Calculus Problems and Solutions
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Determine an antiderivative of $f(x)=x^{16}+3x^5+101$.
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Determine an antiderivative of $f(x)=2x^2-7x+11$.
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Prove that an antiderivative of a polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$ is \begin{eqnarray} F(x)&=&\frac{a_n}{n+1}x^{n+1}+\frac{a_{n-1}}{n}x^n\\ &+&\cdots +a_0x+C. \end{eqnarray}
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Determine an antiderivative of $f(x)=5x^2+3x-10$.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f\,'(x)=0$ on $(a,b)$, then $f(x)$ is a constant.
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Prove that if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$, then there exists a constant such that $F(x)-G(x)=C$.
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Prove that \[ \lim_{x\to a^-}\,[f(x)+g(x)]=L+K \] if \[ \lim_{x\to a^-}\,f(x)=L \] and \[ \lim_{x\to a^-}\,g(x)=K. \]
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Prove that \[ \lim_{x\to a^+}\,[f(x)+g(x)]=L+K \] if \[ \lim_{x\to a^+}\,f(x)=L \] and \[ \lim_{x\to a^+}\,g(x)=K. \]
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Prove that if $F(x)$ is an antiderivative of $f(x)$ and $C$ is a constant, then $G(X)=F(x)+C$ is also an antiderivative of $f(x)$.
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Prove that \[ \lim_{x\to 3}\,f(x)=9 \]where \[ f(x)=\left\{\begin{array}{c}3x^2-18&x\neq 3 \\ 5 & x=3\\\end{array}\right. \]
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Determine the limit \[ \lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}. \]
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Prove that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\lim_{x\to a}\,(f(x)-L)=0.\]
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Prove that if \[\lim_{x\to a}\,(f(x)-L)=0,\] then \[\lim_{x\to a}\,f(x)=L.\]
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Prove that if \[\lim_{x\to a}\,f(x)=L,\] then \[\lim_{x\to a}\,(f(x)-L)=0.\]
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Prove that \[\lim_{h\to 0}\,f(a+h)=L\] if and only if \[\lim_{x\to a}\,f(x)=L.\]
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If $f(x)$ is continuous and $M(x)$ is the maximum value of $f(x)$ in the closed interval $[x,a]$, then prove that \[ \lim_{x\to a^-}\,M(x)=f(a). \]
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If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[x,a]$, then prove that \[ \lim_{x\to a^-}\,m(x)=f(a). \]
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Find the derivative of $f(x)=(x^3-2)(5x+1)$.
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If $f(x)$ is continuous and $M(x)$ is the maximum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,M(x)=f(a). \]
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If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,m(x)=f(a). \]
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Prove that if $f(x)$ is continuous on $[a,b]$, then the function \[ F(x)=\int_a^x\,f(u)du \] is continuous on $[a,b]$, differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$.
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Prove that \begin{equation} \int_a^a\,f(x)dx=0. \end{equation}
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Prove that if $a=bc$, $b\gt 0$ and $c\gt 1$, then $a\gt b$.
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Prove that \[\lim_{x\to 4}\,\sqrt{x}=2.\]
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Prove that \begin{equation} \lim_{h\to 0}\,\frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2\sqrt{x}}. \end{equation}
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Determine $\min(-1,-10)$.
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Determine $\max(-1,-10)$.
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Determine $\min(-1,1)$.
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Determine $\max(-1,1)$.
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Determine $\min(1,-2,34.2,8)$.
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Determine $\max(1,-2,34.2,8)$.
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Prove that if $a\gt 0$ and $b\gt 0$, then $a+b\gt a$ and $a+b\gt b$.
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Prove that $|ab|=|a|\,|b|$.
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Prove that \[ \lim_{h\to 0}\,\sqrt{x+h}+\sqrt{x}=2\sqrt{x}. \]
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Determine the derivative of $f(x)=\sqrt{x}$.
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Prove that if $f(x)$ is continuous on $[a,b]$ and we have a point $e$ such that $a\lt e\lt b$, then \begin{eqnarray}&&\int_a^b\,f(x)dx\\&=&\int_a^e\,f(x)dx+\int_e^b\,f(x)dx.\end{eqnarray}
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Determine the limit \begin{equation} \lim_{x\to 3}\,\frac{x^2-9}{x-3}. \end{equation}
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Show that if we are given two partitions ($P$ and $Q$) that differ by a single point so that $P\subset Q$ then $L(P)\leq L(Q)\leq\int_a^b\,f(x)dx\leq U(Q)\leq U(P)$.
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,ax^n \end{equation} does not exist for all $n\geq 1$.
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Where on the x-axis is $f(x)=x^4-4x^3$ increasing or decreasing?
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Where on the x-axis is $f(x)=1/(x+1)$ increasing or decreasing?
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,x^n \end{equation} does not exist for all $n\geq 1$..
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,x \end{equation} does not exist.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=0$, and $\bar{x}_2=\frac{\pi}{2}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{2}$, and $\bar{x}_2=\pi$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{4}$, and $\bar{x}_2=\frac{3\pi}{4}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=0$, and $\bar{x}_2=\frac{\pi}{2}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{2}$, and $\bar{x}_2=\pi$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{4}$, and $\bar{x}_2=\frac{3\pi}{4}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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