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Copyright 2011 John Wiley & Sons, Inc. All rights
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Calculus Basics
DIFFERENTIAL CALCULUS
What is calculus? Often the simplest questions are the hardest
to answer. Calculus is a branch of
mathematics which uses operators to extract information from
functions.
Branch of Mathematics Mathematical Elements Used
Arithmetic Numbers (0, 1, 101.7, , e, ..)
Algebra Variables (x, y, z, t, L, A, ..)
Analysis Functions (f(x), v(t), a(t), ..)
Calculus Operators (derivative, integral, Laplacian, ..)
There is a very large number of operators but we will focus on
the derivative and integral. There is also a
very large number of functions, but the only ones typically
needed are constant, polynomial, trigonometric,
exponential and logarithmic. These can also be combined by
addition, subtraction, multiplication, division
and/or composition.
The derivative of a function is defined in terms of a limit.
(1.4)
Unless a new function is encountered, the limit definition of
the derivative is never used. Instead, the limits
are determined once for each class of functions, and the
resulting expressions, which are called shortcuts,
are used to calculate them.
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
The derivatives of the other four trigonometric functions
(tangent, cotangent, secant and cosecant) can be
determined by using one or more of the chain rules (see below).
Notice the derivative of ex is itself. This is
a very important property of this very special function. NOW YOU
TRY IT
What is the derivative of ?
(A)
(B)
(C)
(D)
(E) 10x
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When the basic types of functions are combined by addition,
subtraction, multiplication, division and/or
composition, chain rulesare used to determine the derivatives of
the combinations.
(2.0)
(2.1)
(2.2)
(2.3)
EXAMPLE: DETERMINING THE DIVISION CHAIN
RULE
If , then f(x)? Write f(x) as the product of two functions: f(x)
= g(x)(h(x))
-1.
Apply the multiplication and composition chain rules: f(x) =
g(x)(h(x))-1
- g(x)(h(x))-2
h(x).
Multiply the first term by h(x) and divide by h(x): f(x) =
g(x)(h(x))-2
h(x) - g(x)(h(x))-2
h(x).
Factor and write as a fraction:
(2.4)
This is the Division Chain Rule.
EXAMPLE: USING THE CHAIN RULES
If f(x) = 5xe2x
, then f(x) =? The derivative of 5x is 5 and applying the
composition chain rule the derivative of e
2x is
2e2x
.
Applying the multiplication chain rule: f(x) = (5)(e2x
) + (5x)(2e2x
).
Factoring yields: f(x) = 5e2x
(1 + 2x). This is the answer written in its simplest form.
NOW YOU TRY IT
If f(x) = x2sinx, then f(x) =?
(A) 2xsinx
(B) x2cosx
(C) xsinx(2 + x)
(D) xcosx(2 + x)
(E) x(2sinx + xcosx)
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Copyright 2011 John Wiley & Sons, Inc. All rights
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In order to take the derivative of a function that contains the
product of two sub-functions, the product rule
can be applied as follows:
Where u and v are functions of the independent variable x.
EXAMPLE: USING THE PRODUCT RULE
The differential equation f(y) dx = f(x)dy (x and y) and the
corresponding differential
variables (dx and dy) are not on their respective sides of the
equation. Separation of
variables is a technique that uses simple cross multiplication
in order to arrange the
differential equation into the desired form. Therefore:
f(y) dx = f(x)dy
can also be expressed as:
EXAMPLE: USING THE SEPARATION OF
VARIABLES TECHNIQUE
The differential equation that would be used in this case
is:
adx=vdv
Substitute the equation a=v2 into the above equation adx =
vdv:
v2dx = vdv
Using the separation of variables, the equation becomes:
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Copyright 2011 John Wiley & Sons, Inc. All rights
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Integrate both sides of the equation (with integration
limits):
Calculate the integral:
APPLICATIONS OF DIFFERENTIAL CALCULUS
Two applications of the derivative used in physics are
determining the slope of the graph of a function at a
point and the minimum and/or maximum values of a function.
To determine the slope of the graph of a function at a point,
calculate the derivative of the function and
substitute the value of the variable.
EXAMPLE: SLOPE OF A FUNCTION AT A POINT
If f(x) = 5x2 + 2x - 3, then what is the slope at x = 2?
