CSU Homework
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MECH102 Homework 2 MathCAD
a ⋅ x 2 + b ⋅ x + c ＝ 0
Basic Basi c calcula calculations tions - solut solution ion to quadrati quadratic c equation: equation:
a≔ 1
b≔ 2
c≔ 3
xa ≔
−b + b 2 − 4 ⋅ a ⋅ c 2 ⋅ a
xb ≔
xa = − 1 + 1.414i
−b − b 2 − 4 ⋅ a ⋅ c 2 ⋅ a
xb = −1 − 1.41 1.414i 4i
checking results:
≔ a ⋅ x 2 + b ⋅ x + c
f x f xa
0 =
0 =
f xb
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
-1
0
1
2
3
x
⎡ −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 ⎦
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Using units and significant digit display
m ≔ 1 00 ⋅
v ≔ 60 ⋅
p ≔ m ⋅ v
p =
a ≔ 20 ⋅
⋅ 1. 1.217 217 ⋅ 103 ――
2
F ≔ m ⋅ a
F = 62.2
Symbolic algebra
x
clears previous definition of range variable "x"
x = ?
⎡ 6 ⋅ y + ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ − y + 2 ⋅ 3 ⋅ y − 2 − 4 ⎤ solve e,x ⎢ ⎥⎥ x− 2 2 solv x 3 ⋅ y − 2 x y ≔ ―――― ＝ ―― ――→ ⎢⎢ ⎥ y+ 2 2 ⋅ x − 3 ⋅ x ⋅ y y y y − + 2 ⋅ 3 ⋅ − 2 − 6 ⋅ + 4 ⎢ −――――――――― ⎥ 3 ⋅ y − 2 ⎦ ⎣ evaluate solutions:
x 5
0
= 2 + 0.734i
x 5
1
= 2 − 0. 0.7734i
Symbolic calculus
a= 6.096
a a≔ ―
2
strip units off numerical value
a = 6.096
2
a
f x d x
a= 6.096
allows variable "a" serve as a new symbolic variable, while still keeping its numeric value
2 10 ⋅ sin 2 ⋅ x ≔ x − a + ――――
x
d 20 ⋅ cos 2 ⋅ x 10 ⋅ sin 2 ⋅ x ≔ ― f x → 2 ⋅ x − 2 ⋅ a + ―――― − ―――― x x2 dx x ≔ −3 , − −22.9‥ 5 100 80 60
f x
40 20 -3
-2
-1
0
0
1
2
3
4
5
x 5 -3
-2
-1
0 -5
0
1
2
3
4
5
-10
d x
-15 -20 -25 -30
x 2 of 6
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Vector Vector and matrix calculations
vy ≔ −2
vx ≔ −1 ⎡ vx ⎤ v≔ ⎣ vy ⎦
⎡ −1 ⎤ ⎣ −2 ⎦
v=
mag ≔ v
= 2.236
v⋅v= 5
magnitude and dot product
v ≔ vx + 1i ⋅ vy = − 1 − 2i
complex plane representation
angle ≔ arg v
angle of a vector (argument)
= −116.565
v ≔ ma mag g∠ angl angle e = 2. 236 ∠ −116.565°
⎡ −1 −1.1 .182 82 1.09 1.0911 − −0. 0.63 6366 ⎤ ⎢ 0.545 45 −0. −0.273 273 −0. −0.091 091 ⎥ A = 0.5 0.36 3644 − −0. 0.18 1822 0.27 0.2733 ⎦ ⎣ 0.
