Calculus Problems and Solutions
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Prove that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^n}=0. \end{equation}
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Prove that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^2}=0. \end{equation}
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Prove that if $\lim_{x\to\infty}\,f(x)=L$ and $\lim_{x\to\infty}\,g(x)=K$ then \begin{equation} \lim_{x\to\infty}\,f(x)+g(x)=L+K. \end{equation}
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Prove that \begin{equation} \lim_{x\to\infty}\,C=C \end{equation} where $C$ is a constant.
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Assume that \begin{equation} \lim_{x\to\infty}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to\infty}\,g(x)=K \end{equation} and that $K\neq 0$. Prove that \begin{equation}\lim_{x\to\infty}\,\frac{f(x)}{g(x)}=\frac{L}{K}.\end{equation}
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Prove that if \begin{equation} \lim_{x\to\infty}\,f(x)=L \end{equation} and \begin{equation} \lim_{x\to\infty}\,g(x)=K \end{equation} then \begin{equation} \lim_{x\to\infty}\,f(x)\,g(x)=LK. \end{equation}
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Prove that if \begin{equation} \lim_{x\to\infty}\,f(x)=L \end{equation} and $L\neq 0$ then \begin{equation} \lim_{x\to\infty}\,\frac{1}{f(x)}=\frac{1}{L}. \end{equation}
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Prove that \begin{equation}\lim_{x\to \infty}\,\frac{1}{x}=0.\end{equation}
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Prove that if $f(x)$ is integrable between $0$ and $1$, then \begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}
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Prove that $\int_0^2f(x)\,dx=-1$ where \begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }0\leq x\lt 1\\ -2,\,\mbox{if }1\leq x\leq 2\end{array}\right. \end{equation} using Riemann sums.
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Prove that $\int_0^1f(x)\,dx=0$ where \begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation} using Riemann sums.
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Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, and points $\bar{x}_1=\frac{\pi}{8}$, $\bar{x}_2=\frac{3\pi}{8}$, $\bar{x}_3=\frac{5\pi}{8}$, and $\bar{x}_4=\frac{7\pi}{8}$; find $R(P)$ for $\int_0^{\pi}\sin(x)\,dx$.
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Determine the integral $\int_a^e(mx+b)\,dx$ using upper and lower sums.
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Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{1}{16},\frac{4}{16},\frac{9}{16},1\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^1\sqrt{x}\,dx$.
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Given the partition $\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\}$, find $L(P)$ and $U(P)$ for the integral $]\int_0^12x\,dx$.
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Prove that if $0\lt a\lt b$, then $a^2\lt \frac{1}{3}\left(a^2+ab+b^2\right)\lt b^2$.
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Prove that if $0\lt x_1\lt x_2$, then $x^2_1\lt x^2_2$.
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Determine the integral $\int_0^bx^2\,dx$ using upper and lower sums.
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Determine the integral $\int_a^bx\,dx$ using upper and lower sums.
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Determine the lengths of the closed intervals in the partition $\{-10.5,10.5,32,563.4,1000\}$.
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Determine the lengths of the closed intervals in the partition $\{-1,10,99,100,1000\}$.
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Determine the lengths of the closed intervals in the partition $\{-101,-20.3,0.15,10\}$.
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Determine the lengths of the closed intervals in the partition $\{-10,-2.8,4.7,60\}$.
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Determine the lengths of the closed intervals in the partition $\{1,2,3,4,6\}$.
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Determine if the set $\{1.2,3.5,6.8\}$ is a partition of $[1.2,7.2]$.
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Determine if the set $\{0.5,1.2,4.7\}$ is a partition of $[0,4.7]$.
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Determine if the set $\{-20,-19,-10,-5.5,-2,-1,0\}$ is a partiton of $[-19,0]$.
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Determine if the set $\{5,2,4,1,3\}$ is a partition of $[1,5]$.
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Determine if the set $\{-101,-32,0.153,82\}$ is a partition of $[-101,82]$.
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Use the integration symbol to denote the area under $f(x)=-5x^3+2x^2-0.10$ between $-100$ and $2$.
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Use the integration symbol to denote the area under $f(x)=\sin(x)$ between $0$ and $\pi/2$.
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Use the integration symbol to denote the area under $f(x)=x^3$ between $2$ and $5$.
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Solve for $x$ in $5-3x\leq 8+5x$.
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Determine the inflection points of $f(x)=4x^3-\frac{1}{5}x^2-3x+10$.
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Determine the second derivative of $f(x)=4x^3-\frac{1}{5}x^2-3x+10$.
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Determine the derivative of $f(x)=12x^2-\frac{2}{5}x-3$.
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Determine the derivative of $f(x)=4x^3-\frac{1}{5}x^2-3x+10$.
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Prove that $f(x)=x$ has no concavity.
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Determine the second derivative of $f(x)=x^3$.
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Determine the second derivative of $f(x)=x^2$.
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Prove $x^3$ is concave down on the interval $(-\infty,0)$ and concave up on the interval $(0,\infty)$.
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Prove that $f(x)=x^2$ is concave up.
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Prove the Second Derivative Test for maximums.
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Prove the Second Derivative Test for minimums.
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Find the critical points of $f(x)=-x^2$.
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Derivative of $f(x)=-x^2$
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Find the critical points of $f(x)=x^2$.
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