, to the angle
opposite c, θ
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1
Limits and continuity
Limit of a function and limit laws
An informal definition of a limit is given here
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The precise definition is given in §2
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We write
lim f (x) = L
x→c
Limit laws help us work with limits
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That is,
lim (f (x) + g(x)) = lim f (x) + lim g(x)
x→c
x→c
x→c
The other rules are much the same
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These can be
proven from the precise definition
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This also extends to rational
functions
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The sandwich theorem says that if f, g and h are defined on some open
interval containing c such that g(x) < f (x) < h(x) on this interval, except
possibly at c, then limx→c f (x) = L if
lim g(x) = lim h(x) = L
x→c
2
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3
One-sided limits
The left-hand limit of f as x → c is L, and we write limx→c− f (x) = L if for
every ϵ > 0, there exists δ > 0 so that
|f (x) − L| < ϵ whenever c − δ < x < c
The right-hand limit of f as x → c is L, and we write limx→c+ f (x) = L if
for every ϵ > 0, there exists δ > 0 so that
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|f (x) − L| < ϵ whenever c < x < c + δ
A function has a limit at c in an open interval if and only if it has left- and
right-hand limits there and they are equal
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The sandwich theorem establishes two important limits used in the calculus of
trigonometric functions, namely
lim
θ→0
2
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More
precisely, we say that f is continuous at c if
lim f (x) = f (c)
x→c
A function is continuous if it is continuous at every x ∈ Df
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We say that f is
continuous on [a, b] if it is continuous on (a, b), right-continuous at a and leftcontinuous at b
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A function which is not continuous is discontinuous
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A jump discontinuity occurs when the left and right limits disagree
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An infinite
discontinuity occurs when the function grows without bound
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All of the familiar functions (polynomial, rational, trigonometric, exponential
and logarithmic) are continuous wherever they are defined
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(Note: For
f /g, g(c) ̸= 0
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The intermediate value theorem says that for a function f continuous on
[a, b], for any y0 between f (a) and f (b), y0 = f (c) for some c ∈ [a, b]
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Some functions f may not be defined at some c, yet limx→c f (x) = L might still
exist
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5
Limits involving infinity and asymptotes
We say that f has a limit L as x approaches infinity and write limx→∞ f (x) = L
if for all ϵ > 0, there is an M ∈ R for which
|f (x) − L| < ϵ whenever x > M
Similarly, limx→−∞ f (x) = L if for all ϵ > 0, there is an N ∈ R for which
|f (x) − L| < ϵ whenever x < N
The limit laws from §2
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A line y = b is a horizontal asymptote to the graph y = f (x) if limx→−∞ f (x) =
b or limx→∞ f (x) = b
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Infinite limits are ways to describe function behaviour near points of infinite
discontinuity
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8
3
3
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The formula for its slope is
f (a + h) − f (a)
h
This is called a difference quotient with increment h
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The tangent line at the point
(a, f (a)) is the limit of the secant line from (a, f (a)) to (a + h, f (a + h)) as
h → 0
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The derivative of f at a point (x0 , f (x0 )) is given by
f (x0 + h) − f (x0 )
h
Thus, the derivative at a point is the slope of the tangent line at that point
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f ′ (x0 ) = lim
h→0
3
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1, we can obtain a function f ′ (x) whose value
at each x is the value of the derivative of f at x
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If it
exists for all x ∈ Df , f is said to be differentiable
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One-sided derivatives are defined analogously
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For f to be differentiable on [a, b],
it must be differentiable on (a, b), right-differentiable at a and left-differentiable
at b
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That is, if f is differentiable at c, it is
necessary that it is continuous there as well
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A continuous function can fail to be differentiable due to a corner, cusp
or a vertical tangent line, amongst other reasons
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3
Differentiation rules
Using the definition in §3
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The
power rule states that for n ∈ R,
d n
x = nxn−1
dx
The sum and constant multiple rules allow us to take derivatives of constant
multiples and sums of functions
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du
d
du dv
d
(cu) = c
and
(u + v) =
+
dx
dx
dx
dx dx
The product rule says that
dv
du
d
(uv) = u
+
v
dx
dx dx
Note that this can be extended to multiple functions to evaluate (y1 y2
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The quotient rule says that
− u dv
d u v du
= dx 2 dx
dx v
v
′
If f itself is a differentiable function, then we can take the second-order
derivative f ′′ = (f ′ )′
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3
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Consider a particle moving obeying the position function s
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The velocity v = ds
dt is the instantaneous rate of change of position
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10
2
d s
The acceleration a = dv
dt = dt2 is the instantaneous rate of change of velocity
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Other applications of derivatives include in economics and genetics
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5
Derivatives of trigonometric functions
Using the addition formulae and the limits in §2
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3
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Letting
