Math Prelim part 2

8/17/2019 Math Prelim part 2

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Mathematical Methods:

Calculus I

(Sets, Functions, Diff erentiation, Integration)

Godfrey Keller∗

October 2012(minor revisions: 15-Apr-13, 27-Apr-15)

∗[email protected]


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The Greek alphabet

α A alpha

β B beta

γ Γ gamma

δ ∆ delta

ε E epsilon

ζ Z zeta

η H eta

θ Θ theta

ι I iota

κ K kappa

λ Λ lambda

µ M mu

ν N nu

ξ Ξ xi

o O omicron

π Π pi

ρ P rho

σ Σ sigma

τ T tau

υ Υ upsilon

φ Φ phi

χ X chi

ψ Ψ psi

ω Ω omega

Some mathematical symbols

⇒ implies 6⇒ does not imply ⇐⇒ if and only if ∈ is an element of 6∈ is not an element of ∀ for all≈ is approximately equal to ≡ is identically equal to ∃ there exists⊂ subset ∪ union ∩ intersection∅ empty set


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1 Sets and Sequences

1.1 Sets

A set S is a collection of elements .

x ∈ S means: x

is a member of

is an element of

belongs to

the set

S .

T is a subset of S if every element of T is also in S . In mathematical notation:

T ⊂ S if x ∈ T ⇒ x ∈ S

Sometimes sets are defined simply by listing their elements: S = {x1, x2, . . . , xn, . . .}.

Some sets of numbers

• Natural numbers (positive integers): N = {1, 2, 3, . . .}.

• Integers: Z = {. . . , −2, −1, 0, 1, 2, . . .}.

• Real numbers: R – all the numbers on the real line.

• Non-negative real numbers: R+ – all the real numbers greater than or equal to 0.In mathematical notation: R+ = {x ∈ R : x ≥ 0}.

• Strictly positive real numbers: R++ = {x ∈ R : x > 0}.

Sets can have a finite or infinite number of elements. When the elements can be listed,

such as N and Z above, the set is said to be countable ; otherwise, such as R and R+, it

is uncountable .

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Intervals

A subset of the real line containing all the numbers between a and b is called an interval .

Some finite intervals

[a, b] = {x ∈ R : a ≤ x ≤ b}(a, b) = {x ∈ R : a < x < b} – sometimes written ]a, b[[a, b) = {x ∈ R : a ≤ x < b} – sometimes written [a, b[

Some infinite intervals

[a, ∞) = {x ∈ R : x ≥ a} and (−∞, b) = {x ∈ R : x < b}Also (−∞, +∞) = R and [0, ∞) = R+

Set Operations

For two sets S and T :

◦ The union of S and T , written S ∪ T , is the set of elements that are in S or in T or in both:

S ∪ T = {x : x ∈ S or x ∈ T }

◦ The intersection of S and T , written S ∩ T , is the set of elements that are in both:

S ∩ T = {x : x ∈ S and x ∈ T }

◦ If T ⊂ S , then the complement of T in S , written T c, is the set of elements that arein S but not in T :

T

c

= {x ∈ S : x 6∈ T } – other notation: T c

= T

0

= S \T = S − T

◦ The empty set is the set with no elements, ∅ = {}.

◦ S and T are disjoint if S ∩ T = ∅

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Vectors

Rn is the set of vectors x = (x1, x2, x3, . . . , xn), where xk

∈R.

The length or norm of a vector is: ||x|| =q

x21 + x22 + . . . + x

2n

Triangle Inequality: For any x, y ∈ Rn: ||x + y|| ≤ ||x|| + ||y||The distance between two points x, y ∈ R is |x − y|The (Euclidean) distance between two points in Rn is:

||x − y|| =q

(x1 − y1)2 + (x2 − y2)2 + . . . + (xn − yn)2

From the Triangle Inequality: ||x−

z||≤

||x−

y|| + ||y−

z||

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1.2 Sequences

A sequence in Rn is an ordered list of elements of Rn:

x1, x2, . . . , xk, . . . or equivalently {xk}∞

k=1

(It is a countable subset of Rn, in which the order matters.)