Using the shortcut for polynomials combined with the addition
chain rule yields f(x) = 10x + 2. The slope at x = 2 is then 10(2)
+ 2 = 22.
To determine the minimum and/or maximum values of a function,
take its derivative, set it
equal to zero, solve for the variable by factoring and using the
Zero-Product Rule, and
substitute the value or values of the variable into the
function.
EXAMPLE: MAXIMUM STRESS
During normal operation, the main cable of the Golden Gate
Bridge in San Francisco is
subjected to a tensile stress [MPa] described by the following
function:
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f(x) = 50x0.5
- 50x + 100
where f(x) is the tensile stress and x is the strain in the main
cable.
Determine:
(i) The strain at which the maximum stress develops
(ii) The maximum stress the main cable is subjected to at this
strain.
Solution:
(i) To calculate the strain at which the maximum stress
develops, we will take the
derivative of f(x) with respect to x, and set it equal to 0:
f(x) = 50x0.5
- 50x + 100
f (x) = 25x-0.5
- 50
Let f (x) = 0:
25x-0.5
- 50 = 0
25x-0.5
= 50
x-0.5
= 2
x = (1/2)2 = 0.25 . . . strain at which maximum stress is
achieved
(ii) To calculate the maximum stress, substitute x = 0.25 into
f(x).
Maximum stress = f(0.25)
= 50(0.25)0.5
- 50(0.25) + 100
= 112.5 MPa
NOW YOU TRY IT
What's the maximum value of f(x) = 10x(2 - x)?
(A) 1
(B) 2
(C) 5
(D) 10
(E) 20
Evaluating the integral of a function or a set of data points is
often impossible to solve in closed form.
Therefore, numerical integration is required. In this section,
the popular and reasonably accurate
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Copyright 2011 John Wiley & Sons, Inc. All rights
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Trapezoidal Rule will be explained.
First and foremost, it must be stressed that an integral is
simply an infinite sum as follows:
The Trapezoidal Rule is a discreet method that is used to
evaluate an integral; the technique uses the
right hand side of the above equation with .
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Copyright 2011 John Wiley & Sons, Inc. All rights
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Considering a random individual area, it is assumed that a
straight line segment can estimate the
function from fi to fi+1. Notice that there is an error, which
represents the difference between the area
that is bounded by the line segment and the area that is bounded
by f(x). The estimated shape is a
trapezoid. The formula for the area of a trapezoid is:
Estimate the area under the curve by combining the two formulas
above:
Collect terms:
Therefore, in general, the Trapezoidal Rule can be expressed
as:
It is important to note that a smaller will result in a larger
number of areas, which will provide a
more accurate result. This is because a smaller line segment
from fi to fi+1 will more closely represent
the exact value of f(x).
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Copyright 2011 John Wiley & Sons, Inc. All rights
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INTEGRAL CALCULUS
The indefinite integral of a function can also be defined in
terms of a limit, but for our purposes a more
useful definition relates it to the derivative. The indefinite
integral is the inverse of the derivative, and vice
versa. That is to say, the derivative of the indefinite integral
of a function is equal to the function itself and
the indefinite integral of the derivative of a function is equal
to the function itself, within a constant called
the constant of integration.
The symbol for the derivative is d, which is an abbreviation for
difference, and the symbol for the
indefinite integral is , which looks much like an s, which is an
abbreviation for sum. Using these symbols, the inverse
relationships are:
(2.5)
(2.6)
This pair of equations is called the Fundamental Theorem of
Calculus and provides an intuitive way of
calculating the indefinite integrals of many functions. Since
the indefinite integral is the inverse of the
derivative, it's often called the antiderivative.
Unless a new function is encountered, the limit definition of
the indefinite integral is never used. Instead,
the limits are determined once for each class of functions, and
the resulting expressions, which are
called shortcuts, are used to calculate them.
(2.7)
(2.8)
(2.9)
(3.0)
(3.1)
You should verify these shortcuts are correct by applying the
derivative to the indefinite integrals of each
function. If the shortcut is correct, then the derivative of the
indefinite integral will be equal to the original
function. Also, notice the indefinite integral ofex is itself.
