A Quick Review of Differentiation, Integration and Differential Equations

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A Quick Review of Differentiation, Integration
andDifferential Equations

• Shankar Srinivasan
• BINF4000

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Introduction to Differentiation or
Derivatives BINF4000
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Instantaneous Rate of Change
Instantaneous rate of change lim ?x ? 0
?y/?x lim x1 ? x2 ( f(x2) f(x1))
/ (x2 x1)
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CHAPTER 2
Physics
If s f(t) is the position function of a
particle that is moving in a straight line, then
?s /?t represents the average velocity over a
time period ?t and v ds / dt represents the
instantaneous velocity (the rate of change of
displacement with respect to time).
2.4 Continuity
Example A particle moves along the x-axis, its
position at time t given by x(t) t / (1
t2), t gt0, where t is measured in seconds and x
in meters. a) Find the velocity at time t.
b) When is the particle moving
right or moving left?
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CHAPTER 2
If A B ? C is a chemical reaction, then the
instantaneous rate of reaction is Rate of
reaction lim ?t ? 0 ?C / ?t dC / dt
Chemistry
2.4 Continuity
Example The data in the table gives the
concentration C(t) of hydrochloric acid in moles
per liter after t minutes. Find the average rate
of reaction for 2 lt t lt 6.
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CHAPTER 2
Biology
The instantaneous rate of growth is obtained from
the average rate of growth by letting the time
period ?t approach 0 growth rate lim ?t ? 0
?n /?t dn/dt.
2.4 Continuity
Example Suppose that a bacteria population starts
with 500 bacteria and triples every hour.
a) What is the
population after 3 hours? b) What is the
population after t hours?
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Derivatives
CHAPTER 2
Instantaneous rates of change appear so often in
applications that they deserve a special name.
2.4 Continuity
Definition The derivative of a function f at a
value a denoted by f(a), is f(a) lim h ? 0
( f(a h) f(a)) / h if this limit exists.






f(a) lim x ? a ( f(x) f(a)) / (x a).
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The derivative f(a) is the instantaneous rate of
change of y f(x) with respect to x when x a.
The tangent line to y f(x) at (a, f(a)) is the
line through (a, f(a)) whose slope is equal to
f(a), the derivative of f at a. i.e. the
derivative at any point in a curve yields the
slope of the curve at that point
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The Derivative as a Function
CHAPTER 2
2.4 Continuity
The derivative at a number a is a number. If we
let a vary, xa, then the derivate will be a
function as well.
f(x) lim h --gt 0 ( f(x h) f(x)) / h






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The Second Derivative
If f is differentiable function, then its
derivative f is also a function, so f may have
a derivative of its own, denoted by ( f)
f. This new function f is called the second
derivative of f.

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Example Find f for f(x) 3x3 4x2 7 .

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Various notations for nth derivative of the
function yf(x) y n f n (x) (dn y) / (d xn
) Dn x

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Example If f(x) x 3 2x2 9, find 3rd and 4th
derivatives of f.

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CHAPTER 2
2.4 Continuity
What does f Say about f ?
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CHAPTER 2
If f(x) gt 0 on an interval, then f is
increasing on that interval. If f(x) lt 0 on an
interval, then f is decreasing on that interval.
2.4 Continuity
If f(x) gt 0 on an interval, then f is concave
upward on that interval. If f(x) lt 0 on an
interval, then f is concave downward on that
interval.
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CHAPTER 2
A point where a curve changes its direction of
concavity is called an inflection point.
2.4 Continuity
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Example Sketch a graph whose slope is always
positive and increasing sketch a graph whose
slope is always positive and decreasing and give
them equations.
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Example Suppose f(x) x e-x3 . a) On what
interval is f increasing and on what interval is
f decreasing? b) Does f have a
max or a min value?

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CHAPTER 2
Derivative of a Constant Function (d/dx) (c) 0
2.4 Continuity
(d /dx) (x) 1
The Power Rule
If n is a positive integer, then (d /dx) (x n)
n xn-1
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CHAPTER 2

• Example Find the derivatives of the given
  functions.
• f(x) 3x4 5
• g(x) x3 2x 9

2.4 Continuity
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CHAPTER 2
The Power Rule (General Rule) If n is
any real number, then (x n) n xn-1
2.4 Continuity
The Constant Multiple Rule If c is
a constant and f is a differentiable function,
then c f(x) c f(x)
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CHAPTER 2

• Example Find the derivatives of the given
  functions.
• f(x) -3x4
• f(x) ? x .