Some examples in R

• xk = k • xk = 1/k • xk = (−1)k

• Geometric sequence: a,ar,ar2, . . . , a rk−1, . . .

A sequence {xk} in R is said to converge to a limit x if for any ε > 0, there is an integer K such that |xk − x| ≤ ε for all k > K .

Similarly, a sequence {xk} in Rn converges to x if

for any ε > 0, ∃ K such that ||xk − x|| ≤ ε ∀ k > K .Examples

• xk = k does not converge • xk = 1/k converges to zero

• xk = 2 − k2

3k2

+ k

• xk = ark−1: convergence depends on r.

A subsequence of a sequence {xk} is an infinite subset of the elements, in the same order.

• xk = (−1)k does not converge, but has convergent subsequences.

• xk = k has no convergent subsequences.

Series

The sum of the terms of a sequence in R is called a series : x1 + x2 + x3 + . . . =∞

Xk=1

xk.

• Geometric series: If S n is the sum of the first n terms of a geometric sequence:

S n =nX

k=1

ark−1 = a(1 − rn)

1 − r

then {S n}∞

n=1 is also a sequence. If |r| < 1, {S n} converges, so the series converges:

limn→∞

S n = a

1 − r and we write∞X

k=1

ark−1 = a

1 − r

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1.3 Open, Closed, Bounded, Compact and Convex Sets

An open ball Br(x) in Rn is a set defined by

Br(x) = {y ∈ Rn : ||x − y|| < r}

A set S in Rn is open if for all x ∈ S , there is some r > 0 such that Br(x) ⊂ S .A set is closed if and only if its complement is open. (A closed set contains its boundary.)

• The interval (1, ∞) is an open set in R, so (−∞, 1] is closed.Theorem: A set S in Rn is closed if and only if, for any sequence xk such that xk ∈ S for all k and xk

→x, x

∈S .

x is an interior point of S if there is some r > 0 such that Br(x) ⊂ S .A set S in Rn is bounded if there is some r > 0 such that S ⊂ Br(0).A subset of Rn is compact if and only if it is closed and bounded.

Theorem: (Bolzano-Weierstrass) If S is a compact subset of Rn, then every sequence of

points in S has a subsequence that converges to a point in S .

A set S in Rn is convex if, ∀ x, y ∈ S and λ ∈ [0, 1], λx + (1 − λ)y ∈ S .

Examples in R

• The interval [a, b] is closed; (a, b) is open; (a, b] is neither.

All these intervals are bounded, and convex.

• The interval [a, ∞) is closed and convex, but not bounded.

• The set [0, 1] ∪ [2, 3] is compact but not convex.

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1.4 Supremum, Infimum, Maximum and Minimum

Let S be a subset of R.

• If there is a real number b such that x ≤ b ∀ x ∈ S , then b is an upper bound for S ,and the set is bounded above .

• If S is bounded above, then there is an upper bound b∗ such that if b is an upper

bound for S , b∗ ≤ b. It is called the supremum , or least upper bound , b∗ = sup(S ).

• If S is not bounded above, sup(S ) = ∞.

Similarly, S may be bounded below ; inf(S ) is the infimum or greatest lower bound .

• If S = [a, b), S is bounded above and below; inf(S ) = a and sup(S ) = b.

Note that a ∈ S , and min(S ) = inf(S ), but b 6∈ S and max(S ) does not exist.

• S = [a, ∞) is bounded below but not above; sup(S ) = ∞.

• If S = {1/n : n ∈ N}, inf(S ) = 0 but min(S ) does not exist.

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2 Functions of one variable

Let S and T be subsets of R.

A function f from S to T is a rule that associates with each element of S , one and only

one element of T . We write:

f : S → T S is the domain of the function; T is the range of the function.

If x ∈ S , the corresponding element of T is written f (x).Examples

• f (x) = 3x + 2 f : R → R• g(x) = x2 g : R → R+

• h(x) = 1 +√

x h : R+ → R+

When n ∈ N, the function p : R → R given by p(x) = anxn + an−1xn−1 + . . . + a1x + a0is a polynomial of degree n.

In the examples above, f and g are polynomials (of degree one and two, respectively).