u = g(x), we have
dy du
dy
=
dx
du dx
Without having to write out the derivatives explicitly, we can think of the composition as involving an “outside” and “inside” function
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So, we simply differentiate the
“outer” function and evaluate it at the “inner” function, then multiply by the
derivative of the “inner” function
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For y = f ◦ g ◦ h,
dy du dv
dy
=
dx
du dv dx
where u = g(v) and v = h(x)
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3
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We can write the relationship in a formula
F (x, y) = 0
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We can then use algebra to solve for dy/dx
in terms of x and y
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11
3
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We can then use the tools of calculus to analyse how the rate of
change of one or more quantity affects that of the other
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For variables that are not involved in the rate
calculations, equations may be formed to relate them to those that are
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3
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For a differentiable function, the graph of y = f (x) looks like a straight line
locally, so the linearization is a good approximation to f (x) near x = a
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Approximation of errors is discussed in §10
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Often,
we take dx = ∆x
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For small dx, dy is close to ∆y
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If f (a) is known, we can approximate f (a + dx) as
f (a + dx) = f (a) + ∆y ≈ f (a) + dy = f (a) + f ′ (a) dx
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4
4
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It has a absolute minimum on D at c if f (x) ≥ f (c) for all x ∈ D
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A function defined by a
formula can have different extrema depending on the domain D
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This is known as the extreme value
theorem
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A function f has a local maximum at c ∈ D if f (x) ≤ f (c) for all x ∈ D
in some open interval containing c
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The first derivative theorem for local extreme values says that if f has a
local extremum at an interior point c and if f ′ is defined, then f ′ (c) = 0
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Points where this is true
are called critical points
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The smallest and largest of these are the
absolute minimum and maximum respectively
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2
The mean value theorem
Rolle’s theorem states that for a function f (x) continuous over [a, b] and is
differentiable on (a, b), if f (a) = f (b), then there exists c ∈ (a, b) such that
f ′ (c) = 0
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A consequence of the mean value theorem is that functions with the same derivative differ only by a constant
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7)
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3
Monotonic functions and the first derivative test
Suppose f is continuous on [a, b] and differentiable on (a, b)
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If f ′ < 0, then f is decreasing on [a, b]
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The first derivative test for local extrema concerns a critical point c of a
continuous function f , where f is differentiable at all x in some open interval
containing c except possibly at c
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If f ′ changes from negative to positive, f (c)
is a local minimum
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4
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For an interval I, the graph of
f over I is concave up (or convex) if f ′′ > 0 on I
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A point of inflection is where f ′′ changes sign
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Take note that the opposite need not be true
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The second derivative can sometimes be used to determine the nature of local
extrema
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Then, if f ′′ < 0, f (c) is a local maximum
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The test in inconclusive if f ′′ = 0 and the first derivative test is needed
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Combining this with symmetries, intercepts and asymptotes, graphs can be accurately
sketched
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5
Applied optimization
A key application of calculus lies in finding the optimal value of a given function
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Sometimes, the quantity to be optimized may be a function of more than one
variable
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4
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Due to its rapid convergence to a solution for most situations,
most software uses it to find approximations to roots
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From there, the
x-intercept of the tangent line at that point is used as a next approximation
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The recursive formula is given by
xn+1 = xn −
f (xn )
f ′ (xn )
Note that if f ′ (xn ) = 0, the method fails
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It is also
possible for the method to converge, but to the wrong root, which can happen
if the initial guess is too far from the value of the desired root
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Antiderivatives
A function F (x) is said to be an antiderivative of f (x) on an interval I if
F ′ (x) = f (x) for all x ∈ I
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Thus, the family of all antiderivatives of f take the form F (x) + C, where C ∈ R
and F is any antiderivative
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The relation with integral calculus is
discussed in §5
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For
d
tan x = sec2 x, we have
instance, since dx
Z
sec2 x dx = tan x + C
Finding an antiderivative of f is akin to solving for y in the differential equation
dy
= f (x)
dx
Thus, y = F (x) + C is the general solution to the above differential equation
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5
5
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The upper sum is found by taking the height of the rectangles to
be the uppermost value of f (x) over the subinterval
