# Tutor profile: Vyaas S.

## Questions

### Subject: SAT II Mathematics Level 2

If $$ f(x) = \frac{7x + 5}{-4x -13} $$, what value does f(x) approach as x gets infinitely larger?

The best way to approach this problem is to rewrite the definition of function so as to only use negative powers of $$x$$. After doing that, the behavior of negative powers of x can be understood as $$x$$ gets infinitely larger leading us to the solution. We can assume that x is positive and non-zero since x is tending to infinity i.e getting larger. Dividing all the terms by x results in the following equation: $$f(x) = \frac {\frac{1}{x} (7x + 5)} {\frac{1}{x} (-4x - 13)} $$ which is equivalent to the expression $$f(x) = \frac{ 7 + \frac{5}{x}} {-4 - \frac{13}{x}}$$ As x gets infinitely larger, the terms the expressions $$\frac{5}{x} $$ and $$ \frac{-13}{x} $$ approach zero. Hence, the right most terms on the numerator and denominator vanish resulting in the answer $$ f(x) = \frac{-7}{4} $$ as $$x$$ tends to infinity.

### Subject: Calculus

Differentiate $$ y(z) = 3z^{8} -12z^{-3} + 80z - 15 $$

$$ y(z) = 3z^{8} -12z^{-3} + 80z - 15 $$ This question has four terms that need to be differentiated. The first three terms need to be differentiated using the formula $$\frac{d(x^{n})}{dx} = nx^{n-1} $$. The final term is dropped since it is a constant and the differential of a constant is zero. $$y'(z) = 8*3z^{8-1} - 12*(-3)*z^{-3-1} + 80 $$ Simplifying the above equation results in: $$y'(z) = 24z^{7} + 36z^{-4} + 80 $$

### Subject: Differential Equations

Integrate $$ \frac{dy}{dx} = x^{2} + 5x − 3 $$.

$$ \frac{dy}{dx} = x^{2} + 5x − 3 $$ The first step is to multiply both sides of the equation by dx. $$dy = (x^{2} + 5x − 3) dx $$ Now, integrating with respect to both y and x on left and right sides respectively. $$∫dy= ∫(x^{2} + 5x −3)dx $$ Applying the properties of integration for each term on the right hand side, the following equation is reached. $$y = \frac{x^{3}}{3} + \frac{5x^{2}}{2} - 3x + C $$ Note that there is a constant of integration C in the right side which must be included when dealing with indefinite integrals. There is one on the left side as well but it is combined with the right which results in a single constant on the right hand side.

## Contact tutor

needs and Vyaas will reply soon.