A function is one-to-one if for any y ∈ T , there is at most one x ∈ S such that f (x) = y.A function is onto if for any y ∈ T , there is at least one x ∈ S such that f (x) = y.In the examples above, f and h are one-to-one, g is not; f and g are onto, h is not.

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Inverse functions

If f : S

→T is one-to-one and onto, then it has an inverse f −1 : T

→S , for which:

f −1(f (x)) = x

Example: If f (x) = 3x + 2, then f −1(y) = (y − 2)/3

Monotonicity

A function f : S → T is

increasingstrictly increasing if for any x, y ∈ S with y > x, f (y)

≥f (x)

f (y) > f (x)

A decreasing function is defined similarly, and monotonic means increasing or decreasing.

In the examples above, f and h are monotonic, g is not. (More specifically, f and h are

strictly increasing.)

Any strictly monotonic function has an inverse.

Composite functions

If f : S → T and g : T → U , we can define a function h : S → U by h(x) = g(f (x)).h is the composition of f and g , sometimes written g ◦ f .Example: If f (x) = x + 3 and g(y) = y2, then:

(g ◦ f )(x) = g(f (x)) = g(x + 3) = (x + 3)2(f ◦ g)(x) = f (g(x)) = f (x2) = x2 + 3

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2.1 Limits of functions; continuity

Let f : S

→R, where S

⊂R, and x0

∈S . Then f has a limit, y0, at x0 if and only if for

any ε > 0, ∃ δ > 0 such that |x − x0| < δ ⇒ |f (x) − y0| < ε. We write:

limx→x0

f (x) = y0

Some functions have infinite limits at finite values of x, and/or finite limits as x goes to

infinity. For example, if k(x) = 3 + 1/x2, then

limx→0

k(x) = +∞, limx→∞

k(x) = 3

Note that, strictly speaking, this function is not defined at x = 0.

Sometimes the limit of a function at a point depends on whether you approach the point

from below or from above. For example, if f (x) = 1/x, then

f (0+) = limx→0+

x−1 = +∞, f (0−) = limx→0−

x−1 = −∞

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Continuity

The function f (x) is continuous at x0 if and only if limx→x0

f (x) = f (x0).

If a function is continuous at every point in its domain, it is a continuous function .

Example: The function: f (x) =

x if x


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2.2 Convex and concave functions

A continuous function f : R

→R is convex if and only if, for all a, b

∈R:

f (λa + (1 − λ)b) ≤ λf (a) + (1 − λ)f (b) for all λ ∈ (0, 1)

As λ varies from 1 to 0, the left-hand side traces the curve of the function from f (a) to

f (b), while the right-hand side traces the straight line from f (a) to f (b). For a convex

function, the curve lies below the straight line.

The definition of a strictly convex function is the same, but with the inequality being

strict.

To define a concave or strictly concave function, use the inequalities ≥ or >.Alternatively, f is concave ⇐⇒ −f is convex.For a function to be convex or concave, its domain must be a convex set.

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2.3 Some important theorems

The Weierstrass Theorem: A continuous function f : S

→ R, where S is a compact

subset of R, attains a maximum and minimum value on S .

The Intermediate Value Theorem: Suppose f : [a, b] → R is continuous, and f (a) <y0 < f (b). Then ∃ x0 ∈ (a, b) such that f (x0) = y0.Brouwer’s Fixed Point Theorem: Suppose S ⊂ R is a non-empty, compact andconvex set, and f : S → S is a continuous function. Then f has a fixed point: that is∃ x0 such that f (x0) = x0.(Note that a compact and convex set in R is a closed interval [a, b]; but written as above,

Brouwer’s Fixed Point Theorem also applies when S

⊂Rn.)

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2.4 Exponential and logarithmic functions

For any base a > 1 we can define exponential and logarithmic functions.

f (x) = ax

is an exponential function ; f : R → R++, and is continuous, increasing and convex.If a > 1 and x > 0, the logarithm to base a of x, loga x, is defined by:

aloga x = x

loga x is the answer to the question: “To what power must a be raised in order to get x?”

e.g. log2 8 = 3 because 23

= 8.g(x) = loga x

is a logarithmic function ; g : R++ → R, and is continuous, increasing and concave.A logarithmic function loga x is the inverse of an exponential function a

x:

y = ax ⇐⇒ x = loga y

• The most important exponential and logarithmic functions are the ones with base

e = 2.71828 . . .

