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Mathcad Homework
MathCad functions and commands explained with numerical method of root finding...
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MECH102 Homework 2 MathCAD
a ⋅ x2 + b ⋅ x + c= 0
Basic Basic calculations calculations - solution solution to quadratic quadratic equation: equation:
a≔1
b ≔2
c≔3
−b + b 2 − 4 ⋅ a ⋅ c xa ≔ 2⋅a
−b − b 2 − 4 ⋅ a ⋅ c xb ≔ 2⋅a
xa = −1 + 1.414i
xb = −1 − 1.41 1.414i 4i
checking results:
f x ≔ a ⋅ x 2 + b ⋅ x + c f xa = 0
f xb = 0
Plotting a function with automated ranges and number of points (which can be changed)
-10 -8
-6
-4
-2
130 120 110 100 90 80 70 60 50 40 30 20 10 0
f x 0
2
4
6
8
10
x
Plotting a function using a range variable, and changing plot display options:
x ≔ −5 , −4. −4.95 95‥ 3 18 16 14 12
f x
10
x=
8 6 4 2 -5
-4
-3
-2
x
-1
0
1
2
3
⎡ −5 ⎤ ⎢ −4.95 ⎥ ⎢ ⎥ ⎢ −4.9 ⎥ ⎢ −4.85 ⎥ ⎢ −4.8 ⎥ ⎢ −4.75 ⎥ ⎢ −4.7 ⎥ ⎢ ⎥ −4.65 ⎢ ⎥ ⎢ −4.6 ⎥ ⎢ −4.55 ⎥ ⎢ −4.5 ⎥ ⎢ ⎥ ⎢ −4.45 ⎥ ⎣ ⋮ ⎦
x=
⎡⋮ ⎤ ⎢ 2.45 ⎥ ⎢ ⎥ ⎢ 2.5 ⎥ ⎢ 2.55 ⎥ ⎢ 2.6 ⎥ ⎢ 2.65 ⎥ ⎢ 2.7 ⎥ ⎢ ⎥ 2.75 ⎢ ⎥ ⎢ 2.8 ⎥ ⎢ 2.85 ⎥ ⎢ 2.9 ⎥ ⎢ ⎥ ⎢ 2.95 ⎥ ⎣3 ⎦
Full Name
S#
Using units and significant digit display
m ≔ 100 ⋅
v ≔ 60 ⋅
a ≔ 20 ⋅
⋅ p = 1.217 ⋅ 103 ――
p≔m⋅v
2
F ≔ m ⋅ a
F = 62.2
Symbolic algebra
x
x = ?
clears previous definition of range variable "x"
⎡ 6 ⋅ y + ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ ⎤ − y + 2 ⋅ 3 ⋅ y − 2 − 4 ⎢ ⎥ 2 solve , x ⎢ x x−2 3⋅y−2 ⎥ x y ≔ ――――= ―― ――→ ⎢ ⎥ 2⋅x−3⋅x⋅y y+2 − y+2 ⋅ 3⋅y−2 −6⋅y+4 ⎥ ⎢ −――――――――― 3⋅y−2 ⎣ ⎦ evaluate solutions:
x 5 = 2 + 0.734i
x 5 = 2 − 0.734i
0
1
Symbolic calculus
a = 6.096
a a≔―
2
strip units off numerical value
a = 6.096
2
a
f x ≔ x − a
a = 6.096
allows variable "a" serve as a new symbolic variable, while still keeping its numeric value 2
10 ⋅ sin 2 ⋅ x + ―――― x
d 20 ⋅ cos 2 ⋅ x 10 ⋅ sin 2 ⋅ x d x ≔ ― f x → 2 ⋅ x − 2 ⋅ a + ―――― − ―――― x x2 dx x ≔ −3 , −2.9 ‥ 5
100 80 60
f x
40 20 -3
-2
-1
0
0
1
2
3
4
5
x
-3
d x
-2
-1
5 0 0 -5 -10 -15 -20 -25 -30
1
2
3
4
5
Full Name
S#
Vector and matrix calculations
vx ≔ −1 v≔
vy ≔ −2
⎡ vx ⎤
v=
⎣ vy ⎦
⎡ −1 ⎤ ⎣ −2 ⎦
mag ≔ v = 2.236
v ⋅ v =5
magnitude and dot product
v ≔ vx + 1i ⋅ vy = −1 − 2i
complex plane representation
angle ≔ arg v = −116.565
angle of a vector (argument)
v ≔ mag ∠ angle = 2.236 ∠ −116.565°
entering and displaying a vector in polar form