2.4 Continuity
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CHAPTER 2
The Sum Rule
If f and g are both differentiable, then
f(x) g(x) f(x) g(x)
2.4 Continuity
The Difference Rule
If f and g are both differentiable, then
f(x) - g(x) f(x) g(x)
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CHAPTER 2

• Example Find the derivatives of the given
  functions.
• y (x2 3) / x
• f(x) x2 _ 7x 55.

2.4 Continuity
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CHAPTER 2
Derivative of the Natural Exponential Function
( ex ) ex .
2.4 Continuity
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CHAPTER 2

• Example Differentiate the functions
• y x2 2 ex
• y e x1 1.

2.4 Continuity
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CHAPTER 2
The Product and Quotient Rules
2.4 Continuity
Though the derivative of the sum of two functions
is the the sum of their derivatives, an analogous
statement is not true for products, nor for
quotients.
f(x) g(x) ? f(x) g(x)
f(x)/ g(x) ? f(x) /g(x)
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CHAPTER 2
The Product Rule
If f and g are both differentiable, then f(x)
g(x) f(x) g(x) f(x) g(x)
2.4 Continuity
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CHAPTER 2
Problem If f(3) 4, g (3) 2, f(3) 6 and
g(3) 5, find the following numbers.
a)( f
g)(3)
b) (f g)(3)
2.4 Continuity
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Definition A function f has an absolute maximum
(or global maximum) at c if f (c) gt f (x) for
all x in D, where D is the domain of f. The
number f(c) is called the maximum value of f on
D. Similarly, f has an absolute minimum at c if
f (c)lt f (x) for all x in D and the
number f (c) is called the minimum value of f on
D. The maximum and the minimum values of f are
called the extreme values.
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Definition A function f has a local maximum (or
relative maximum) at c if f (c) gt f (x) when
x is near c. This means that f (c) gt f (x) for
all x in some open interval containing c.
Similarly, f has a local minimum at c if f (c) lt
f (x) when x is near c.
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Example Find the absolute and local maximum and
minimum values of f(t) 1 / t , 0 lt t lt 1
and sketch the graph.
CHAPTER 2
2.4 Continuity
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• Increasing / Decreasing Test
• If f (x) gt 0 on an interval, then f is
  increasing on that interval.
• If f (x) lt 0 on an interval, then f is
  decreasing on that interval.

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The First Derivative Test Suppose that c is a
critical number of a continuous function f.
animation
a)If f changes from positive to
negative at c, then f has a local maximum at c.
b)If f changes from negative to positive at c,
then f has a local minimum at c. c)If f doesnt
change sign at c, then f has no local maximum or
minimum at c.
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A function (or its graph) is called concave
upward on an interval I if f is an increasing
function on I and concave downward on I if f is
decreasing on I.
Concavity Test
a) If f(x) gt 0 for all x in I, then the
graph of f is concave upward on I. b) If
f(x) lt 0 for all x in I, then the graph of f is
concave downward on I.
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• The Second Derivative Test Suppose f is
  continuous near c.
• If f(c) 0 and f(c) gt 0, then f has a local
  minimum at c.
• If f(c) 0 and f(c) lt 0, then f has a local
  maximum at c.