The exponential function: f (x) = ex (sometimes written f (x) = exp(x)).

The natural log function: g(x) = loge(x) (often written g(x) = ln x).

Reminders:

anam = an+m, a−n = 1/an, (ab)n = anbn, (am)n = amn, a1

n = n√

a, a0 = 1

loga(xy) = loga x + loga y, loga xn = n loga x, loga(1/x) = − loga x, loga 1 = 0

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3 Diff erentiation

All the functions in this section are assumed to have domain and range in R – they are

real-valued functions of a single variable.

3.1 Definition of a derivative

The slope of a function f at a point x is approximately

f (x + h) − f (x)h

where h is small. If this fraction has a limit as h→

0 then we say that f is diff erentiable

at x, and the value of the limit is called the derivative of f at x.

For a function to be diff erentiable at x, its graph must be continuous (no jump) and

smooth (no kink) at x. If a function is diff erentiable at all points in its domain (its graph

is a continuous smooth line) then we say that it is a diff erentiable function.

The derivative of f at x is denoted by: f 0(x) or df

dx or Df (x)

and we often write:

y = f (x) ⇒ dydx

= f 0(x)

The Mean Value Theorem: Suppose f : [a, b] → R, and f is a diff erentiable function.Then there is a point c ∈ (a, b) such that

f (b) − f (a) = f 0(c)(b − a)

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3.2 Basic rules for diff erentiating simple functions

If f and g are functions and k is a constant:

y = f (x) + g(x) ⇒ dydx

= f 0(x) + g0(x)

y = kf (x) ⇒ dydx

= kf 0(x)

You should know the derivatives of common functions:

f (x) = c ⇒ f 0(x) = 0f (x) = xn ⇒ f 0(x) = nxn−1

f (x) = ln x ⇒ f 0

(x) = 1/x = x−1

f (x) = ex ⇒ f 0(x) = ex

What about: f (x) = |x|, the absolute value of x?

When x > 0 f (x) = x, so f 0(x) = 1.

When x


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3.3 First derivatives and monotonicity

The derivative of a function f (x) is a measure of how the function changes when x changes.

If f (x) is diff erentiable at the point x0, then:

f 0(x0) ≥ 0 ⇐⇒ f is increasing at x0, f 0(x0) > 0 ⇒ f is strictly increasing at x0;f 0(x0) ≤ 0 ⇐⇒ f is decreasing at x0, f 0(x0) < 0 ⇒ f is strictly decreasing at x0.

So for example, if f 0(x) > 0 ∀ x, then f is strictly monotonic, and an increasing function.Note, however, that a strictly increasing function may have a zero first derivative at some

point. An example is x3.

3.4 Diff erentials

If y = f (x) we sometimes write

dy = f 0(x) dx

where dx is the di ff erential of x and represents an infinitesimal change in x.

This equation tells us that when x changes by an infinitesimal amount dx, the size of the

corresponding change in y , called the di ff erential of y , is given by f 0(x) dx.

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3.5 Rules for diff erentiating more complicated functions

Product Rule: d

dx

(f (x)g(x)) = f 0(x)g(x) + f (x)g0(x)

e.g. h(x) = x2ex ⇒ h0(x) = 2xex + x2ex

Quotient Rule: d

dx

f (x)

g(x)

!=

f 0(x)g(x) − f (x)g0(x)g2(x)

e.g. h(x) = x2

x

−1

⇒ h0(x) = 2x(x − 1) − x2

(x

−1)2

= x2 − 2x(x

−1)2

(The quotient rule is derived from the product rule by writing f (x)

g(x) as f (x) g(x)−1, then

applying the chain rule below.)

