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Nicolaus S.
Third Year Mathematics Major at UC Davis
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Pre-Calculus
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Question:

What is $$sec(45)$$?

Nicolaus S.

Secant is the opposite of cosine. Cosine is $$\dfrac{adjacent}{hypotenuse}$$, so Secant is $$\dfrac{hypotenuse}{adjacent}$$. Knowing the special right triangle with angles 45, 45, and 90 degrees, the adjacent side is $$\sqrt{2}$$ and the hypotenuse is $$2$$. so, the Secant of 45 is $$\dfrac{2}{\sqrt2}=\sqrt2$$

Calculus
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Question:

Find the integral of the one variable function $$f(x)=3x^2+(1/x)$$. Show your work.

Nicolaus S.

Treat this as two different functions, $$g(x)$$ and $$h(x)$$, with $$g(x)=3x^2$$ and $$h(x)=\dfrac{1}{x}$$. So, $$f(x)=g(x)+h(x)$$. Also, the integral of $$f(x)$$, let's call it $$F(x)$$, equals the integral of $$g(x)$$, called $$G(x)$$, plus the integral of $$h(x)$$, called $$H(x)$$, and also plus constant $$C$$ because the the derivative of a constant is $$0$$. To find the integral of $$g(x)$$, all you do is add $$1$$ to the exponent, and divide the whole thing by the new exponent. So, now we have $$G(x)=\dfrac{3(x^{2+1})}{2+1}=\dfrac{3x^3}{3}=x^3$$ Now, to find $$H(x)$$. $$h(x)$$ can be rewritten as $$x^{-1}$$. If we try to do the same thing as $$g(x)$$, we will get $$\dfrac{x^{-1+1}}{-1+1}=\dfrac{x^0}{0}$$, which is undefined. The integral of $$x^{-1}$$ is actually a special integral, equal to $$ln(x)$$. That just has to be memorized. Now we add $$G(x)$$, $$H(x)$$ and $$C$$ to get $$F(x)$$. so, $$F(x)=x^3+ln(x)+C$$

Algebra
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Question:

What is the slope between the points $$(1,0)$$ and $$(5,11)?$$ Show your work.

Nicolaus S.

Rise Over Run $$(y2-y1)/(x2-x1)$$ In this case, $$(11-0)/(5-1)$$ Which simplifies to $$11/4$$, or $$2.75$$

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