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Hanzhi Z.

Freshman at NYU Stern

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Physics (Newtonian Mechanics)

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Question:

A particle's velocity can be modeled with the function v(t) = 4t^3 + 3. a) What's the particle's acceleration at t = 1? b) How far did the particle travel from t = 1 to t = 3?

Hanzhi Z.

Answer:

a) because acceleration is the derivative of velocity, it can be modeled by a(t) = 12t^2, and substituting t = 1, we get the acceleration is 12 at t = 1. b) displacement is the integral of velocity, so it can be modeled by x(t) = t^4 + 3t, and substituting in t = 1 and t = 3, we get 4 and 90, respectively. Subtracting the latter from the former, we get the displacement of 86.

Calculus

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Question:

What is the limit of (6x + sinx)/(5x^3 - 2sin(2x)) as x approaches 0?

Hanzhi Z.

Answer:

-7/4, substituting x with 0, we get the indeterminate answer of 0/0. At this point we can use L'Hospital's Rule to evaluate, so taking the derivative of the numerator and denominator we get (6 + cosx)/(15x^2 - 4cos(2x)). Once again substituting x with 0, we get -7/4.

ACT

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Question:

Bill got 2 more lollipops on day 2 than on day 1. On day 3, he got triple the amount of lollipops he got on day 2. If Bill got 12 lollipops on day 3, how many lollipops did he get on day 1? A) 4 B) 5 C) 8 D) 2 E) None of the above

Hanzhi Z.

Answer:

D. If we use a to represent the number of lollipops Bill got on day 1, b to represent day 2, and c to represent day 3, we can use the first sentence to create the equation 2 + a = b, and the second sentence to create 3b = c. If c = 12, we can backtrack to get b = 4, and using 2 + a = b, we get a = 2.

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