⎡1 2 3⎤ A ≔ ⎢ 2 1 5 ⎥ ⎣0 −2 3⎦
⎡ −1.182 1.091 −0.636 ⎤ A = ⎢ 0.545 −0.273 −0.091⎥ ⎣ 0.364 −0.182 0.273 ⎦ −1
⎡1 0 0⎤ A ⋅ A = ⎢ 0 1 0 ⎥ ⎣0 0 1⎦ −1
Programming
f x ≔ ‖ if x < 1 ‖ ‖ x ‖ else if 1 ≤ x ≤ 3
|| || || || ‖ ‖ || ‖ − x−1 2 +1|| ‖ ‖ || x > 3 else if ‖ || ‖ −3 | || ‖
piecewise function
x ≔ −2 , −1.9‥ 5
2 1 0
f x
-1 -2 -3 -4
-2
-1
0
1
x
2
3
4
5
Full Name
S#
General programming problem example: find the sum of first N numbers divisible by 3
N ≔ 10000
| results ≔ ‖ i ← 0 ‖ | n ← 0 ‖ | ‖ total ← 0 | ‖ while n < N || ‖ ‖ || ‖ ‖i←i+1 || ‖ ‖ remainder ← mod i , 3 || ‖ ‖ total ← total + ‖ remainder = 0| | | | ‖ ‖ ‖ if |||| ‖ ‖ ‖ i |||| ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
|||| ||| |||| || || | || |
‖ else ‖ ‖ ‖ 0 ‖ ‖ ‖ if remainder = 0| ‖ n←n+1 | ‖
i total
results = 3 ⋅ 10 4 1.5 ⋅ 108
largest ≔ results sum ≔ results
largest = 3 ⋅ 10 4
0,0
sum = 1.5 ⋅ 108
0,1
alternative solution (for this problem)
i ≔ 3 ,6 ‥3 ⋅ N
sum = 1.5 ⋅ 108
sum ≔ ∑ i i
other (even better) alternative solutions (for this problem): N
3 ⋅ ∑ i = 1.5 ⋅ 108
3⋅
or
i=1
N ⋅ N + 1 = 1.5 ⋅ 108 2
Finding roots
f x ≔ 2 ⋅ x 2 − 4 ⋅ sin x − 2 x≔1
f x , x = 1.725
x ≔ −1
f x , x = −0.423 x ≔ −1 , −0.9 ‥ 2
f x
4 3 2 10 0 -1 -2 -3 -4
marker added at 0 on vertical axis
-1
-0.5
0
x
0.5
1
1.5
2
Full Name
S#
Solving a set of nonlinear equations s e u l a V s s e u G
x≔1 y≔1 x=2 − y2
s t n i a r t s n o C r e v l o S
y=
sin x +x⋅y x ⎡ 0.252 ⎤ x,y = ⎣ 1.322 ⎦
solution ≔
x ≔ solution = 0.252
y ≔ solution = 1.322
0
1
Checking results (solving symbolically and plotting)
x,y
⎤ ‾‾‾ x − 2 ⋅ 1i ⎥ fa x ≔ x = 2 − y ――→ ⎢ ⎣ − x − 2 ⋅ 1i ⎦ 2
solve , y ⎡
⎛ ⎞ solve , y sin x sin x fb x ≔ y = ―― + x ⋅ y ――→ −――― x x⋅ x−1 ⎝ ⎠ fa x = 1.322
fb x = 1.322
1
x ≔ 0.01, 0.05 ‥ 0.5
2 1.8
fa x
1
1.6 1.4
fb x
1.2 1 0
0.1
0.2
x
0.3
0.4
0.5
Iterative calculations with subscripts
i ≔ 1 ‥ 20 x ≔x i
i−1
x
x ≔1 0
y ≔1 0
+2
i−1
+x
y ≔ ―― i 2
i
x=
⎡ 1⎤ ⎢ 3⎥ ⎢ ⎥ ⎢ 5⎥ ⎢ 7⎥ ⎢ 9⎥ ⎢ 11 ⎥ ⎢ 13 ⎥ ⎣ ⋮⎦
y=
⎡ 1⎤ ⎢ 2⎥ ⎢ ⎥ ⎢ 4⎥ ⎢ 6⎥ ⎢ 8⎥ ⎢ 10 ⎥ ⎢ 12 ⎥ ⎣ ⋮⎦
Full Name
S#
Finding an optimal solution given constraints
z x , y ≔ x − 1 s e u l a V s s e u G s t n i a r t s n o C
r e v l o S
2
− x ⋅ sin y
x≔1 y≔1 x > −2 x < 2 ⋅ y2 + 3 −3 < y < 5
z_max_loc ≔
z , x , y =
⎡ −2 ⎤ ⎣ 1.571 ⎦
x ≔ z_max_loc = −2 0
y ≔ z_max_loc = 1.571 1
z x , y = 11
×
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