The cost of production of an item is given as $\displaystyle C=50x - 5x^2 + \frac{x^3}{6}$ where x is number of items to be produced. If the average cost and marginal cost are equal then, what quantity of items should be produced?

Find the value of $\displaystyle \lim_{x \to 4} \frac{x^2 - 2x - 8}{x - 4}$

Evaluate: $\displaystyle \int_2^4 (3x - 2)^2 dx$

Determine $\displaystyle f(x)$, given that $\displaystyle f'(x) = 12x^2 - 4x$ and $\displaystyle f(-3) = 17$

Find $\displaystyle \frac{dy}{dx}$ where $\displaystyle x = \frac{e^t + e^{-t}}{2}$ and $\displaystyle y = \frac{e^t - e^{-t}}{2}$

What is the differential function of $\displaystyle \sqrt{x^2 + 2}$ ?

What is the value of $\displaystyle \lim_{y \to 2} \frac{y^2-4}{y-2}$