⎡1 2 3⎤ A ≔ ⎢ 2 1 5 ⎥ ⎣ 0 −2 3 ⎦ −1
A ⋅ A
entering and displaying a vector in polar form
−1
⎡1 0 0⎤ =⎢0 1 0⎥ ⎣0 0 1⎦
Programming
f x
‖ if x < 1 || ≔ ‖ || x ‖ || ‖ else if 1 ≤ x ≤ 3 | | ‖ ‖ || 2 ‖ − x − 1 + 1 | | ‖ ‖ || ‖ else if x > 3 || ‖ −3 | | | ‖
piecewise function
x ≔ −2 , − −11.9 ‥ 5
2 1 0
f x
-1 -2 -3 -4 -2
-1
0
1
2
3
4
5
x 3 of 6
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General programming programming problem example: find the sum of first N numbers divisible divisible by 3
N ≔ 10000
| | | | || || ‖ i ← i + 1 || mod i , 3 | | ‖ remainder ← || ‖ if remainder ＝ 0 | || ‖ ‖ total ← t otal + i | || | ‖ ‖ | n ← n + 1 | || ‖ ‖ i tota totall |
‖i← results ≔ ‖ 0 0 ‖n← ‖ total ← 0 ‖ while n < N ‖ ‖ ‖‖ ‖ ‖ ‖ ‖ ‖
results =
3 ⋅ 10 4 1. 1.55 ⋅ 108
large lar gest st ≔ resu results lts
largest = 3 ⋅ 10 4
su sum m ≔ re resu sult ltss
sum = 1.5 ⋅ 10 8
0,0
0,1
alternative solution (for this problem)
i ≔ 3 , 6 ‥ 3 ⋅ N
sum = 1. 1.5 5 ⋅ 108
sum ≔ ∑ i i
other (even better) alternative solutions (for this problem): N
3 ⋅ ∑ i = 1. 1.5 5 ⋅ 108
N ⋅ N + 1 3 ⋅ ――― = 1.5⋅ 10 8 2
or
i= 1
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 , − −00.9 ‥ 2
4 3 2 1
f x
0
marker added at 0 on vertical axis
0 -1 -2 -3 -4 -1
-0.5
0
0.5
1
1.5
2
x 4 of 6
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Solving a set of nonlinear equations s e u l a V s s e u G
x≔ 1 y≔ 1
s t n i a r t s n
x ＝ 2 − y 2
o C
y ＝ ―― x
r e v l o S
sin x
+x⋅y ⎤ ⎡ x , y = 0.252 ⎣ 1.322 ⎦
solution ≔
x ≔ solution
0
= 0.252
y ≔ solution
1
= 1.322
Checking results (solving symbolically and plotting)
x,y fa x
2
solve so lve , y
≔ x ＝ 2 − y ――→
⎡
x− 2 ⋅ 1i
⎤
x− 2 ⋅ 1i ⎦ solve solve , y sin x
⎣− fb x
⎞ ⎛ sin x + x ⋅ y ――→ −――― ≔ y ＝ ―― x x ⋅ x − 1 ⎠ ⎝ fa x
1
= 1.322
fb x
= 1.322
x ≔ 0.01 0. 01, 00.0 .055 ‥ 0.5 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
i
≔x
i− 1
x y i
x
0
≔1
y
0
≔1
+2
i− 1
+x
≔ ―― 2
i
x
=
⎡ 1⎤ ⎢ 3⎥ ⎢ ⎥ ⎢ 5⎥ ⎢ 7⎥ ⎢⎢ 9 ⎥⎥ 11 ⎢ 13 ⎥ ⎣ ⋮⎦
y
=
⎡ 1⎤ ⎢ 2⎥ ⎢ ⎥ ⎢ 4⎥ ⎢ 6⎥ ⎢⎢ 8 ⎥⎥ 10 ⎢ 12 ⎥ ⎣ ⋮⎦
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Finding an optimal solution given constraints
z x , y 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 − 1 − x ⋅ sin y
x≔ 1 y≔ 1 x> −2 x < 2 ⋅ y 2
+3
−3 < y < 5 z_max_loc ≔
z , x , y
⎡ ⎤ = −2 ⎣ 1.571 ⎦
x ≔ z_max_loc
= −2
y ≔ z_max_loc
= 1.571
0
1
z x , y
= 11
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