Chain Rule: d

dxf (g(x)) = f 0(g(x))g0(x)

Another way of expressing the chain rule is to say that if z = f (y) and y = g(x), then we

can think of z as a function of x, with derivative:

dz

dx =

dz

dy

dy

dx

e.g. (i) f (x) = (3x + 2)4 ⇒ f 0(x) = 4(3x + 2)3 × 3 = 12(3x + 2)3

(ii) g(x) = exp(x2) ⇒ g0(x) = 2x exp(x2)

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3.6 Diff erentiating an inverse function

Suppose the function f (x) has an inverse. Then:

y = f (x) ⇒ x = f −1(y)

Inverse Function Rule: dx

dy = 1

,dy

dx

Alternatively we can express this as:

(f −1)0(y) = 1

f 0(x) =

1

f 0(f −1(y))

For example, we can use the inverse function rule to work out the derivative of ax:

y = ax ⇒ ln y = x ln a⇒ x = ln y

ln a

⇒ dxdy

= 1

y ln a

Applying the rule: dy

dx = y ln a

Writing the answer in terms of x: dy

dx

= ax ln a

3.7 Implicit diff erentiation

If y is a function of x but is not expressed in the form y = f (x), we can find the

derivative by implicit diff erentiation. For example, ln x + 2 ln y = k is an indiff erence

curve. Diff erentiating w.r.t. x gives:

1

x +

2

y

dy

dx = 0,

2

y

dy

dx = −1

x,

dy

dx = − y

2x

as the slope of the indiff erence curve.

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3.8 Higher order derivatives

In most cases, the derivative of a function f (x) is itself a function, and also diff erentiable.

Just as the first derivative tells us how the value of the function changes with x, the

second derivative tells us how the slope of the function changes with x.

Notation: y = f (x), dy

dx = f 0(x),

d

dx

dy

dx

!=

d2y

dx2 = f 00(x),

d3y

dx3 = f 000(x) etc.

Examples:

f (x) = ax2 + bx + c f 0(x) = 2ax + b f 00(x) = 2a

g(x) = ex g0(x) = ex g00(x) = ex

h(x) = ln x h0(x) = x−1 h00(x) = −x−2

When a function has continuous derivatives up to and including order n, we say that itis of class C n.

Polynomials, exp, and ln are examples of functions that are C ∞.

3.9 Second derivatives and convexity, concavity

We can check whether a function f (x) is convex or concave by looking at the second

derivative:

f 00(x) ≥ 0 ∀ x ⇐⇒ f is convex, f 00(x) > 0 ∀ x ⇒ f is strictly convex;f 00(x) ≤ 0 ∀ x ⇐⇒ f is concave, f 00(x) < 0 ∀ x ⇒ f is strictly concave.

Note, however, that a strictly convex function may have a zero second derivative at some

point. An example is x4.

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3.10 Taylor’s Theorem

Notation: The k th derivative may be denoted by f (k)(x).

Taylor’s Theorem:1 Suppose f : S → R, where S is a convex subset of R, is a C k+1function. Then for any a, x ∈ S , there exists z ∈ (a, x) such that:

f (x) = f (a) + (x − a)f 0(a) + (x − a)2

2 f 00(a) + . . . +

(x − a)kk!

f (k)(a) + (x − a)k+1

(k + 1)! f (k+1)(z )

The remainder, Rk(x, a) = (x − a)k+1

(k + 1)! f (k+1)(z ), converges to zero faster than (x − a)k.

Taylor Approximations:

First-order: f (x) ≈ f (a) + (x − a)f 0(a)Second-order: f (x) ≈ f (a) + (x − a)f 0(a) + (x − a)

2

2 f 00(a)

kth-order: f (x) ≈ f (a) + (x − a)f 0(a) + (x − a)2

2 f 00(a) + . . . +

(x − a)kk!

f (k)(a)

Maclaurin’s Theorem:

f (x) = f (0) + xf 0(0) + x2

2 f 00(0) + . . . +

xk

k!f (k)(0) +

xk+1

(k + 1)!f (k+1)(z )

Taylor Series: If f is C ∞ we have an infinite series, and if it converges we can write:

f (x) = f (a) + (x − a)f 0(a) + (x − a)2

2 f 00(a) + . . . +

(x − a)kk!

f (k)(a) + . . .

When the Taylor series converges, it is the Taylor expansion of f about a.