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The actual value is somewhere
between the lower and upper sums
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+ f (xn )∆x
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It may
not be obvious if this is an under- or over-approximation
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2
Sigma notation and limits of finite sums
Sigma notation is a compact way of writing sums
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+ an =
n
X
ak
k=1
The index of summation k tells where the sum starts and ends
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Finite sums obey algebraic rules such as the sum rule, the difference rule and
the constant multiple rule
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3
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Consider a bounded function f over [a, b]
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, xn },
where x0 = a, xn = b and x0 < x1 <
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Let ∆xk = xk − xk−1 be the width of the kth subinterval
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We can then construct the sum
SP =
n
X
f (ck )∆xk
k=1
The sum SP is called a Riemann sum for f on [a, b]
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For each partition P , we define
its norm, ||P ||, as the largest subinterval width
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3
Definite integrals
For some functions, in considering SP as ||P || approaches 0, the corresponding
Riemann sum approaches a limiting value J, regardless of the choice of P and
ck
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We write
Z
b
f (x) dx = lim
J=
||P ||→0
a
n
X
f (ck )∆xk
k=1
The integral notation above is suggestive of the summation, with the differential
dx in place of ∆x and the continuous function f (x) in place of f (ck )
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Sufficiently discontinuous functions cannot
be approximated by increasingly thin rectangles and thus are not integrable
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1, we discussed areas
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4
b
Z
1
av(f ) =
b−a
f (x) dx
a
The fundamental theorem of calculus
The fundamental theorem of calculus links the definite integral to the derivative
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The mean value theorem for definite integrals says that for a function f
continuous on [a, b], there exists some c ∈ [a, b] for which
f (c) =
1
b−a
Z
b
f (x) dx
a
The first
R xpart of the fundamental theorem of calculus involves a function
F (x) = a f (t) dt which is continuous on [a, b] and differentiable on (a, b) when
f is continuous on (a, b)
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That is,
Z x
d
′
f (t) dt = f (x)
F (x) =
dx a
The second part of the fundamental theorem of calculus (evaluation
theorem) gives that
Z
b
f (x) dx = F (b) − F (a)
a
Thus, to evaluate an integral, we need only find an antiderivative and evaluate
it at the endpoints of the interval
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The two parts of the fundamental theorem essentially show that differentiation
and integration are “inverse” operations
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The
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integral over the negative areas will yield a negative number, so we need to sum
the absolute values
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5
Indefinite integrals and the substitution method
We previously defined the indefinite integral of f to be the set of all antiderivatives of f
...
d
f (g(x)) = f ′ (g(x))g ′ (x), we get that
From the chain rule, dx
Z
f (g(x))g ′ (x) dx = F (g(x)) + C
Letting u = g(x), this can be written as
Z
Z
du
f (u)
dx = f (u) du = F (u) + C
dx
The method relies on finding the right substitution, which may be hard
...
Alternatively, we can try to let u be the most complicated portion of the integrand
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6
Definite integral substitutions and the area between
curves
In performing the substitution method on a definite integral, we could find an
antiderivative as in §5
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Alternatively, we
have that
Z
b
f (g(x))g ′ (x) dx =
Z
g(b)
f (u) du
a
g(a)
We can also exploit symmetry to evaluate integrals on a symmetric interval
[−a, a]
...
Sometimes, regions may be easier to define using functions of y, in which case
we can integrate with respect to y in the above
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1
Applications of definite integration
Volumes using disks and washers
Consider a solid S in space
...
Then, we define the volume of S to
be
b
Z
A(x) dx
V =
a
To find volumes, we first sketch the solid and its typical cross-section
...
Lastly, find the limits of integration and the
integrate to find the volume
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A solid of revolution is the solid generated by rotating a planar region about
an axis in the plane
...
R(x)
is the radius of the disk, the distance from the axis to the region’s boundary
...
In which case, we have a washer rather
than a disk as the cross-section
...
If the axis is parallel to the y-axis, we can use the formulae above, replacing x
with y
...
2
Volumes using shells
Using the disk or washer methods can be inconvenient
...
Then, integrating over
the appropriate interval, we obtain the volume of the solid
...
For revolution about a vertical line, we obtain the formula
Z
V =
b
2π(shell radius)(shell height) dx
a
21
To find the variable of integration, draw a line segment parallel to the axis
...
Again, for revolution
about a horizontal line, replace x with y
...
3
Arc length
A curve y = f (x) is smooth if f has a continuous derivative at every point of
[a, b]
...
This leads to a Riemann sum
...
Then,
we may be able to integrate with respect to y instead
...
Define
the arc length function s to be the arc length from an initial point (a, f (a))
to an arbitrary point (x, f (x))
...
4
R
ds
...
Since a piece of a smooth curve can be approximated by a line
segment, we can approximate surface area as a sum of areas of frustums, given
by A = π(y1 + y2 )L, with y1 and y2 being the radii of both sides of the frustum
and L being the slant height
...
22
6
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For a nonconstant force, since F is relatively constant over a small distance, we
again form a Riemann sum
...
For a fluid standing still, the pressure p at depth h is given by p = wh, where
w is the weight-density of the fluid
...
However, for a vertical plate, integration is required to find the fluid force
...
Then, we have
Z
b
w · (strip depth) · L(y) dy
F =
a
6
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We can sum individual moments
...
