1. Fundamentals of Engineering REVIEW COURSE For Civil Engineers
Mathematics
Probability and Statistics
Computational Tools

2. Non Calculus Math
Mensuration of Areas and Volumes
Graphs of Linear and Quadratic Equations
Conic Sections
Trigonometry
Matrices
Vectors
X Centroids and Moments of Inertia
X Progression of Series
X Non Separable Differential Equations

3. Mensuration of Areas and Volumes
A spherical float for a level control is to be designed so that the ratio of the volume of the float in m3 to the area of the float in m2 will be .06 m. What is the diameter of the float?
V = (4/3) π r3 A = 4 π r2
V /A = (1/3)r2 = .06 m3 d = .36m

4. Mensuration of Areas and Volumes
A highway salt shed is designed in the shape of a paraboloid. Its floor area is 50 m2 and has a maximum height of 5m. What is the maximum amount of salt in m3 that can be stored in the shed?
V = π d2 h/8 Abase = π d2/4
V = Abase h/2 = 125 m3
Page 26 (NCEES)

5. Graphs of Linear and
Quadratic Equations

6. Where does the straight line between (1, 5) and (12, 3) cross the x axis?
 m = (y2 - y1)/(x2 - x1 )= (3 - 5)/(12 - 1) = -.182
 y= x + b 5 = -.182(1) +b
y= -.182x
At y =0
x = /.182 = 28.5

7. What is the length of a line segment with a slope of 4/3, measured from the y-axis to the point (6, 4)?
(y2 - y1) = m (x2 - x1) = (y - 4) = (4/3)(x - 6)
Y = (4/3)x – 4 => (6, 4), (0, -4)
(x2−x1) 2 + (y2 −y1) 2
(6−0) 2 + [(4− −4 ] 2
(36) 2 + (64) 2 =10 10 25 50 75

8. What are the roots of the equation x2 – 4x – 21 = 0?
4 ± = 4 ± = (7, -3)
What are the roots of the equation x2 – 4x – 20 = 0?
x=(6.9, 2.9)

9. Where does the curve y = x2 – 4x - 21 intersect the line y = -
Where does the curve y = x2 – 4x - 21  intersect the line y = -.182x ?
yc = yL = x2 – 4x-21 = x  
x x – = 0
x= ± => x1 = x2 = 3.55
y1 = -.182(7.35) = 3.84
y2 = -.182(3.55) = 4.54
=> (7.35, 3.84), (3.55, 4.54)

10. So ellipse but A = C and B =0
Conic Sections
Page 27 (NCEES)
What volume is mapped by 360 degree rotation of this equation about its symmetric axis?
𝑥 𝑦 2 −6𝑥−4𝑦=4
B2 - AC = 0 – 1(1) < 0
=> paraboloid
So ellipse but A = C and B =0
So it’s a circle…a special case of an ellipse => sphere

11. Conic Sections
Page 27 (NCEES)
What is the focus of the geometric shape defined by this this equation with vertices at (0, 0)?
Y = 𝑥 2 −𝑥
Page 26 (NCEES)
B2 - AC = 0
=> parabola
(y – k)2 = 2p(x – h); Center at (h, k) is the standard form of the equation. When h = k = 0, Focus: (p/2, 0); Directrix: x = –p/2
Focus = (p/2, 0)

12. Trigonometry - Visualization
What are the left and right components of the force holding the 50N ball?
FLx = FRx FLx = FLy + FRy = Fg
FLx = (50N)cos(30) FRx = (50N)cos(50)

13. Trigonometry - Visualization Law of Sines
𝑎 sin 𝐴 = 𝑏 𝑠𝑖𝑛 𝐵 = 𝑐 sin 𝐶
Page 26 (NCEES)
Solution:
a=c sin(A)/sin(C)
a =10(.55)/.13 = 42.3
H = a cos(41) = 31.9m

14. Trigonometry - Visualization
If the rectangle coordinates of a point are (-3, -5.2), what are its polar coordinates (r, θ )?
(-6, 120°)
(6, -120°)
(6, 120°)
(6, -150°)
𝑟= a 2 + b 2
3 2 + (−5.2) 2 = 6
tan (θ) = (5.2/3)
θ = tan-1 = 60°
Page 24 (NCEES)

15. Trigonometry Identities and Formulas
cos(α - β) = cos(α)cos(β) + sin(α)sin(β)
Page 24 (NCEES)
α = c, β = π/4
cos(c - π/4) = cos(c)cos(π/4) + sin(c)sin(π/4)
cos(c - π/4) = .707 cos(c) sin(c)

