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Mathematics 6.pdf
- 2. MATHEMATICS
Calculus
• Derivatives
• Critical Points
• Partial Derivatives
• Curvature
• Gradient, Divergence, and Curl
• Limits
• Integrals
• Centroids and Moments of Inertia
• Differential Equations
• Laplace Transformations
© 2018 Professional Publications, Inc. 124
Lesson Overview
- 5. MATHEMATICS
derivatives
also used to locate
• maxima
• minima
• inflection
© 2018 Professional Publications, Inc. 127
Critical Points
0
f a
0
0
f a
f a
0
0
f a
f a
Critical Points
- 9. MATHEMATICS
partial derivative
• used when a function depends on two
or more independent variables (e.g., x
and y)
• differentiation must be taken with
respect to each separately
© 2018 Professional Publications, Inc. 131
Partial Derivatives
- 12. MATHEMATICS
curvature, K
change in arc angle (Δα) over arc length
(Δs) as the latter goes to zero
reciprocal of radius of curvature, R
© 2018 Professional Publications, Inc. 134
Curvature
Curvature
- 17. MATHEMATICS
limit
value that a function approaches as its
independent variable approaches a target
value a (typically 0 or ∞)
© 2018 Professional Publications, Inc. 139
Limits
- 18. MATHEMATICS
For the limit to exist, f(x) must exist (be continuous) in the region x=a.
© 2018 Professional Publications, Inc. 140
Limits
Existence of Limits
- 19. MATHEMATICS
L’Hôpital’s rule
used to find a limit of an expression when
both the numerator and denominator are
indeterminate (both zero or both infinite)
at a limit point α
If f(x) and g(x) are differentiable around
the limit point α, then
The rule may be applied repeatedly as
long as the numerator and denominator
are both determinate.
© 2018 Professional Publications, Inc. 141
Limits
lim lim
x x
f x f x
g x g x
- 22. MATHEMATICS
integral
• inverse of differentiation
• two kinds: definite integrals and
indefinite integrals
definite integral
• defined by the fundamental theorem
of calculus:
• represents the area under a
continuous function f(x) between
two limits, a and b
• a and b are limits of integration
© 2018 Professional Publications, Inc. 144
Integrals
- 23. MATHEMATICS
indefinite integrals
• limits of integration are not specified
• sometimes called antiderivatives
techniques of solving
• table of indefinite integrals
• integration by substitution
• integration by parts
© 2018 Professional Publications, Inc. 145
Integrals
- 29. MATHEMATICS
integration by substitution
to simplify the solution process, may
change
• variable of integration
• integrand
• limits of integration
For example:
• let u = f(x)
• then du = (du/dx)dx
© 2018 Professional Publications, Inc. 151
Integrals
- 32. MATHEMATICS
integration by parts
• substitution method
• useful when the integrand is a product
of functions (e.g., of x)
© 2018 Professional Publications, Inc. 154
Integrals
- 35. MATHEMATICS
centroids
• centroid of an area is analogous to
center of gravity of a homogenous
body
• integral of function f(x) between two
limits, (a, b), calculates the area
under the function (the zeroth
moment)
• centroid requires area and the first
moment
© 2018 Professional Publications, Inc. 157
Centroids and Moments of Inertia
- 36. MATHEMATICS
moment of inertia of an area
• resistance of a cross-sectional area
to bending—important in mechanics
problems
• also called area moment of inertia or
second moment of area
parallel axis theorem
• used to calculate moments of inertia
about axes other than x = 0 and y = 0
© 2018 Professional Publications, Inc. 158
Centroids and Moments of Inertia
- 37. MATHEMATICS
mass moment of inertia
• analogous to area moment of inertia
• resistance of a mass to rotation
around an axis
parallel axis theorem
© 2018 Professional Publications, Inc. 159
Centroids and Moments of Inertia
2
I r dm
2
parallelaxis G
I I md
- 38. MATHEMATICS
A rod with mass of 2 kg/m uniformly
distributed along its length is normal to
y- and y′-axes as shown. Find the bar’s
mass moment of inertia about each axis.
© 2018 Professional Publications, Inc. 160
Example: Centroids and Moments of Inertia
- 39. MATHEMATICS
A rod with mass of 2 kg/m uniformly
distributed along its length is normal to
y- and y′-axes as shown. Find the bar’s
mass moment of inertia about each axis.
Solution
Mass is
From table of mass moments of inertia in
NCEES Handbook, the mass moment of
inertia around the y-axis is
© 2018 Professional Publications, Inc. 161
Example: Centroids and Moments of Inertia
kg
5 m 2 10 kg
m
m
2
2
2
10 kg 5 m
83.33 m kg
3 3
y
mL
I
- 40. MATHEMATICS
A rod with mass of 2 kg/m uniformly
distributed along its length is normal to
y- and y′-axes as shown. Find the bar’s
mass moment of inertia about each axis.
Solution (continued)
To find the mass moment of inertia
around the y′-axis, use the parallel axis
theorem.
© 2018 Professional Publications, Inc. 162
Example: Centroids and Moments of Inertia
2
2 2
2 2
2
12
10 kg 5 m 5 m
10 kg 2 m
12 2
223.33 m kg
c
y y
mL
I I md md
- 41. MATHEMATICS
Calculus
• Derivatives
• Critical Points
• Partial Derivatives
• Curvature
• Gradient, Divergence, and Curl
• Limits
• Integrals
• Centroids and Moments of Inertia
• Differential Equations
• Laplace Transformations
© 2018 Professional Publications, Inc. 163
Lesson Overview