By dividing the plate into strips, the moments about the
x− and y−axes, Mx and My are given by
Z
Z
Mx = y˜ dm, My = x
˜ dm
where dm = δdA, with dA being the area of a strip and (˜
Rx, y˜) being the centre
of mass of the strip
...
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The centroid of an object is its center of mass assuming constant density
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If we know the centroid of a vertical plate, we have a convenient formula for the
fluid force
¯
F = whA
¯ is the distance from the surface of the fluid to the centroid of the plate
where h
and A is the area of the plate
...
Pappus’ theorem for surface areas says that for a smooth arc of length
L revolved around a line that does not cut it, the surface area of the solid of
revolution is then
S = 2πρL
24
7
7
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The horizontal line test states that a function is one-to-one if
and only if no horizontal line intersects it graph more than once
...
If f is one-to-one on D, the inverse function f −1 , defined by f −1 (b) = a if
f (a) = b, exists
...
Note that f −1 ̸= 1/f
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Hence, to obtain a formula for f −1 (x), start from y = f (x)
and solve for x = f −1 (y)
...
Since the graphs are reflections, we would expect the slope at a point on the
graph y = f −1 (x) to be the reciprocal of the slope at the corresponding point
of y = f (x)
...
2
1
f ′ (f −1 (a))
Natural logarithms
We can now define the natural logarithm function, ln x(x > 0), by the following integral:
Z x
1
dt
ln x =
1 t
From this definition and the mean value theorem, we obtain many of the familiar
logarithm properties, such as ln 1 = 0, ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b
and ln(ar ) = r ln a
...
We define Euler’s number e to be the unique number so that:
Z e
1
dt = 1
ln e =
1 t
By the fundamental theorem of calculus,
d
1
ln x =
dx
x
We also have the integral formula:
25
Z
dx
= ln |x| + C
x
With this formula, we can find the integrals of the tangent, secant, cotangent
and cosecant functions
...
By first taking the
natural logarithm of both sides, we can transform products and quotients into
sums and differences
...
Lastly, we multiply through by y
...
7
...
As in, ex = exp x = ln−1 x
...
That is,
Z
d x
e = ex and
ex dx = ex + C
dx
From the definition of ex and the logarithm laws, we can derive the familiar
exponent laws for base e
...
It is related to natural logarithms by the equation loga x = ln x/ ln a
...
We can write e as a limit in order to compute its numerical value
...
26
7
...
This yields the initial value problem:
dy
= ky, y(0) = y0
dt
The solution to this is y = y0 ekt , and the quantity is said to undergo exponential change
...
If k < 0,
it undergoes exponential decay
...
The above differential equation is an example of a separable differential equation, which takes the form
dy
= g(x)h(y)
dx
We can write this in differential form and integrate to get the implicit solution
Z
Z
dy
dy = g(x) dx
h(y)
Exponential functions are good models for many real situations
...
Radioactive decay follows exponential decay
...
It is given
by
ln 2
k
Heat transfer also obeys exponential decay
...
Half-life =
7
...
Examples of indeterminate forms include 0/0, ∞/∞, 0·
∞, ∞ − ∞, ∞0 and 1∞
...
If neither form holds, we may be able to rearrange the expression
to put it in these forms
...
7
...
However, we can restrict them to an interval to allow the existence of the inverse
...
g
...
The arcsine of x, y = arcsin x, is defined as the number y ∈ (−π/2, π/2) so
that sin y = x
...
Some arcsine and arccosine identities are useful, including
arccos(−x) = π − arccos x and arcsin x + arccos x = π/2
We can apply the formula from §7
...
Using the derivative formulae, we get the useful integrals
Z
x
dx
√
+C
= arcsin
a
a2 − x2
Z
x
dx
1
=
arctan
+C
a2 + x2
a
a
Z
x
dx
1
√
= arcsec
+C
a
a
x x2 − a 2
28
7
...
For example, we
have cosh2 x − sinh2 x = 1
...
We get
d
d
sinh x = cosh x and
cosh x = sinh x
dx
dx
The other derivatives are found through the quotient rule
...
6)
...
For example, arcosh x is the number in [0, ∞) whose hyperbolic cosine
is x
...
For example, arsech x = arcosh(1/x)
...
1
...
The above formulae give useful integral formulae, like in §7
...
29
7
...
Exponential and logarithmic functions are important because of how
quickly and slowly they grow respectively
...
We say that f grows faster than g if
limx→∞ f (x)/g(x) = ∞ or equivalently limx→∞ g(x)/f (x) = 0
...
We say that f grows at the same rate as g
if limx→∞ f (x)/g(x) = L > 0
...
Vice versa, the logarithmic function with base
a > 1 grows slower than all xn , n > 0
...
We write f = o(g)
...
We write f = O(g)
...
