Calculus Problems and Solutions
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Prove that $f(x)=mx+b$ is uniformly continuous on the entire $x$-axis.
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Prove that $f(x)=1/x$ is uniformly continuous on the interval $[1,\infty)$.
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Prove that $f(x)=1/x$ is continuous for $x\lt 0$.
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Prove that \[ \lim_{x\to a}\frac{1}{x}=\frac{1}{a} \] for any $a$ in $(-\infty,0)$.
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Prove that $f(x)=1/x$ is continous for $x\gt 0$.
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Prove that \[ \lim_{x\to a}\frac{1}{x}=\frac{1}{a} \] for any $a$ in $(0,\infty)$.
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Prove that \[|xa|\left|\frac{1}{x}-\frac{1}{a}\right|=|x-a|.\]
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Prove that $x^2$ is uniformly continuous on the domain $[-10,10]$.
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Prove that $x^2$ is continuous.
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Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.
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Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$, then there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$.
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Prove for a bounded function on $[a,b]$ that if there exists a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$, then $\phi=\Phi$.
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Prove for a bounded function on $[a,b]$ that $\phi=\Phi$ if and only if for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.
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Prove, for a bounded function on $[a,b]$, that if \[ \forall\epsilon\gt 0(\exists\delta\gt 0(\forall P(||P\,||\lt\delta\Rightarrow U(P)-L(P)\lt\epsilon))) \] is true, then $\phi=\Phi$.
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Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$, then for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.
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Prove that every upper sum has a greatest lower bound.
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Prove that the set of all lower sums has a least upper bound.
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Determine the derivative of $\frac{1}{x}$.
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Prove $f(x)=\frac{1}{x}$ is concave down on the open interval $(-\infty,0)$ and concave up on the open interval $(0,\infty)$.
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Determine the derivative of \[\frac{ax+b}{ex+f}.\]
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Use the product rule to determine the derivative of $f(x)=(x^2-3)(5x^3-7)(-10x^4+8)$.
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Prove that \[\lim_{h\to 0}\,\frac{h^n}{h}=0\]where $n$ is a positive integer greater than $1$.
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Prove that \[\left(\begin{array}{c}n\\0\end{array}\right)=1.\]
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Prove that \[ \frac{d}{dx}x^n=nx^{n-1} \]using the Binomial Theorem where $n$ is a positive integer.
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Prove that \[ \left(\begin{array}{c}n\\1\end{array}\right)=n.\]
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What is \[ \frac{n!}{1!(n-1)!} \]where $n$ is a positive integer?
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Use the Binomial Theorem to solve $(x+y)^2$.
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What is $6!$?
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What is $5!$?
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Determine the derivative of $f(x)=(x-1)(x^2-4)$.
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Determine the tangent line of $f(x)=(x-1)(x^2-4)$ at $x=-1$.
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Determine \[ \lim_{x\to 2}\frac{x^2+2x-24}{x-2}. \]
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Determine \[ \lim_{x\to 3}\,-10x^4+7x^2+300. \]
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Determine \[ \lim_{x\to -2}\frac{x^2-4}{x^2-4}. \]
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Prove that if $f(x)$ is bounded on $[a,b]$, then \[\lim_{||P\,||\to 0}\,L(P)\] is the least upper bound of all lower sums.
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Determine \[ \lim_{x\to -3}\frac{x^3+27}{x+3}. \]
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Prove that $x^3+y^3=(x+y)(x^2-xy+y^2)$.
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Prove that if $f(x)$ is bounded on $[a,b]$, then \[\lim_{||P\,||\to 0}\,U(P)\] is the greatest lower bound of all upper sums.
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What is the formula for the tangent line of $f(x)=-4x^2+10x$ at $x=-3$?
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What is the formula for the tangent line of $x^3$ at $x=2$?
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Determine the function limit \[ \lim_{x\to 1}\,\frac{x^2-1}{x^2-2x+1}. \]
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Prove that $\phi\leq\Phi$ where $\phi$ is the least upper bound of all lower sums and $\Phi$ is the greatest lower bound of all upper sums.
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Prove that if $P$ and $Q$ are two arbitrary partitions of $[a,b]$, then $L(P)\leq U(Q)$.
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Determine \[\int_2^54x+4\,dx.\]
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Determine \[\int_0^1\,x^3+3x^2-8x-2\,dx.\]
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Prove the Fundamental Theorem of Integral Calculus for continuous functions.
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Determine an antiderivative of $f(x)=125x^{24}+32x^2-16x+2$.
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Determine an antiderivative of $f(x)=16x^3+6x^2+2x+7$.
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Determine an antiderivative of $f(x)=x^3+3x^2-8x-2$.
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Determine an antiderivative of $f(x)=4x+4$.
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