LEGO 150% at Magnificata Alignment Fonts Style MathWorld I Demonstrations I Wolfram Con MATHEMATICA STUDENT EDITION 12, 13. 11. 10, 9. 8. 6. 3. 5. 2. V Cos [t]? Cos [t]? + Sin [t]? + Sin[t]? |f" (x) | (1+(f' (x))²)3/2 (b) Plot K(x) for 0sxs1. (c) Find the point (x,y) 3. (a) Find the curvature function k(x) for y = cos x using the formula K = %D where the curvature has its maximum. T' (t) ||T' (t)|| * 3 л 4. Determine the unit normal vectors to r(t) = ti+ sin tj att = and t = " using the formula N' (t) %3D %3D ||T' (t)|| ', (b) ||r' (t)|| 5. Determine the curvature and torsion of the vector-valued function r(t) = (3 + 2 t) i + 6tj+ (5 - t) k (a) using the formulas K = %3D using the built-in function ArcCurvature, (3) using the built-in function FrenetSerretSystem. ||T' (t)|| (b) using the built-in 6. Find the curvature and torsion of the general helix r(t) = a cos ti + a sin tj+ctk (a) using the formulas K = %3D ||r' (t)|| function ArcCurvature, (3) using the built-in function FrenetSerretSystem. Inf- r[t_] := {a * Cos [ t], a * Sin[[t], c* t} %3D = Sqrt[r'[t].r'[t]] Inf-)= ndr[t_] Ve?+ a? Cos (t] + a? Sin[t]? Outf 1= Inf J ndr2 = Simplify [%, Element [t, Reals]] %3D Outf ) Va? + c? Int 1 T[t_] =r'[t] / ndr2 a Sin [t] a Cos [t] dutf 1- MacBook Air 000 esc F3 F8 F10 F12 F4 F6 FR %24 4. 6. 9 7. delete T. E R. fab
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Question 3: A, B, C
![LEGO
150%
at
Magnificata
Alignment
Fonts
Style
MathWorld I
Demonstrations I
Wolfram Con
MATHEMATICA STUDENT EDITION
12,
13.
11.
10,
9.
8.
6.
3.
5.
2.
V Cos [t]?
Cos [t]? + Sin [t]?
+ Sin[t]?
|f" (x) |
(1+(f' (x))²)3/2
(b) Plot K(x) for 0sxs1. (c) Find the point (x,y)
3. (a) Find the curvature function k(x) for y = cos x using the formula K =
%D
where the curvature has its maximum.
T' (t)
||T' (t)|| *
3 л
4. Determine the unit normal vectors to r(t) = ti+ sin tj att =
and t = " using the formula N' (t)
%3D
%3D
||T' (t)||
', (b)
||r' (t)||
5. Determine the curvature and torsion of the vector-valued function r(t) = (3 + 2 t) i + 6tj+ (5 - t) k (a) using the formulas K =
%3D
using the built-in function ArcCurvature, (3) using the built-in function FrenetSerretSystem.
||T' (t)||
(b) using the built-in
6. Find the curvature and torsion of the general helix r(t) = a cos ti + a sin tj+ctk (a) using the formulas K =
%3D
||r' (t)||
function ArcCurvature, (3) using the built-in function FrenetSerretSystem.
Inf- r[t_] := {a * Cos [ t], a * Sin[[t], c* t}
%3D
= Sqrt[r'[t].r'[t]]
Inf-)= ndr[t_]
Ve?+ a? Cos (t] + a? Sin[t]?
Outf 1=
Inf J ndr2 = Simplify [%, Element [t, Reals]]
%3D
Outf ) Va? + c?
Int 1 T[t_] =r'[t] / ndr2
a Sin [t]
a Cos [t]
dutf 1-
MacBook Air
000
esc
F3
F8
F10
F12
F4
F6
FR
%24
4.
6.
9
7.
delete
T.
E
R.
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