# probability Question (answer)

The following are the possible outcomes and their sums Roll 1 ...

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The following are the possible outcomes and their sums Roll 1 ...

To make it continuous, all you need is equate consecutive formulas at their boundaries. (x3 - B) / (x - 2) and Ax+B must agree at x=0. You get (03 - B) / (0 - 2) = A·0+B which is B/2 = B. This implies that B=0. Ax+B and 2x2 must agree at x=4...

Assuming degrees, remember that if a±b=90±360n then, sin a = cos b. Hence you have (x - 6) + (3x - 4) = 90 + 360n ⇒ 4x - 10 = 90 + 360n ⇒ x = 25 + 90n or (x - 6) - (3x - 4) = 90 - 360n ⇒ -2x - 2 = 90 - 360n ⇒ x = 180n - 46.

a and c are numeric since their values are numbers. b is categorical, as you major in a subject, which is not a number.

a and b are continuous since within a reasonable interval (e.g. (0,20) for a), any real number is possible. c is discrete as there are finite gaps between consecutive possible values, which in this case must be non-negative integers 0,1,2,3,...

This is just the area of a quarter of a circle of radius 2 so you get (1/4)*π*22=π Or you can use trig substitution: x = 2 sin θ, dx=2 cos θ dθ. ∫02√(4 - x2) dx = ∫0π/2 √(4 - sin2 θ) * 2 cos θ dθ = ∫0π/2 4 cos2 θ dθ = ∫0π/2 (2+2 cos 2θ) dθ =...

Assuming the smallest rectangle, there are two options. Arrange the circles in a line, or in a 2 by 2 square arrangement. Here they are: OOOO or OO OO In the former you can see that the rectangle's width is four diameters,...

4x2 + y2 - 8x + 2y + 4 = 0 4x2 - 8x + 4 + y2 + 2y + 1 = 1 4(x - 1)2 + (y+1)2 = 1 We have: a = 1/2 (semiminor axis), b=1 (semimajor axis) Compute c = √(b2 - a2) = (1/2)√3 Finally, ε = c/b...

Divide both sides by x2 and use the quotient rule with (u/v)' = (u'v-uv')/v2 where u=f(x) and v=x. x·f'(x) - f(x) = x (f'(x)·x - f(x)·1)/x2 = x-1 (f(x)/x)' = x-1 f(x)/x = ln |x| + C f(x) = x ln |x| +Cx Now...

You want to set each factor equal to zero and then solve Step 1: z = 0 or z-1 = 0 or z+3=0 Step 2: z = 0 or z = 1 or z = -3 For the other problem, you must make the right side 0 by subtracting 2 from both sides to get x2 - x - 12 = 0 Now...

The answer must be the same, 5, because you can use a u-substitution: u=x+c, du=dx. It yields ∫12--cc f(x) dx = ∫12 f(u-c) dx = 5

You can use the angle difference formula for sin 15° sin 15° = sin(60° - 45°) = sin 60° cos 45° - cos 60° sin 45° = (√6 - √2) / 4 cos 15° = cos(60° - 45°) = cos 60° cos 45° + sin 60° sin 45° = (√6 + √2) / 4 tan 15° = sin 15°/cos 15°) = = (√6 -...

sec2x + csc2x / csc2x(1+tan2x) = sec2x +1/(1+tan2x) = sec2x +1/sec2 x = sec2 x + cos2 x

sec2 x = 1+tan2 x = 1+(4/11)2 = 137/121. sec x = (-1/11)√137 cot x = 1/tan x = 11/4 csc2 x = 1+cot2 x = 137/16 csc x = (-1/4)√137 sin x = 1/csc x = -4/√137 cos x = 1/sec x = -11/√137

You want f'(x) ≥ 0. You get 1-1/x2 ≥ 0 1 ≥ 1/x2 x2 ≥ 1 |x| ≥ 1 x ∈ (-∞,-1] ∪ [1,∞)

∫-13 f(x) dx = ∫-12 (8-x2) dx + ∫23 x2 dx = [8x - x3/3] -12 + [x3/3]23 = 65/3 + 19/3 = 84/3 = 28 Answer: D

Rewrite as f(x) = ∫ (2x sin x + x2 cos x) dx Note that the integrand has the form d/dx (uv) = u'v+uv' where u(x) = x2 and v(x) = sin x. Hence f(x) = uv = x2 sin x + C

Solution using R. 1. Copy and paste this data in Notepad and save. Let's say you saved it as "WellData.txt". 2. Launch RStudio. 3. On the menu bar, select Tools > Import Dataset > From Text File... 4. Select WellData.txt...

Goal: Minimize D=√[x2+(y-1/2)2] Constraint: G=x2+2y=0 Let's minimize S=D2 instead. We will use Lagrange Multipliers. S = x2+y2-y+1/4 We need ∇S=λ∇G We get the system (i) 2x = 2λx (ii) 2y - 1 = 2λ (iii)...

If you don't know things like the addition identity, you can derive the formula by using the law of cosines. Start with an isosceles triangle with legs of unit length and vertex angle 2t. Let x be the length of the base. By the law of cosines [c2=a2+b2-2ab cos θ],...