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CHAPTER 2
2.4 Continuity
Integrals or Antiderivatives
A Function F is called an antiderivative of f on
an interval I if F(x) f (x) for all x in I.
The operation of finding the antiderivative is
also called Integration. ?ab f (x) dx ?ab F
(x) dx F (x) over the interval a,b
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Theorem If F is an antiderivative of f on an
interval I, then the most general antiderivative
or Indefinite Integral of f on I is F(x)
c where c is an arbitrary constant.
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CHAPTER 2
2.4 Continuity
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Example Find f (x) for f (x) 3e x 5 sin x
.
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Example Find a function f such that f(x)
x 3 .
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CHAPTER 2
2.4 Continuity
Areas and Distances
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The area problem Find the area of the region S
that lies under the curve y f (x) from a to b.
This means that S, is bounded by the graph of a
continuous function f, the vertical lines xa
and xb, and the x-axis.
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Example Estimate the area under the graph of f
(x) x32 from x -1 to x2 using three
rectangles and right endpoints. Sketch the curve
and the approximating rectangles.
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CHAPTER 2
2.4 Continuity
f (x)
?x
xi
xi1
animation
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Definition The area A of the region S lies under
the graph of the continuous function f is the
limit of the sum of the areas of approximating
rectangles A lim n?00 Rn lim n?00 f (x1)
?x f (xn) ?x
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Example Use definition to find an expression for
the area under the curve y x3 from 0 to 1
as a limit.
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The Definite Integral
CHAPTER 2
2.4 Continuity
f (x)
?x
xi
xi1
xi
? i1n f (xi) ?x
Riemann Sum
animation
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Definition of a Definite Integral If f is a
continuous function defined for b ? x ? a we
divide the interval a,b into subintervals of
equal width ?x (ba)/n. We let x0 ( a), x1, x2
xn ( b) be the endpoints of these
subintervals and we choose sample points x1,
x2 xn, so xi lies in the ith subinterval
xi-1, xi. Then the definite integral of f from
a to b is b ?ab f (x) dx lim n?0 ? i1n f
(xi) ?x .
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Example If f (x) x2, 0 lt x lt 1, evaluate the
Riemann sum with n 4, taking the sample points
to be right endpoints.
Right endpoints
animation
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Example If f (x) x2, 0 lt x lt 1, evaluate the
Riemann sum with n 4, taking the sample points
to be left endpoints.
Left endpoints
animation
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Example If f (x) x2, 0 lt x lt 1, evaluate the
Riemann sum with n 4, taking the sample points
to be midpoints.
Midpoints
animation
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Midpoint Rule ?ba f (x) dx ? ? ni 1 f (xi)
?x ?x f (x1) f (xn)
where ?x (b a) / n and xi ½ (xi-1
xi) midpoint of xi-1, xi.
_
_
_
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• Properties of the Integral
• ?ab c dx c(b a), where c is any constant.
• ?ab f (x) g(x)dx ?ab f (x) dx ?ab g(x)
  dx
• 3. ?ab c f (x) dx c ?ab f (x) dx ,
• 4. ?ab f (x) - g(x)dx ?ab f (x) dx - ?ab
  g(x) dx.
• 6.

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Example Solve ?-13 (3x 5) dx.
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Differential Equations

• Definition
• A differential equation (D.E) is a relationship
  between an independent variable (say x), and a
  dependent variable (say y), and one or more
  differential operators of y w.r.t x

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Order of a differential equation

• The order of a D.E. is given by the highest
  derivative involved in the equation
• Examples

1st Order
2nd Order
3rd Order
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Degree of a differential Equation.

• The degree of a D.E. is given by the largest
  power of the highest-order derivative in the
  equation
• Examples

2nd Order 3rd Degree
3rd Order 1st Degree
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Ordinary Differential Equations(ODE)

• Definition An ODE has only one independent
  variable
• Examples

x is the independent var.
? is the independentvar.
t is the independent var.
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Partial Differential Equations (PDE)

• Definition PDE has two or more independent
  variables
• Examples

Wave Equation has x and t as indep. vars.
Heat Equation has x and t as indep. vars.
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Linear and Non Linear ODE

• Definition A D.E in which the dependent variable
  and every derivative involved is to the first
  degree only, and in which no products or
  quotients of the dependent variable and/or its
  derivatives occur, is said to be linear.

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Linear and Non Linear ODE

• Examples

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Solution of a Differential Equation

• A function is said to be a solution of a DE, if
  its substitution of the dependent variable in the
  differential equation results in a relationship
  which is true for all the values of the
  independent variable.
• Other ways of putting this are to say that such a
  function satisfies the differential equation or
  that it is an integral of the equation. Solving a
  DE is sometimes referred to as integrating the
  equation.