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Ephraim E.
Physicist and Mathematician, Experienced Tutor (7 years exp.)
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Physics (Newtonian Mechanics)
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Question:

A 3 kg mass starts from rest and is pushed by a horizontal 9 N force along a frictionless, horizontal surface. How far will the block go in 2 seconds?

Ephraim E.

Note that by Newton's second law, we can get the acceleration of the block, a = F/m as F = 9 N, m = 3 kg, then a = 9/3 m/s^2 = 3 m/s^2 [1] Also, for uniformly accelerated motion, note that d = vo t + 1/2 a t^2 [2] where d = the displacement of the object vo = initial velocity = 0 m/s (as it started from rest) t = time of travel a = the acceleration = 3 m/s^2 (from equation [1]) Hence, the displacement is d = (0 m/s) (2 s) + (1/2) (3 m/s^2) (2 s)^2 = 6 m [ANSWER] DONE!

Calculus
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Question:

Find the derivative of f(x) = (x^3) sin x.

Ephraim E.

Note that the product rule states that if f(x) = g(x) h(x) [1] then f'(x) = g(x) h'(x) + g'(x) h(x). Hence, if we let g(x) = x^3 h(x) = sin x Then g'(x) = 3x^2 (by power rule) h'(x) = cos x (elementary derivative) Thus, by product rule [Equation 1], f'(x) = x^3 cos x + 3x^2 sin x. [ANSWER] DONE!

Statistics
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Question:

In how many ways can you distribute 10 identical candies to 3 children?

Ephraim E.

This is called a "balls and urns" problem in statistics. Note that for x candies and y children, the number of ways to distribute the candies is # ways = C(x, y-1) = (x + y - 1)!/[x! (y-1)!] Hence, as there are x = 10 candies and y = 3 children here, #ways = C(10 + 3 - 1, 3 - 1) = C(12, 2) = 12!/[10!2!] Evaluating this expression, = 66 ways. [ANSWER] DONE! :)

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