This is another very important property of this
very special function. DEFINITE INTEGRAL
In many engineering applications, a physical quantity, f, can be
expressed as a function of another
independent variable, x. Examples include:
Velocity, v, as a function of time, t
o For instance, v(t) = t2+2t (velocity of a car on a
highway)
Force, F, as a function of displacement, s
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o For instance, F(s) = 3s (in the case of the stretching of a
spring)
Temperature, T, as a function of time, t
o For instance, T(t) = 100(1 - e-t) (temperature of a glass of
ice water being microwaved)
Electricity usage, P, in a house as a function of the number of
people, n, in a house
o For instance, P(n) = 2n + 4
The definite integral is a technique used in aggregating the
product of f(x) and x between a lower and an upper limit. For
instance, when velocity, v(t), of a car is integrated over the time
interval between 1 second to 3 seconds, we obtain the displacement
of the car between 1 second and 3
second.
Properties of the Definite Integral
1. When the limits of an integral are reversed, a negative sign
needs to
be assigned.
2. Integrating between similar limits results in 0.
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3. Multiple definite integrals can be decoupled and
aggregated separately.
4. If n is a constant, it can be taken out of the operation.
5. A definite integral can be broken down into steps and
aggregated, as long as the lower and upper limits are completely
covered in the operation.
Some Common Definite Integrals
EXAMPLE 1:
f (x) = 1
EXAMPLE 2:
f (x) = x
EXAMPLE 3:
f(x) = nxn, where n is a constant
EXAMPLE 4:
f(x) = ex
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EXAMPLE 5:
f(x) = enx
EXAMPLE 6:
EXAMPLE 7:
f(x) = sin x
EXAMPLE 8:
f(x) = cos x
EXAMPLE 9:
f(x) = tan x
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Copyright 2011 John Wiley & Sons, Inc. All rights
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Copyright 2011 John Wiley & Sons, Inc. All rights
reserved.
NOW YOU TRY IT
What is the indefinite integral of ?
(A)
(B)
(C)
(D)
(E) 10x
(F) 10x + C
APPLICATIONS OF INTEGRAL CALCULUS
Two applications of the integral used in physics are determining
area on the graph of a function and the
average value of a function.
The area on the graph of a function is defined as the area
enclosed by the graph of the function, the
horizontal axis, and two vertical lines. (See Figure 0.6.)
Figure 0.6
If the area is above the horizontal axis, then it's positive and
if it's below, then it's negative. If the two
vertical lines are located in the same place, then the area is
zero.
One way to determine this area is to calculate the definite
integral of the function. This integral is the
same as the indefinite integral except it results in a specific
value rather than a generalized function. For
this kind of integral, the constant of integration is omitted,
the two values of the variable defining the
vertical lines are substituted and the results are
subtracted.
The two values defining the vertical lines are called limits of
integration. The larger value is called
the upper limit and the smaller is called the lower limit. The
result using the lower limit is subtracted
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Copyright 2011 John Wiley & Sons, Inc. All rights
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from the one using the upper limit. The subset of the domain on
the horizontal axis between the two
vertical lines is called the interval.
EXAMPLE: THE AREA UNDER A CUBIC
If f(x) = x3, then what is the area enclosed by the graph of the
function, the horizontal axis,
and vertical lines at x = 1 and x = 2? (See Figure 0.7.)
Figure 0.7
Using the shortcut for polynomials, the indefinite integral of
the function (omitting the
constant of integration) is . Substituting and .
The area is the difference: .
NOW YOU TRY IT
What is the area enclosed by the graph of , the horizontal axis,
and vertical
lines at x = 1 and x = 2?
(A) 1
(B)
(C) 5
(D)
(E) 10
To determine the average value of a function over some interval,
calculate the definite integral over the
interval and divide by the length of the interval.
The average value of the function can be thought of as the
average height of the graph, measured with
respect to the horizontal axis. This method finds the average
height by calculating the enclosed area and
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dividing by the length of the interval. Since area is length
multiplied by height, dividing the area by the
length of the interval gives the average height.
EXAMPLE: THE AVERAGE VALUE OF A CUBIC
Determine the average value of f(x) = x3 over the interval 1
< x < 2.
Using the shortcut for polynomials, the indefinite integral of
the function (omitting the
constant of integration) is . Substituting and .
The area is the difference: . The length of the interval is 2 -
1 = 1. The
average value of the function over the interval is then .
NOW YOU TRY IT
What is the average value of over 1 < x < 2?
(A) 1
(B)
(C) 5
(D)
(E) 10