Alex P.

Senior at University of the Pacific

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Pre-Calculus

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Question:

what is the derivative of x^3

Alex P.

Answer:

To find this, we must use the limit definition. Since the limit definition is stated as lim as x->0= (F(x+h)-F(x))/h we must use this formula. so f(x+h) = (x+h)^3. Multiplying this out would become x^3 +3x^2h+3xh^2 +h^3. Take this answer subtracting it with F(x) which is x^3 will give the equation. 3x^2h +3xh^2 and h^3. We know this is correct because every term is divisible by h. So when we divide (3x^2h+3xh^2+h^3)/h we are left with 3x^2 +3xh+ h^2. Lastly, since h is approaching zero, we assume that h is 0. Replacing all h with 0 gives 3x^2 + 3x(0) + (0)^2. Simplifying everything, the final answer is found to be 3x^2.

Calculus

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Question:

Find the derivative of f(x) = sin(x^3) dx in respects to y.

Alex P.

Answer:

This is a simple use of U substitution. The first thing we need is to set u = x^3 as it is the easier function to derive. This gives the function f(x)= sin(u). Now we take the derivative of this function which gives the answer f(x)= cos(u). Next we must use the chain rule application which states that we derive the inside function also. so the derivative of the inside function will be 3x^2. now we are given f(x)=cos(u)*(3x^2), The last step we need to do is replace u with x^3 so the final answer will be F(x)= cos(x^3)*(3x^2)

Algebra

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Question:

Given that a triangle's angle are 30, 60, 90, determine the hypotenuse of the triangle with sides 5 and 12

Alex P.

Answer:

This could be determined in two different ways. 1) We could use the Pythagorean Theorem which states A^2 + B^2 = C^2. Given that we have two sides, it is safe to assume that 5^2 + 12^2 = C^2. Since 5^2= 25 and 12^2 = 144, adding the two up will give 169. The square root of 169 is 13 so the missing length is 13. 2) We can use the property of SohCahToa. We are given two sides, so it is appropriate that we choose Soh or Cah (Sin = opposite/ hypotenuse/ Cos= adjacent/hypotenuse). A fun note to notice is that the larger the angle, the larger the number. Using this information, 30 degrees would correlate with the length 5 and 60 degrees would correlate with the length 12. Given this we can use sin(30)= 5/x. We can rewrite this as x= 5/sin(30). so the answer will be again 13.

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