Basic Applications of Calculus

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 $\displaystyle \lim_{y \to 2} \frac{y^2-4}{y-2}$

If $\displaystyle x = at^2$ and $\displaystyle y = a(t^3 - t)$ then $\displaystyle \frac{dy}{dx} =$

The $\displaystyle \lim_{x \to 2} \frac{x^2-4x+4}{x-2} =$

The marginal revenue function for a product $\displaystyle MR = 5 - 4x + 3x^2$. Then the total revenue function is

Evaluate : $\displaystyle \lim_{x \to -3} \frac{x^2 + 4x + 3}{x^2 + 6x + 9}$

$\displaystyle \int (2x + 5)^7 dx$

If $\displaystyle f(x) = 3e^{2x}$, then $\displaystyle f'(x) - 2xf(x) + \frac{1}{6}f(0) - f'(0)$ is equal to

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?

The cost function of a company is given by $\displaystyle C(x) = 600x - 10x^2 + \frac{x^3}{2}$ where x denotes the output. Find the level of output (in nearest integer) at which average cost is minimum.

Evaluate the integral $\displaystyle \int_0^1(2x^2-x^3)dx$

Find $\displaystyle \frac{dy}{dx}$ for $\displaystyle x^2y^2+y = 0$.

The cost function of an organisation as $\displaystyle C(x) = 500-5x^2+\frac{x^3}{3}$, where $\displaystyle x$ denotes the output. Find the level of output at which marginal cost is the minimum.

The value of $\displaystyle \int_{0}^{4} \frac{x+3}{x+2} dx$ is

The value of $\displaystyle \int_{3}^{4} \frac{2x}{1+x^2} dx$ is