A sequential search
algorithm, where a list is searched through in order, is O(n)
...
30
8
8
...
There is no fixed
method to find an indefinite integral
...
5) and trigonometric identities,
with the antiderivatives of our basic functions and the formulae in §7, to find
indefinite integrals
...
8
...
3, we can derive the integration by parts formula
Z
Z
′
u(x)v (x) dx = u(x)v(x) − v(x)u′ (x) dx
For definite integrals, we can put in the appropriate bounds in the formula above
...
If the
resulting integral is still not something we can evaluate, we can perform integration by parts again
...
In this case, we can solve for the integral
using algebraic manipulation
...
In some situations, we can take v ′ = 1 if the integrand is
only a single function
...
This is called a reduction formula
...
3
Trigonometric integrals
We study some special integrals involving trigonometric functions
...
If m is odd, we save a copy
of sin x, converting the rest to cos x using sin2 x = 1 − cos2 x
...
An analogous method is used for odd n
...
31
R√
Next, we consider cases involving square roots, such as
1 + cos 4x dx
...
In this case, we can use
cos 4x = 2 cos2 2x − 1
...
Similar to cosine and sines,
we can use the identities sec2 x = tan2 x + 1, (sec x)′ = sec x tan x and (tan x)′ =
sec2 x
...
Lastly, we consider integrands of the form sin mx sin nx, sin mx cos nx and
cos mx cos nx
...
4
sin mx sin nx =
1
[cos(m − n)x − cos(m + n)x]
2
sin mx cos nx =
1
[sin(m − n)x + sin(m + n)x]
2
cos mx cos nx =
1
[cos(m − n)x + cos(m + n)x]
2
Trigonometric substitutions
We can sometimes make a substitution where
√ x becomes
√ a trigonometric
√ function of θ
...
Since we want our final answers in terms of x, we want the substitution to
be reversible
...
For x = a sec θ, we either use
θ ∈ (0, π/2) or θ ∈ (π/2, π)
...
8
...
The method of partial fractions involves
breaking down a rational function into a sum of simpler functions that can be
easily integrated
...
e
...
If not, use division and
perform the process on the remainder
...
A factor is irreducible when it has no real roots
...
If g(x) contains the irreducible quadratic factor (x2 + px + q)m , then the associated partial fractions are
m
X
i=1
Bi x + Ci
(x2 + px + q)i
The partial fraction representation of f (x)/g(x) is then the sum of all of the
above
...
Then, we can use various methods, such as comparing coefficients, differentiating and substituting in specific values of x, to solve for the
unknowns
...
6
Integral tables and computer algebra systems
An integral table lists general integration results
...
It can be assumed that the constants can take on any value
within reason (i
...
no division by zero or taking even roots/logarithms of negative numbers)
...
A computer algebra system, like Mathematica, can be used to find integrals
...
It may also give
a different representation of an answer calculated by hand
...
We can use infinite series (§10) to find a formula for the integral
...
7)
...
7
Numerical integration
Rb
We may seek a numerical approximation to a definite integral a f (x) dx if it
is hard or impossible to find an antiderivative of f
...
2)
is an example of an approximation
...
For the sake of convenience, we assume n equally-spaced subintervals with
∆x = (b − a)/n
...
The trapezoidal rule involves using areas under trapezoids in each subinterval
...
+ 2yn−1 + yn )
2
Simpson’s rule involves using areas under parabolas
...
The formula for each
parabolic region’s area is ∆x(yk−1 + 4yk + yk+1 )/3, so the approximation is
given by
T =
∆x
(y0 + 4y1 + 2y2 +
...
The error is the difference between the approximated and actual value of the
integral
...
Then, M is an upper
bound of max |f ′′ | on [a, b]
...
Then, M is an upper bound of max |f (4) | on [a, b]
...
|ET | ≤
Notice that the error for the trapezoidal rule varies with the inverse square of
n, whereas the Simpson’s rule varies with the inverse fourth power of n
...
Also, the error formulae show
that Simpson’s rule gives the exact value of the integral for constant, linear,
quadratic and cubic functions
...
8
...
34
A type I improper integral has infinite limits of integration
...
Define
∞
Z
b
Z
f (x) dx = lim
b→∞
a
Z
b
Z
∞
Z
c
f (x) dx
∞
−∞
a
∞
Z
f (x) dx +
f (x) dx =
f (x) dx
a→−∞
∞
a
b
Z
f (x) dx = lim
f (x) dx and
c
A type II improper integral has a vertical asymptote in the interval of integration
...
Define for a discontinuity at a and b
Z
a
b
Z
f (x) dx = lim+
c→a
b
Z
f (x) dx and
a
c
b
Z
f (x) dx = lim−
c→b
c
f (x) dx
a
respectively
...
Else, the integral diverges
...
