ENGR 1990 Engineering Mathematics
Geometry: Some Useful Definitions and Concepts
Figure 1: Two Parallel and Two Perpendicular Lines
o Parallel lines do not intersect. Lines L1 and L2 are parallel.
o Perpendicular lines intersect at a 90o angle. Lines L3 a

ENGR 1990 Engineering Mathematics
Applications of Derivatives ME 2560, ME 2570
Example #1
Consider a long slender beam of
length L with a concentrated load P
acting at distance a from the left end.
Due to this load, the beam experiences an
internal bendin

ENGR 1990 Engineering Mathematics
The Derivative of a Function as a Function
Previously, we learned about the meaning of the derivative of a function f ( x) at some
arbitrary point x0 . The derivative f ( x0 ) was simply the slope of the tangent line at x

ENGR 1990 Engineering Mathematics
The Derivative of a Function An Introduction
Consider a continuous function y f ( x) .
Y
The derivative of the function at any point P
is simply the slope of the function at that
point. The line that has the same slope as

ENGR 1990 Engineering Mathematics
The Integral of a Function as a Function
Previously, we learned how to estimate the integral of a function f ( x) over some
interval a x b by adding the areas of a finite set of trapezoids that represent the area
under f

ENGR 1990 Engineering Mathematics
Sine and Cosine Functions of Time
Y
P
Arm OP rotates so P moves in a circular path.
From trigonometry, the coordinates of P are
x = cos( )
y = sin( )
x
O
If the bar completes one revolution (2 radians) in
one second, then

ENGR 1990 Engineering Mathematics
The Integral of a Function An Introduction
Definite Integral
Consider a continuous function y = f ( x) .
Y
y = f ( x)
The integral of f ( x) from x = a to x = b is
simply the area under the curve between
those two points.

ENGR 1990 Engineering Mathematics
Homework #5 Application of Complex Numbers in Electrical Engineering
1. A voltage v(t ) = 110 cos (120 t + 3 ) volts is applied to the RC series
circuit with R = 80 and C = 50 f . Given that the total impedance is
R
Z = Z

ENGR 1990 Engineering Mathematics
Homework #2 Quadratic Equations
50 (ft/s)
Y
4
60
1. A ball is thrown off a tower at a height of 60 (ft) at a speed of
50 (ft/s) and strikes the hill at some point (x, y) as shown. The
X and Y positions of the ball are giv

ENGR 1990 Engineering Mathematics
Homework #3 Geometry/Trigonometry
1. The polar coordinates of an object O are r 2700 (ft) , 60 (deg) . Find the Cartesian coordinates x
and y of O using: a) a calculator to evaluate the trig functions, and b) the values g

ENGR 1990 Engineering Mathematics
Homework #4 Two Dimensional (2D) Vectors
1. A force F has a magnitude F 250 (lb) and makes an angle 135 (deg) with the X axis.
Express the force F in terms of the unit vectors i and j .
2. A force F has a magnitude F 100

ENGR 1990 Engineering Mathematics
Homework #1 Applications of Lines
1. A weight W0 10 (lb) is hanging from a spring of stiffness k (lb/in).
When weights of 50 and 75 pounds were added to W0 , it displaced by
the additional amounts shown in the table. Use

ENGR 1990 Engineering Mathematics
Lab/Recitation #14 Review for Final Exam
1. a) A force F has a magnitude of 500 (lb) and acts at an angle of 120 (deg) to the positive Xaxis. Express F in terms of the unit vectors i and j .
b) A unit vector n is pointed

ENGR 1990 Engineering Mathematics
Homework #6 2D Vectors and Simultaneous Equations
1. For the double-loop DC circuit shown, the
currents I1 and I 2 can be found by solving the
following simultaneous equations.
R2
J
I2
I1
V1
( R1 + R3 ) I1 + ( R3 ) I 2 =

ENGR 1990 Engineering Mathematics
Homework #8 Derivatives
1. The stiffness (k) of a spring at a given displacement ( x0 ) is the
slope (derivative) of the force-displacement curve measured at
that displacement. Given a hardening spring having the forcedis