• ex = 1 + x + x2

2! +

x3

3! + . . . +

xn

n! + . . . (valid for any x)

• ln(1 + x) = x − x22

+ x33

+ . . . + (−1)n+1 xnn

+ . . . (valid for −1 < x ≤ 1)

• If n is a positive integer, we have the (finite) Binomial expansion of (1 + x)n:

(1 + x)n = 1 + nx + n(n − 1)

2! x2 + . . . + n C kx

k + . . . + xn

For other values of n we get an infinite series, valid only for |x| < 1.

1An English mathematician at last! However, the result had been discovered about 40 years earlierby Gregory – a Scot.

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3.11 L’Hôpital’s Rule

L’Hôpital’s Rule is a method for evaluating limx→a

f (x)

g(x)

(where a can be finite or infinite),

if either both functions are zero or both functions are ± ∞ at a:

limx→a

f (x)

g(x) = lim

x→a

f 0(x)

g0(x)

If this limit cannot be evaluated, we look at second derivatives . . .

Justification: replacing f and g by their Taylor expansions about a:

f (x)

g(x) =

f (a) + (x − a)f 0(a) + 12(x − a)2f 00(a) + . . .g(a) + (x

−a)g0(a) + 1

2(x

−a)2g00(a) + . . .

• limx→0

ex − 1x

• CRRA utility function with parameter r is: u(y) = y1−r − 1

1 − r if r 6= 1.But what if r = 1?

limr→1

y1−r − 11 − r = limr→1

−y1−r ln y−1 = ln y

Using L’Hôpital’s Rule we can prove two important results. For any k > 0:

• limx→∞

xk

ln x = ∞

• limx→∞

xk

ex = 0

(Also true for other bases a > 1.)

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4 Integration

Integration is the reverse of the process of diff erentiation.

• Suppose we start with a function f (x).

• If we can find another function F (x) for which F 0(x) = f (x),

then we also know that d

dx (F (x) + c) = f (x) for any constant c,

and we can write:Z

f (x) dx = F (x) + c.

Z f (x) dx is called the integral of f with respect to x; f (x) is called the integrand .

4.1 Basic rules for integration

If f and g are functions and k is a constant:

Z (f (x) + g(x)) dx =

Z f (x) dx +

Z g(x) dx and

Z kf (x) dx = k

Z f (x) dx

The integrals of common functions:

Z xn dx = 1n+1 x

n+1 + c (∀ n ∈ R, n 6= −1)Z x−1 dx = ln x + cZ eax dx = 1a e

ax + c where a is a constant

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4.2 Definite and Indefinite Integrals

The integrals above are indefinite integrals .

• If we know that the indefinite integral of f (x) is F (x) + c,

then we also have the definite integral :Z b

af (x) dx =

hF (x)

iba

= F (b) − F (a),where a and b are constants, called the limits of integration.

a bx

f (x) So to evaluate a definite integral, integrate,

evaluate the answer at the limits, and then

take the diff erence.

Interpretation :

The definite integralZ b

af (x) dx is the area

under the graph of f (x), between x = a and

x = b.

It follows that:

Z ab

f (x) dx = −Z b

af (x) dx and

Z ba

f (x) dx =Z q

af (x) dx +

Z bq

f (x) dx

The Mean Value Theorem for Integrals: If f : [a, b] → R is continuous, ∃ c ∈ [a, b]such that: Z b

af (x) dx = f (c)(b − a)

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4.3 Methods for integrating more complicated functions

There are few rules for integration.

• Either you recognise (or guess) that the integrand is the derivative of a particular

function (this takes practice).

• Or you try one of the methods in this section to see if it works.

• Or it may be that there is no analytical solution i.e. you can’t write the answer as

a function, so the only possible method is to find a numerical approximation.

Integration by parts:

The Product Rule, for diff erentiating the product of two functions u and v , is:

d

dx

u(x)v(x)

= u(x)v0(x) + v(x)u0(x)

Integrating this we get:

u(x)v(x) =Z

u(x)v0(x) dx +Z

v(x)u0(x) dx

and rearranging gives the formula:

Z u(x)v0(x) dx = u(x)v(x) −

Z v(x)u0(x) dx

For definite integrals:

Z ba

u(x)v0(x) dx =hu(x)v(x)

iba−Z b

av(x)u0(x)dx

Hence, if we start with an integral that can be written in the form R

u(x)v0(x) dx, then

we can evaluate it using this rule provided that we know how to evaluate

R v(x)u0(x) dx.