Some improper integrals are nonelementary, so we need methods to determine
if they converge
...
Similarly,
R∞
R∞
f (x) dx diverging implies that a g(x) dx also diverges
...
Note that the above tests are stated for type I improper integrals, but they also
hold for type II improper integrals
...
9
Probability
For events involving more than one outcome, a probability is assigned to the
likelihood of an outcome occurring
...
If there are finitely many outcomes, X is discrete
...
For continuous random variables, we define their probability distribution
function f to have the properties that f is defined
R ∞ on R, has finitely many
discontinuities, is nonnegative and is such that −∞ f (x) dx = 1
...
The expected
value (or mean) of X is given by
Z ∞
xf (x) dx
µ = E(X) =
−∞
The variance of X is the expected value of (X −µ)2
...
In symbols,
Z ∞
p
Var(X) =
(x − µ)2 f (x) dx and σX = Var(X)
−∞
Certain phenomenon can be modelled by exponentially decreasing probability
distribution functions
...
In this case,
(
0 if x < a or x > b
f (x) =
1
b−a if a ≤ x ≤ b
Many situations involve the normal distribution
...
Then,
f (x) =
2
2
1
√ e−(x−µ) /(2σ )
σ 2π
36
9
9
...
A solution of the equation is a function y(x) that satisfies y ′ (x) =
f (x, y(x))
...
It always contains an arbitrary constant
...
If the condition is of the form
y(x0 ) = y0 , the equation is called an initial value problem
...
To draw one, draw a line segment at selected points (x, y)
with slope f (x, y)
...
It helps us visualise solution curves without having to solve the
equation
...
Euler’s method is a numerical method of solving first-order differential equations
...
We
consider the initial problem dy/dx = f (x, y), y(x0 ) = y0
...
Hence, a recursive formula
for the approximation of y is
yi = yi−1 + f (xi−1 , yi−1 ) dx
If the step size is small enough, Euler’s method yields a good approximation to
the actual value of y
...
2
First-order linear equations
A first-order linear differential equation can be written in the form
dy
+ P (x)y = Q(x)
dx
This is called its standard form
...
37
The integration factor v(x) turns out to be e
be summarised as follows:
R
P (x) dx
...
Then, find e P (x) dx
...
Lastly, divide out to find y
...
3
Applications
One application of linear equations is in modelling current in an RL circuit
...
In physics, if a family of curves dy/dx = f (x, y) represents a potential, then its orthogonal trajectories dy/dx = −1/f (x, y) represent equipotential
lines
...
Let y be the amount of a chemical in a container of liquid (with volume V )
...
We assume that the mixture
is kept uniform so that the concentration is the same throughout
...
4
Graphical solutions of autonomous equations
A differential equation is autonomous if dy/dx = g(y)
...
If we wish to find a graphical solution (a representative set of solution curves),
we can use a phase line
...
Then, find the sign of y ′ on each interval
...
This gives us the shape of the graph without
solving the differential equation
...
The values
of y appear to go away from c, so we call c an unstable equilibrium
...
Many differential equations of use are autonomous
...
The population model in §7
...
The logistic growth model takes this into account
...
Note that 0 and M are equilibrium values, and
are unstable and stable respectively
...
9
...
A general system of two such equations is given by:
dy
dx
= F (x, y) and
= G(x, y)
dt
dt
It is said to be autonomous as the right hand sides do not involve the dependent variable t
...
A solution curve through a specified point is called a
trajectory of the system
...
Phase plane diagrams give us a way of graphically representing solutions to
such a system
...
The solution points are
called equilibrium points
...
By looking at the directions of the trajectories near the equilibrium points, we
can check for stability
...
If
there is at least one trajectory that goes away from the point, it is unstable
...
Notably, with only direction, we do
not know the actual shape of the trajectory
...
This is called a limit cycle
...
1
Infinite sequences and series
Sequences
A sequence {an } is a list of numbers a1 , a2 ,
...
Each of
the ai are known as terms of the sequence and i is the index
...
Sequences can be understood as functions mapping positive integers i to ai
...
We can define convergence for sequences analogously to limits for functions (§2)
...
We can define divergence to infinity and negative infinity analogously to §2
...
We now discuss how to find limits of sequences
...
1) have a corresponding
counterpart for sequences
...
The continuous function theorem for sequences tells us that if an → L
and if f is a function continuous at L and defined at all an , then f (an ) → f (L)
...
A consequence is
that we can use L’Hˆ
opital’s rule (§7
...
Sequences can be defined recursively instead of explicitly
...
The Fibonacci numbers are a wellknown example
...
A sequence {an } is bounded above if there exists M such that an ≤ M for
all n
...
If there are no smaller upper bounds than M , it
is a least upper bound
...