ENGR 1990 Engineering Mathematics
Homework #9 Derivatives
1. For the under-damped spring-mass-damper system shown, the spring
stiffness is k = 25 (lb/ft) , the damping coefficient is c = 3 (lb-s/ft) , the
mass is m = 0.25 (slug) , the initial position is

ENGR 1990 Engineering Mathematics
Homework #10 Integrals
1. A hardening spring has the force-displacement function f ( x) = 100 + 10 x + x 2 (lb) . The work done by
the spring as it is stretched over some displacement interval is the negative of the integ

ENGR 1990 Engineering Mathematics
Homework #7 Exponential, Natural Logarithms, and Trigonometric Functions
x(in)
1. The displacement x of a spring-mass-damper system over a one
second interval is shown by the red line in the plot below. The
downward displ

ENGR 1990 Engineering Mathematics
Application of Sine, Cosine, and Exponential Functions in ME 3600
Vibration
Equilibrium
Vibration refers to oscillatory motion of a body or structure Position
about an equilibrium position. In the case of a simple spring

ENGR 1990 Engineering Mathematics
Application of Integration in Electrical Engineering
CurrentVoltage Relationships for Resistors, Capacitors and Inductors
The voltage across and the current through a resistor are related
simply by its resistance.
i (t )

ENGR 1990 Engineering Mathematics
Applications of Integration in ME 2580
a(t )
2
a(t ) (m/s )
Example 1: Acceleration profiles
0.3
Given: A car has an acceleration profile as
shown. Its initial position and velocity are
zero. s(0) v(0) 0
15
5
20
10
t (se

ENGR 1990 Engineering Mathematics
Application of Quadratic Equations in Electric Circuits
Example #1
+
Given: A 100 watt light bulb is connected in series with a
resistor R = 10 (ohms) . The applied voltage is
V = 120 (volts) . The power used by the light

ENGR 1990 Engineering Mathematics
Application of Lines ME 2580 Dynamics
Example #1
Given: Consider a car moving with velocity v(t ) .
v(t )
For a constant braking force, the velocity of the
car satisfies the equation:
v (t ) = v0 + a0 t
(1)
Here, v0 is th

ENGR 1990 Engineering Mathematics
Application of Lines ME 2560 Statics, ME 2570 Mechanics of Materials,
ME 2580 Dynamics
Example #1
Given: Consider a weight W0 = 17.3 (lb) which
is supported by a linear spring of stiffness k.
The length u is the unstretch

ENGR 1990 Engineering Mathematics
Application of Lines in Electric Circuits
DC Circuit with a Single Resistor
A direct current (DC) circuit with a single resistor is shown
in the diagram. The symbol V represents the applied voltage,
the symbol R represent

ENGR 1990 Engineering Mathematics
Application of Geometry/Trigonometry ME 258 Dynamics
vC
The motion of a rigid body at any instant of
time as it moves in the XY plane may be described
as pure rotational motion about an instantaneous
center (IC) of zero v

ENGR 1990 Engineering Mathematics
Application of Quadratic Equations ME 2580 Dynamics
Example #1
Given: A golfer hits a ball with an
initial velocity of v0 = 96 (ft/s) at
an angle of = 50 (deg) . If we
neglect
air
resistance,
the
following equations descr

ENGR 1990 Engineering Mathematics
Application of Trigonometric Functions in Mechanical Engineering: Part II
Problem: Find the coordinates of the end-point of a
two link planar robot arm.
Y
B ( x, y )
2
Given: The lengths of the links OA and AB and the
ang

ENGR 1990 Engineering Mathematics
Application of Trigonometric Functions in Mechanical Engineering: Part I
Y
Given: The position of an object O is to be found
relative to the base B. The distance r and the angle
have been found using radar.
O
r
Find: Fin

ENGR 1990 Engineering Mathematics
Application of Two-Dimensional Vectors ME 2560, ME 2570, ME 2580
Scalars and Vectors
A scalar is a quantity represented by a positive or negative number. It contains a
magnitude (its absolute value) and a sign. They are s