16. Trigonometry Identities
Simplify the expression
𝑡𝑎𝑛𝜙 𝑠𝑒𝑐𝜙(1− 𝑠𝑖𝑛 2 𝜙) 𝑐𝑜𝑠𝜙
Page 24 (NCEES)
𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙 =𝑡𝑎𝑛 𝜙, 1− 𝑠𝑖𝑛 2 𝜙= 𝑐𝑜𝑠 2 𝜙, 𝑐𝑠𝑐𝜙= 1 𝑐𝑜𝑠𝜙
𝑡𝑎𝑛𝜙 𝑠𝑒𝑐𝜙(1− 𝑠𝑖𝑛 2 𝜙) 𝑐𝑜𝑠𝜙 = 𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜙 ( 𝑐𝑜𝑠 2 𝜙) cos⁡(𝜙)
= 𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙 = tan 𝜙

17. Matrices
Addition =
Multiplication =
(2 x 3)
(2 x 2)
(3 x 2)

18. Matrices
+ − + 4 − −2 0 0
Determinant - 3x3 Checker
Board Method
−2 0
−2 0
4(0 - 0) + 1(0+6) + 1(0 + 10) = 16

19. Vectors – Dot Product The magnitude of the vector A =|A|
= 𝐴𝑥𝐵𝑥+𝐴𝑦𝐵𝑦+𝐴𝑧𝐵𝑧 = 𝐴·𝐵 = a number.
𝐴·𝐵 is the scalar projection of B onto A times |A|
𝐴·𝐵 =|A||B| cos(θ)
The vector projection of B onto A is
𝐴·𝐵 |A| 𝐴

20. What is the dot product (a.b) of? a = (2i + 4j + 8k), b= (-2i + j -4k)
Vectors
What is the dot product (a.b) of?
a = (2i + 4j + 8k), b= (-2i + j -4k)
a.b = ai bi + aj bj + ak bk
a.b = (2)(-2)+ (4)(1)+ (8)(-4)
= -32

21. Vectors Page 35 (NCEES) A. -84.32° B. 84.32° C. 101.20°
What is the angle between the two vectors?
A = 4i + 12j +6k
B = 24i – 8j + 6k
A●B = |A||B| cos(θ)
A●B = (4)(24)+(12)(-8)+(6)(6) = 36
|A| = (4) 2 + (12) 2 + (6) 2 = 14
|B| = (24) 2 + (−8) 2 + (6) 2 = 26
|A| |B| = 364
cos(θ) = A●B 𝐴 |𝐵| = => θ = 84.32°
Page 35 (NCEES)
A °
B °
C °

22. Vectors Normalize a = (2i + 4j + 8k) (convert it to a unit vector)
U=a/ ꓲaꓲ =
2𝑖 4𝑗 8𝑘

23. Vectors

24. Vectors
The cross product AxB gives another vector C that is orthogonal
to both A and B.
The direction of C into or out of the plane containing A and B is
resolved by the right hand rule.
The magnitude of C is equal to the area of the parallelogram
bounded by A and B.

25. -3 18 -46 =( i, j, k) Find the cross-product (axb) of
Vectors
Find the cross-product (axb) of
a=(7i,3j,-4k) b=(1i,0j,6k)
7 3 −4 7 3 −
18 -0 18
-46
0 -3 -3
=( i, j, k)

26. Vectors
What is the area of a parallelogram bounded by a = (7i, 3j, -4k) and b = (1i, 0j, 6k)?
1801
1650
2001
981
From the previous slide
a x b = (18i, 46j, -3k)
la x bl = =

27. Vectors What are the components of the matrix (x) in Ax = b
− −0.44
b =
A =
xi xj xk
− −0.44 x =
.3xi +.52xj + xk = -.01
.5xi + xj +1.9xk = .67
.1xi + .3xj +.5xk = -.44
Xi = -.07, xj = .06, xk =.05

28. X Curvature in Rectangular Coordinates
Calculus
Limits
Integration
Differentiation
Laplace Transforms
X Curvature in Rectangular Coordinates
X Partial Fractions for Integration

29. Calculus – Limits L’Hospital’s Rule
What is the value of the following limit?
lim 𝑥→0 sin⁡(5𝑥) 𝑥 => 0 0
  lim 𝑥→ d(sin⁡(5𝑥)) dx 𝑑𝑥 𝑑𝑥
lim 𝑥→0 cos⁡(5𝑥) = 5
.87 1 5

30. Calculus - Limits
? => -2x/-1 = 8

31. Calculus - Integration
Evaluate the following integral
𝑡 𝑒 3𝑡 𝑑𝑡 = ?
𝑥 𝑒 3𝑥 𝑑𝑥= 𝑒 𝑎𝑥 (𝑎𝑥−1) 𝑎 2
𝑡 𝑒 3𝑡 𝑑𝑡 = 𝑒 3𝑡 (3𝑡−1) 9
Page 27 (NCEES)
a = 3, x =t

32. Calculus - Integration
𝑥 1+𝑥 𝑑𝑥= ?
Let u = 1+𝑥 then du = 𝑑𝑥 𝑥 => dx = 2u du
Also u2 – 1 = x
𝑥 1+𝑥 𝑑𝑥= 𝑢 2 −1 2 𝑢 𝑢 𝑑𝑢= 𝑢 2 −1 𝑑𝑢
= 2( 𝑢 u) = 2[ (1+𝑥) 3/2 3 − (1+𝑥) 1/2 ]