If a sequence has both a
lower and upper bound, it is bounded
...
A sequence is nonincreasing if an ≥ an+1 for all n
...
A sequence that is either nonincreasing or nondecreasing is called
40
monotonic
...
Specifically, if it is bounded above and nondecreasing, it converges to the least upper bound and if it is bounded below and
nonincreasing, it converges to the greatest lower bound
...
2
Infinite series
Given
Pn an infinite sequence {an }, its nth partial sum sn is given by sn =
i=1 ai
...
If it exists, S converges
...
Sometimes, we just write
P
an if the bounds are understood or unimportant
...
It takes the form a + ar +
...
=
P∞
n−1
...
If |r| ≥ 1, then the
n=1 ar
series diverges
...
Partial fractions can be useful in evaluating
telescoping series
...
Note that even if limn→∞ an = 0,
the series need not be convergent
...
Adding or deleting any number of finite terms from a convergent series does not
change convergence behaviour, though the value of the series usually changes
...
3
∞
X
an−h =
n=1+h
∞
X
an+h
n=1−h
The integral test
P∞
As a result of the monotone sequence theorem, n=1 an of nonnegative terms
converges if and only if its partial sums are bounded above
...
This
leads to the integral test
...
P∞
The p-series is a series of the form n=1 1/np
...
Using the integral test, we can show that the series converges
for p > 1 and diverges for p ≤ 1
...
In many situations, finding the
value of S is impossible and we have to approximate using sn
...
Under the same conditions as
the integral test, we have the following bounds:
Z ∞
Z ∞
f (x) dx
f (x) dx ≤ Rn ≤
n+1
n
10
...
Comparison tests help us establish the convergence behaviour of
these series
...
Then,
a
converges
if
b
converges
...
However, the test does not give any conclusions if a series is larger than a convergent series or if it is smaller than a divergent series
...
Suppose
an , bP
n > 0 for all n ≥ N and let
P
limn→∞ an /bn = c
...
If c = ∞
P diverge
...
10
...
It can be shown
that absolute convergence implies convergence
...
The ratio test uses the ratio of terms to measure how quickly a series grows
...
Then, the series absolutely converges if ρ < 1 and
42
diverges if ρ > 1
...
The test is useful for expressions containing powers involving n and factorials
...
Suppose
P
an is any series and suppose that limn→∞ n |an | = ρ
...
The test is inconclusive if
ρ = 1
...
6
Alternating series and conditional convergence
A series whose terms alternate between positive and negative is called an alternating series
...
The alternating series test
P∞gives the conditions needed for an alternating
series to converge
...
converges
if each un > 0, the un terms are eventually nonincreasing (i
...
un ≥ un+1 ≥
...
The nonincreasing criterion can be hard to establish directly
...
P
If the alternating series (−1)n+1 un converges to L, then L lies between sn
and sn+1 , so |L − sn | < un+1
...
A series which is convergent but not absolutely convergent is conditionally
convergent
...
In fact, we can
rearrange a conditionally convergent series to make it equal any real number r
...
7
Power series
A power series looks like an infinite
P∞ polynomial
...
A power series defines a function where it converges and its partial
sums are polynomial approximations to the function
...
5)
...
It is possible for there to be a positive number R such
that the series diverges for |x − a| > R and converges absolutely for |x − a| < R
...
The constant R is called the radius of convergence
and the interval from a − R to a + R, possibly containing the endpoints, is the
43
interval of convergence
...
1-§10
...
In the intersection of the intervals of convergence of two power series, we can add
and subtract the power series term-by-term
...
We usually find coefficients in a product by writing out the first few terms of
each series and then multiplying them out
...
Note
that this may not hold for series which are not power series
...
8
Taylor and Maclaurin series
One way of generating a power series for a function of the form 1/[1 − f (x)] is
by using the geometric series formula (§10
...
However, we need a more general
way to represent functions with a power series
...
This yields the formula for the coefficients, an = f (n) (a)/n!
...
The special case for a = 0
is called the Maclaurin series generated by f
...
However, we do not know if the Taylor series converges to f (x)
on some interval containing a
...
9
...
The nth partial sum is called the Taylor polynomial of order n
...
9 is the first-order Taylor polynomial
...
10
...
2)
...
The Rn is called the remainder term
...
We can often estimate the error term Rn
...
Then, the remainder term satisfies
|Rn (x)| ≤ M
|x − a|n+1
(n + 1)!
If all the other conditions of Taylor’s theorem is met and this inequality holds
for all n, then the series converges to f (x)
...
10
Applications of Taylor series
The Taylor series generated by f (x) = (1 + x)m is called a binomial series
...
On this interval,
(1 + x)
m
=1+
∞
X
m
k=1
where the binomial coefficient
m
k
k
xk
= m(m − 1)(m − 2)
...
