A
Acute angle, 39
Alternating current, 32
Angle or phase, 35
Angle sum of triangle, 44
Angles, 36–54
and trigonometry, 42–54
theorems, 37–42
types, 43
Antiderivative, integral calculus, 113–115
B
Bisection method, 77
Black-box diagram, 5–7
C
Cantilever beam and weight, 3–4
Cartesian vector form, 22–23
Chain rule, 107–108
Circumference, 38
Coefficient of determination (CoD), 72
Complementary angles, 40
Complex conjugate, 35
Complex numbers, 32–34
arithmetic of, with circuits, 35–36
Composition, function, 60–64
Continuity, function, 83
Coordinate system, 9–36
angle or phase, 35
Cartesian vector form, 22–23
complex numbers, 32–34
complex plane, 34
cross product, 26–29
defined, 10
dot product, 24–26
imaginary unit, 33
manipulating space, 29–32
modulus or magnitude, 34–35
normal, 26
ordered pair, 10–11
origin, 11
polar coordinates, 14–17
position vectors, 19–21
real plane, 11
rectangular coordinates, 10
reorientation of, 30–31
scalars, 18
simple product, 23–24
third dimension and, 11–13
unit vector, 21–22
vectors, 17–18
addition and subtraction, 23
multiplication, 23
Corresponding angles, 41
Cos (θ), special values of, 50
Cosine curve, 47
Cross product, 26–29
area, to find, 28–29
D
Data set, function, 69–75
Definite integral, integral calculus, 113
Degree, 37
Dependent variable, 71
Derivative, 98
rules, 104
Determining, function, 57–59
Differential calculus, 95–111
chain rule, 107–108
derivative, 98
rules, 104
general power rule, 99
higher order derivatives, 100–101
optimization, process of, 108–111
position function
complicated, 101–104
derivative of, 99–100
product rule, 104–106
quotient rule, 106–107
tangent line, 96
Dimensional analysis, 38
Dirac’s delta function, 148–150
Displacement, 96
Distance formula, 49
Dot product, 24–26
E
Electromagnetics, 63
End behavior, function, 89
Engineering
abstraction in, 2–4
black-box diagram, 5–7
definition, 1–2
mathematical model, 4–5
statics, 4
system, 5
Equilateral triangle, 45
Even and odd functions, inputs and outputs, 133, 136–137
F
Finite sums, inputs and outputs, 153
Flip, inputs and outputs
delay in, 146–147
Flipping a function, inputs and outputs, 141–142
Forcing a function, inputs and outputs, 142–144
Fourier series, 157–163
square wave, finding, 157–163
Friction (on a Tire), 14–15
Function
behavior, 83–94
composition, 60–64
continuity, 83
data set, 69–75
defined, 55
dependent variable, 71
determining, 57–59
evaluation, 59
independent variable, 71
indeterminate or undefined, 85
inverse, 64–68
limit, 84
one-to-one, 68
relation, 56
residual, 74
tolerance, 79
visualizing, 55
Fundamental theorem, integral calculus, 114
G
General power rule, 99
Graphical relationships
angles, 36–54
types, 43
degree, 37
equilateral triangle, 45
isosceles triangle, 45
radian, 37–42
right triangle, 45
Scalene triangle, 44–45
sines and cosines, laws of, 44–46
trigonometry, ideas in, 42–52
unit circle, 46–49
H
Heaviside unit step function, 144–145
Higher order derivatives, 100–101
I
Imaginary part, 33
Imaginary unit, 33
Impulse function, inputs and outputs, 148–150
sampling property, 151–152
Independent variable, 71
Indeterminate or undefined, function, 85
Infinite series, 152–163
Initial phase, 126
Inputs and outputs, 121–163
classifications, 121–137
common manipulations, 137–144
moving (t) using, 150–151
Dirac’s delta function, 148–150
even and odd functions, 133, 136–137
finite sums, 153
flip
delay in, 146–147
of a switch, 145–146
flipping a function, 141–142
forcing a function, 142–144
Fourier series, 157–163
square wave, finding, 157–163
graphical subtraction, 148
heaviside unit step function, 144–145
impulse function, 148–150
sampling property, 151–152
infinite series, in terms of, 152–163
left and right, moving a function, 139–140
power series, 154–157
pulse, 147
scaling a function
horizontally, 141
vertically, 140
signal, 121
amplitude, 125
angular frequency, 125–127
anticausal, 128
causal, 128
deterministic, 122
energy, 129–131
even, 133–134
everlasting, 127–128
noncausal, 129
nondeterministic, 122
odd, 134–136
periodic, 123
power, 133
sinusoidal, 124
special inputs, 144–152
transformation, 137
translation, 138
up and down, moving a function, 138–139
windowing a function, 148
Integral calculus, 111–119
antiderivative, 113–115
by parts, 118–119
definite integral, 113
fundamental theorem, 114
LIATE rule, 117
simple substitution, 115–118
Inverse, function, 64–68
Isosceles triangle, 45
K
Kilonewtons, 18
L
LIATE rule, 117
Limit, function, 84
Manipulating space, coordinate system and, 29–32
Mathematical model, 4–5
Modulus or magnitude, 34–35
Moment, 26
Motional electromotive force (emf), 63
N
Normal, coordinate system, 26
Normalizing the vector
real plane, 11
O
Obtuse angle, 38
One-to-one, function, 68
Optimization, process of, 108–111
Ordered pair, 10–11
Origin, 11
P
Phase. see angle or phase
Phone (as system), 6
Polar coordinates, 14–17
Position function
complicated, 101–104
derivative of, 99–100
Position vectors, 19–21
Power series, inputs and outputs, 154–157
Product rule, 104–106
Pulse, inputs and outputs, 147
Purely imaginary number, 33
Purely real number, 33
Pythagorean Theorem, 16
for sine and cosine, 49–51
Q
Quotient rule, 106–107
R
Radian, 37–42
Real part, 33
Real plane, 11
Rectangular coordinates, 10, 16
Reflex angle, 39
Residual, function, 74
Right triangle, 45
Roots of function, 75–83
S
Scalars, 18
Scalene, 44
Scalene triangle, 44–45
Signal, inputs and outputs, 121
amplitude, 125
angular frequency, 125–127
causal, 128
deterministic, 122
energy, 129–131
even, 133–134
everlasting, 127–128
noncausal, 129
nondeterministic, 122
odd, 134–136
periodic, 123
power, 133
sinusoidal, 124
Simple product, 23–24
Simple spring and mass system, 6–7
Simple substitution, integral calculus, 115–118
Sine (θ), special values of, 50
Sine curve, 47
Sines and cosines, laws of, 44–46
Small-angle approximation, 92
Special inputs, 144–152
Statics, 4
Supplementary angles, 40
Tangent line, 96
Tensile test, 18
Third dimension and coordinate system, 11–13
Tolerance, function, 79
Transformation, inputs and outputs, 137
Translation, inputs and outputs, 138
Trigonometry, 42–52
U
Unit circle, 46–49
Unit vector, 21–22
Up and down, moving a function, 138–139
V
Vectors, 17–18
addition and subtraction, 23
multiplication, 23
Versors, 22
Vertical angles, 41
W
Windowing a function, 148