Find \(\operatorname*{lim}\limits_{x \rightarrow +\infty} \dfrac{x^3 - 5}{4 x^3 + x + 1}\).

Find \(\operatorname*{lim}\limits_{x \rightarrow -\infty} e^x\).

\(f'(x)\) can be defined as:

Let \(f(x) = \ln x \cdot \cos x\). Find \(f'(x)\).

Let \(y = 4 e^{\tan x}\). Find \(\dfrac{d y}{d x}\).

Let \(f(x) = \sin^{-1} x\). Find \(f'(0)\).

The equation of the line tangent to the graph of \(f(x) = x^2 + 5 x\) at the point with x-coordinate \(x = 2\) is:

Let \(f(x) = x^3 - 3 x\). Which of the following statements are true?
I. \(f(x)\) has local maxima at both \(x = -1\) and \(x = 1\).
II. \(f(x)\) has a local minimum at \(x = 1\) and an inflection point at \(x = 0\).
III. \(f(x)\) has both a local minimum and an inflection point at \(x = 0\).

A commercial nursery has \(1000\) yards of fencing which the owners plan to use to enclose as large a rectangular garden as possible. The garden will be bounded on one side by a barn, so no fencing is needed on that side. How large will the garden be (in square yards)?

The width of a rectangle is increasing at a rate of \(2\) cm/sec, and its length is increasing at a rate of \(3\) cm/sec. At what rate is the area of the rectangle increasing when its width is \(4\) cm and its length is \(5\) cm?

A rock is dropped from a height of \(400\) feet and falls toward the earth in a straight line; \(t\) seconds after it is dropped, it has fallen a distance of \(s(t) = 16 t^2\) feet. At what speed is the rock traveling when it hits the ground?

Which of the following gives the area between the curves \(y = x^2\) and \(y = 2 x\) over the interval \([-2, 2]\)?

Suppose that \(f(x)\) is a continuous function with the following properties: \(f''(x) = \cos x\), \(f'(\pi) = 2\) and \(f(0) = 4\). What is \(f(\pi)\)?

Suppose that the function \(f(x)\) is defined by \(f(x) = \displaystyle\int_{1}^{x} \dfrac{e^t}{t} d t\). Find \(f'(x)\).

Suppose that \(f(x) = \dfrac{x}{x^2 + 1}\). Find \(\displaystyle\int_{0}^{2} f'(x) d x\).

Which of the following statements about indefinite integrals are true?
I. \(\int f(x) + g(x) d x = \int f(x) d x + \int g(x) d x\)
II. \(\int f(x) g(x) d x = \int f(x) d x \cdot \int g(x) d x\)
III. \(\int f'(g(x)) g'(x) d x = f(g(x)) + C\)
IV. \(\int [f(x)]^n d x = \dfrac{[f(x)]^{n+1}}{n + 1} + C\)

Find the volume of the solid obtained by rotating the region bounded by \(y = x^2\) and \(y = x\) over the interval \([0, 1]\) around the x-axis.

The integral \(\int \dfrac{1}{x \ln x} d x\) can be found by

The integral \(\int x \sin x d x\) can be found by

Find \(\displaystyle\int_{0}^{\ln \sqrt{3}} \dfrac{e^x}{1 + e^{2 x}} d x\).

Find \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{x - \sin x}{x^3}\).

Find \(\displaystyle\int_{1}^{\infty} \dfrac{1}{x^2} d x\).

Which of the following improper integrals converge to a finite value? (I) \(\displaystyle\int_{1}^{\infty} e^{-x} d x\) (II) \(\displaystyle\int_{-\infty}^{\infty} x^3 d x\) (III) \(\displaystyle\int_{-\infty}^{\infty} \dfrac{1}{1 + x^2} d x\)

The second order Taylor polynomial at \(x = 0\) for \(f(x) = e^{-x}\) is

Which of the following series converge? (I) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2}\) (II) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n}\) (III) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{2^n}\)

The radius of convergence of the power series \(\displaystyle\sum_{n=0}^{\infty} x^n\) is