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For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.


Find the derivative of $x^x$.


Let a differentiable function $f(x)$ satisfy $$f(x)\cos{x} + 2\int_0^xf(t)\sin{t}dt = x+1$$

Find $f(x)$.


Find the value of $\displaystyle\lim_{n\to\infty}\sin^2\left(\pi\sqrt{n^2+n}\right)$.


Let $f(x)$ be a twice differentiable continuous function, and $f(0)=f'(0)=0$, $f''(0)=6$. Find the value of $$\lim_{x\to 0}\frac{f\left(\sin^2{x}\right)}{x^4}$$

Find the value of $$I=\int\frac{e^{-\sin{x}}\sin(2x)}{(1-\sin{x})^2}dx$$


Compute the value of $$\lim_{x\to\pi}\frac{\ln(2+\cos{x})}{\left(3^{\sin{x}}-1\right)^2}$$


Compute the value of $$\lim_{n\to\infty}n^2\left(1-\cos\frac{\pi}{n}\right)$$


Let $$f(x)=\left\{\begin{array}{ll} \cos{x} &, x\in[-\frac{\pi}{2}, 0)\\e^x&,x\in[0,1] \end{array}\right.$$

Compute $\displaystyle\int_{-\frac{\pi}{2}}^{1}f(x)dx$.


Determine whether or not these two series converge: $$(A)\ \ \sum_{n=1}^{\infty}\sin\left(\frac{\cos{n}}{n^2}\right)\qquad (B)\ \  \sum_{n=1}^{\infty}\cos\left(\frac{\sin{n}}{n^2}\right)$$


The equation $x^y=y^x$ describes a curve in the first quadrant of the plane containing the point $P=(4, 2)$. Compute the slope of the line that is tangent to this curve at $P$.


Compute $$\int\frac{1}{\sin{x}}d{x}$$


Compute $$\int_0^{\frac{\pi}{4}}\frac{1}{\sin{x}+\cos{x}}d{x}$$


If water is poured into a right cone whose height is $H$ cm and base's radius is $R$ cm at a speed of $A$ $cm^3$ per second, what is the speed the water is rising when the depth of water is half of the cone's height?


Let $f(x)=x^2\cos(ax)$ where $a$ is a constant. Find the $50^{th}$ order derivative of $f(x)$, i.e. $f^{(50)}(x)$.


Estimate the value of $\sqrt[4]{10018}$.


Compute $$\int\frac{1}{ax+b}d{x}$$


Compute $$\int\frac{1}{x^2-a^x}d{x}$$


Compute $$\int\frac{1}{\sqrt{a^2-x^2}}d{x}$$


Compute $$\int\frac{x}{1+x^2}dx$$


Compute $$\int\frac{\ln{x}}{x}dx$$


Compute $$\int\sin^5{x}dx$$


Compute $$\int\sec{x}dx$$


Compute $$\int\sec{x}dx$$


Compute $$\int\frac{1}{\sqrt{x^2+1}}dx$$