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Tiffany C.

Economics Student at Princeton University

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Calculus

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Question:

What do the first and second derivatives tell you about a graph's shape?

Tiffany C.

Answer:

If the first derivative is positive, then that means that the function's value increases with $$x $$, i.e. $$ f(x) > f(x')$$, granted that $$ x > x'$$. If the first derivative is negative, that means that the function's value decreases or appears to be going downhill if you move to the right along the x-axis. The second derivative tells you about the concavity of the function. If the second derivative is positive, its overall shape is concave up, in which it looks like a cup or a U sitting on the x-axis. If the second derivative is negative, it is concave down, in which it looks like an upside down U on the x-axis.

Algebra

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Question:

Expand $$(x+4)(x-4)$$.

Tiffany C.

Answer:

Whenever you have a quantity that is in the form $$ (x + a)(x - a)$$, the resulting expansion will always be equal to $$ x^2 - a^2$$. Therefore, the expansion of this equation would be $$ x^2 - 16$$

Physics

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Question:

If you were standing on a scale on an elevator, would the scale reading change depend on if the elevator is moving? Why?

Tiffany C.

Answer:

According to Newton's laws, you need a force in order to accelerate a system. Therefore, there must be an imbalance of forces if, in the elevator, you go from resting to moving upwards. If you were accelerating upwards, the normal force of the elevator floor on your body would have to be greater than your actual weight (which is $$mg$$) so that there is a net force going upwards, therefore the scale weight would greater than your actual weight (oh no!).

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