Example

To evaluateZ 10

xe2x dx, put u(x) = x and v0(x) = e2x. Then u 0(x) = 1 and v(x) = 12e2x.

So: Z 10

xe2x dx =h12

xe2xi10−Z 10

12

e2x dx = 12e2 −

h14

e2xi10

= 14

e2 + 1

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Integration by substitution:

(Compare the Chain Rule for diff erentiation.)

Consider f (y) where y = g(x). Then dy = g 0(x) dx, and so

f (y) dy = f (y)g0(x) dx

Integrating this we get:

Z f (y)g0(x) dx =

Z f (y) dy or

Z f (y)

dy

dx dx =

Z f (y) dy

For definite integrals the formula is:

Z b

af (y) dy

dx dx =

Z g(b)

g(a)f (y) dy

So, if we can find a function y = g(x) so that an integral can be written in the formZ f (y)

dy

dx dx, then we can integrate it using this formula –

that is, by a change of variable or “substituting y for x”.

(This method is easier to use in practice than to describe in general.)

Example

To evaluateZ k0

2x

x2 + 3 dx where k is a constant, try the substitution y = x2 + 3.

In this case, dy = 2x dx and so:

Z k0

2x

x2 + 3 dx =

Z k2+33

1

y dy =

hln y

ik2+33

= ln

k2 + 3

− ln 3 = ln

1 + 13k2

A Useful Rule

The integral above is an example of a more general case that is worth remembering. If the

integrand can be written in the form g 0(x)/g(x) for some function g, then the substitution

y = g(x) gives us: Z g0(x)g(x)

dx = ln g(x) + c

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4.4 Improper Integrals

If one (or both) of the limits of integration is infinite, then it is an improper integral .

Z ∞

af (x) dx means lim

b→∞

Z ba

f (x) dx

If this limit exists (is not infinite), then we say that the integral converges. If not, we say

that the integral does not exist.

Examples

• Z b

1

x−2 dx = h−x−1i

b

1

=

−b−1 + 1, so Z

∞

1

x−2 dx = 1.

•Z b1

x−1 dx =h

ln xib1

= ln b, soZ ∞

1x−1 dx does not exist.

We also have an improper integral if the integrand has an infinite discontinuity at the

upper or lower limit of integration, or between them. For example:

•Z 1

ax−1 dx =

hln x

i1a

= − ln a, soZ 10

x−1 dx does not exist.

• Z 1

a

x−1/2 dx = h2x1/2i1

a

= 2(1

−

√ a), so Z

1

0

x−1/2 dx = 2.

If the infinite discontinuity is at q ∈ (a, b), then we considerZ pa

f (x) dx andZ b

rf (x) dx as p → q − and r → q +.

Only when both limits exist can we say that the integral from a to b exists.

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4.5 Repeated Integrals

If f is a function of two variables x and y , we can integrate with respect to each in turn.

If f is continuous, the order of integration doesn’t matter:

Z dc

Z ba

f (x, y) dx

!dy =

Z ba

Z dc

f (x, y) dy

!dx

This is the volume under the surface f (x, y) for a ≤ x ≤ b and c ≤ y ≤ d.The repeated integral may be written:

Z dc

Z ba

f (x, y) dxdy (though some authors use the opposite convention . . . )

4.6 Diff erentiating under the integral sign

Let f be a continuous function of two variables x and y , and let p and q be diff erentiable

functions of y .

Consider the integral I (y) =Z q(y)

p(y)f (x, y) dx. Then

dI dy

=Z

q(y)

p(y)

∂

∂ yf (x, y) dx + f (q (y), y)

dq dy

− f ( p(y), y) dpdy

Example

I (y) =Z y2

y2xy dx ⇒ I 0(y) =

Z y2y

2x dx + 2y3.2y − 2y2.1 =hx2iy2

y+ 4y4 − 2y2 = 5y4 − 3y2

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Math Prelim part 2