Mathcad Homework

Full Name

S#

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