33. Calculus – Integration
1 2 ((sin x−cos x)e2) + C

34. Calculus - Integrationx x x

35. Calculus - Integration

36. Calculus – Integration - 1

37. Calculus – Integration - 2
10 2 3
5 2 3 %
9 2 3

38. Differentiation
What is f`(x) = if f(x) = 3ln x
𝑑( 𝑎 𝑢 ) 𝑑𝑥 = ln 𝑎 𝑎 𝑢 𝑑𝑢 𝑑𝑥
a = 3 u = ln x
𝑑(3ln x ) 𝑑𝑥 = ln ln 𝑥 𝑑𝑙𝑛 𝑥 𝑑𝑥
𝑑𝑙𝑛 𝑥 𝑑𝑥 =1/𝑥
𝑑(3ln x ) 𝑑𝑥 = ln ln 𝑥 /𝑥
Page 29 (NCEES)

39. Differentiation

40. Differentiation
What is the slope of the following curve as it crosses the positive part of the x axis?
y =10 x2 -3 x - 1
0 = 10 x2 -3 x – 1
= 3 ± −3 −4(40)(−1) 2 (10) = > x = 1/2, -1/5
𝑑𝑦 𝑑𝑥 =20 𝑥−3 => 20(1/2) – 3 = 7
3/120
1/5
1/3 7

41. Differentiation – Maximum, Minimum and reflection points
What is the maximum value of the following function
on the interval x< 0
-210
-36 -5
210
𝑓 𝑥 =2 𝑥 𝑥 2 −30𝑥+10
𝑓 ′ 𝑥 =6 𝑥 2 +24𝑥−30
0= 𝑥 2 +4𝑥−5= 𝑥+5 𝑥−1
=>𝑥=1, −5
𝑓 −5 =2 − −.5 2 −30 −5 +10=210
Check
𝑓 ′′ 𝑥 =12𝑥+24 ,⇒ 𝑓 ′′ −5 =12 −5 +24=−36

42. Find the Laplace transform of
Laplace Transforms
Find the Laplace transform of
1 𝑠− 𝑠
𝑒 4𝑡 +5=
Page 34 (NCEES)

43. Laplace Transforms Find the Laplace transform for 𝑒 𝛼 𝑐𝑜𝑠𝛽 for α= 0
𝐿[𝑒 𝛼 𝑐𝑜𝑠𝛽]= 𝑠+𝛼 𝑠+𝛼 2 + 𝛽 2
= 𝑠 𝑠 2 + 𝛽 2

44. Differential Equations
Find y as a function of x if 2y(x -1)dy = xdx

45. Differential Equations

46. Probability Permutation – a particular sequence of events
Combination – a precise number of events without any order
If there are four triangles and three rectangles in a box, what is the probability that you will draw two triangles on the first two draws and then a rectangle on the third?
Pt1 = (4/7) Pt2 = (3/6), Pr3 = (3/5)
Pttr = (4/7)(3/6)(3/5) = .17
What is the probability of drawing two triangles and one rectangle in three tries?
Pttr + Ptrt + Prtt = = .51

47. Probability
What is the probability of drawing at least one diamond in four draws from a deck of cards?
P (of no diamonds) = (39/52)(38/51)(37/50)(36/49) = .304 or 30.4 %
is the probability of getting no diamonds. The total probability has to be 1.0 so
The probability of getting at least one diamond in four draws if or 69.6%

48. Probability
What is the probability of drawing at least one King in four draws from a deck of 52 cards?
.306
.016
.106
.056
4/52+4/52+4/52 +4/52 = .306 is the probability of drawing at least one.

49. Probability
What is the probability of drawing at least one King in four draws from a deck of 52 cards?
.306
.016
.106
.056
4/52+4/52+4/52 +4/52 = .306 is the probability of drawing at least one.

50. Statistics
What is the likelihood that a two-sigma event will occur, given a random normal distribution of events?
<1%
<3%
<10%
<30%
Probability that an event will occur outside the probability (area)
of 2 standard deviations from the mean.
Page 46 (NCEES)

51. Statistics
What is the standard deviation of the following measurements (3.2, 3.22, 3.24, 3.26, 3.27)?

52. Statistics Page 40 (NCEES)
Page 40 (NCEES) Equation for Correlation Coefficient R also Provided

53. Statistics

54. Statistics Use linear regression to fit y = c 𝑒 −𝑘𝑥
First take the natural log ln y = ln c –k x
The equation for ln y is linear with a slope of –k x
And an intercept of ln c.
So proceed with linear regression as usual

55. Statistics
Page 39 (NCEES)

56. Computer Tools

57. Computer Tools