R
We can find nonelementary integrals using power series
...
Taylor series can also be used to approximate the numerical value of constants
...
This approximation is called Leibniz’s formula:
∞
X
(−1)n x2n+1
π
=
4
2n + 1
n=0
Limits in indeterminate form (§7
...
By substituting x = iθ, where i =
ex , we get Euler’s identity
√
−1 and θ is real into the Taylor series for
eiθ = cos θ + i sin θ
This leads us to a definition of a complex exponent to be consistent with the
above identity
...
1
Parametric equations and polar coordinates
Parametrizations of plane curves
The path of a moving particle in the plane can be represented by a pair of
parametric equations x = f (t) and y = g(t) over some interval I
...
t is the parameter and typically
represents time
...
The equations constitute a parametrization of
the curve
...
The natural
parametrization x = t and y = f (t) gives a way to parametrize such functions
...
To sketch the graph of a parametric curve, we can sometimes eliminate the parameter to obtain an equation in terms of x and y only
...
A cycloid is a parametric curve that is traced out by a point on the circumference of a rolling circle of radius a
...
An inverted cycloid is a brachistochrone for the origin and its minimum point, a curve connecting two points
that minimizes the time taken for a frictionless bead to travel along it
...
It is also a tautochrone, a
curve along which a frictionless bead takes the same amount of time to reach
the lowest point irregarding where it started
...
2
Calculus with parametric curves
A parametric formula for the derivative for a pair of parametric equations is
given by
dy/dt
dy
=
dx
dx/dt
provided that dx/dt ̸= 0
...
By repeating this process, we can develop a formula for the
second derivative and beyond
...
If it is a definite integral, the bounds have to be adjusted accordingly
...
2
...
46
Further assuming f ′ and g ′ to be continuous, and that the curve is traversed
exactly once from a to b, the arc length is
s
Z b 2 2
dy
dx
+
dt
L=
dt
dt
a
Note that the natural parametrization shows that the p
formula in §6
...
The arc length differential ds = dx2 + dy 2 also holds for
parametric curves
...
3
Polar coordinates
A polar coordinate system is an alternative to the Cartesian plane
...
Then, for each point P
in the plane, we assign a polar coordinate pair (r, θ), where r is the directed
distance from the origin and θ is the angle measured counterclockwise from the
initial ray
...
If (r, θ) represents
P , then so do (r, θ + 2nπ) and (−r, θ + (2n + 1)π), where n ∈ Z
...
However, one of these forms may be simpler to work with
...
4
Graphing polar coordinate equations
Symmetries can be useful in graph sketching
...
Then, if (r, −θ) is also on the graph, the graph is symmetric about the x−axis
...
If
(−r, θ) is also on the graph, the graph is symmetric about the origin
...
This tells us the shape of the graph near the origin
...
47
11
...
The area of
the region is then given by
Z
β
A=
α
1 2
r dθ
2
Note that for this formula to hold, we assume that the region does not sweep
an angle of more than 2π and that r ≥ 0 on the interval
...
If the region is bounded by θ = α, θ = β, r = r1 and r = r2 , with r1 ≥ r2 , the
area is
Z
β
A=
α
1 2
r1 − r22 dθ
2
Determining the points of intersection of polar curves may be difficult due to
the non-uniqueness of polar coordinates, so a sketch may be helpful
...
6
Conic sections
A conic section is a curve formed by the intersection of a double cone and a
plane
...
The translations in §1
...
A parabola is a set of points equidistant from a point, called the focus, and
a line, called the directrix
...
The
vertex for the above parabola is the origin
...
The line through the foci is called the focal axis
...
The major axis is the line segment of length 2a connecting the vertices (±a, 0)
...
a and b are called the semimajor and semiminor axes
respectively
...
A hyperbola is a set of points whose distances from two points, the foci, form a
constant difference
...
√
y2
x2
−
=1
a2
b2
c2 − a2
...
Cutting the double cone with a plane can also lead to a point, a line or a pair
of lines
...
11
...
The eccentricity e = c/a for a conic is a measure of how “squashed” it is
...
6 by
√
√
a2 − b2
a2 + b2
and e =
e=
a
a
respectively
...
Although we only defined the directrix for a parabola, we define two directrices
for the ellipses and hyperbolas above to be x = ±a/e
...
This leads to a general polar equation which represents a general
conic with a focus at the origin and a directrix at x = k :
ke
1 + e cos θ
For other orientations, the plus symbol can be replaced with a minus symbol
and the cosine and be replaced with the sine
...
This reduces the formula to r = a(1 − e2 )/(1 + e cos θ)
Title: Calculus Notes
Description: 50 pages worth of calculus notes that will help you in class.
Description: 50 pages worth of calculus notes that will help you in class.