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Liam H.

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Pre-Calculus

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Question:

Determine the implied domain of the function $$ f(x)=sqrt{x+1} $$.

Liam H.

Answer:

To determine the implied domain of a function, we need to think about where the function is defined. The square root function is only defined for non-negative numbers, so we require that $$ x+1>=0 $$. That is, we need the inside of the root to be positive or zero, else our function won't be defined. Now we just need to solve $$ x+1>=0 $$ ===> x>=-1. And so the implied domain of f is [-1,infinity)

Calculus

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Question:

Find the derivative of the function $$ f(x)=x^{2} \sin(x) $$

Liam H.

Answer:

This particular function is the *product* of two elementary functions. Whenever we want to find the derivative of the product of the functions, we can simply apply the product rule. The product rule is the following: $$ \frac {d}{dx} (u*v)=u*\frac {d}{dx} (v) + v*\frac {d}{dx} (u). So to differentiate f(x), we just use the above formula: $$ f'(x) = sin(x)*\frac {d}{dx} (x^{2}) + x^{2}*\frac {d}{dx} (sin(x)) $$ Since \frac {d}{dx} (x^{2}) = 2x, and \frac {d}{dx} (sin(x))=cos(x), we conclude that $$ f'(x) = 2xsin(x) + x^{2}*cos(x) $$

Algebra

TutorMe

Question:

Find all solutions to the quadratic equation $$ 3x^{2}+7x+2=0 $$.

Liam H.

Answer:

There are various methods one can use to solve quadratic equations. In my experience, the most reliable method for solving these equations is to use the quadratic formula. What is the quadratic formula you ask? Well, given *any* equation of the form: $$ ax^{2}+bx+c=0 $$, we can find the solutions by evaluating the following formula: $$ x=\frac{-b+-sqrt{b^{2}-4ac}}{2a} $$. Now, in our problem, a=3, b=7, and c=2, so all we need to do is substitute these values into the above formula: $$ x=\frac{-7+-sqrt{7^{2}-4(3)(2)}}{2(3)} $$ And so x=$$ x=\frac{-7+-sqrt{25}}{6} $$ $$ x=\frac{-2+-5}{6} $$ And so either x=(-2+5)/6=1/2 or x = (-2-5)/6=-7/6 Done!

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