Calculus Problems and Solutions
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Prove that if $f(x)$ is continuous at $a$ then there is an open interval around $a$ for which $f(x)$ is bounded.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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Prove that $f(x)=|x|$ is not differentiable at $x=0$.
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Show that \[\lim_{h\to 0}\,\frac{|h|}{h}\] does not exist.
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The derivative of $f(x)=mx+b$.
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Find the derivative of $[f(x)]^n$ where $f(x)$ is differentiable and $n$ is an integer.
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Determine the derivative of $(x^2-1)^{100}$.
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Determine $(f\circ g)'(x)$ if $f(x)=(x-1)/(x+1)$ and $g(x)=x^2$.
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Determine $(f\circ g)(x)$ if $f(x)=(x-1)/(x+1)$ and $g(x)=x^2$.
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Determine $(f\circ g)'(x)$ if $f(x)=2x$ and $g(x)=3x$.
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Determine $(f\circ g)(x)$ if $f(x)=2x$ and $g(x)=3x$.
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Derivative of $x^n$.
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The Derivative of the Sum of a Finite Number of Differentiable Functions.
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What is the derivative of the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$?
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Prove that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $f(g(a))$ then the function composition $(f\circ g)(x)$ is continuous at $a$.
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Chain Rule - Supporting Problem 1
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Chain Rule - Supporting Problem 2
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Chain Rule - Supporting Problem 3
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Prove that if $f(x)$ is continuous at $a$ then there is a closed interval around $a$ for which $f(x)$ is bounded.
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If $f(x)$ is continuous in the interval $[a,b]$, then $f(x)$ is bounded on $[a,b]$.
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Prove that for any real numbers $a$ and $\delta\gt 0$ that the interval $(a-\delta,a+\delta)$ contains the closed interval $[a-\delta/2,a+\delta/2]$.
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Prove that $|a|=0$ if and only if $a=0$.
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Prove that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $f(g(a))$ then the function composition $(f\circ g)(x)$ is continuous at $a$.
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Prove that if \[\lim_{h\to 0}\,g(h)=0\] and $g(h)\neq 0$ for some interval $(-\bar{\delta},0)\cup (0,\bar{\delta})$ where $\bar{\delta}\gt 0$ then \[ \lim_{h\to 0}\,g(h)=\lim_{g(h)\to 0}\,g(h). \]
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Prove the Chain Rule.
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Prove that $f(x)$ is differentiable at $x$ if and only if \[ \lim_{t\to x}\,\frac{f(t)-f(x)}{t-x} \] exists. When the limit exists it is equal to $f'(x)$.
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Suppose that $f(x)=x+2$ and $g(x)=x^2$. Find $f\circ g$ and $g\circ f$.
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Derivative of $x^n$ where $n$ is a negative integer.
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Prove that if $f(x)$ is continuous at $a$ and $f(a)\neq 0$ then $1/f(x)$ is continuous at $a$.
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Prove the Quotient Rule
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Derivative of $x^n$ where $n$ is a positive integer.
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Prove that if \[ \lim_{x\to a}\,\left[f(x)+g(x)\right]=M\mbox{ and }\lim_{x\to a}\,f(x)=L \] then \[ \lim_{x\to a}\,g(x) \] exists and is equal to $M-L$.
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Prove that if \[\lim_{h\to 0}\,f(a+h)=f(a)\] then $f(x)$ is continuous at $a$.
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Prove that if \[\lim_{h\to 0}\,f(a+h)=L, \] then \[\lim_{x\to a}\,f(x)=L.\]
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Prove that if \[\lim_{x\to a}\,f(x)=L,\] then \[\lim_{h\to 0}\,f(a+h)=L.\]
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Prove that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\quad\lim_{h\to 0}\,f(a+h)=L.\]
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Prove that if $f(x)$ is continuous at $a$ then \[\lim_{h\to 0}\,f(a+h)=f(a).\]
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Prove that $f(x)$ is continuous at $a$ if and only if \[\lim_{h\to 0}\,f(a+h)=f(a).\]
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Prove that if a function is differentiable at $a$ then it is also continuous at $a$.
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Prove the product rule.
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What is the derivative of $f(x)+g(x)$ where $f(x)$ and $g(x)$ are differentiable?
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What is the derivative of $C\,f(x)$ where $C$ is a constant and $f(x)$ is differentiable.
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Prove that \[\lim_{h\to 0}\,\frac{h^2}{h}=0.\]
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Prove that \[\lim_{h\to 0}\,\frac{h}{h}=1.\]
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Find the derivative of $f(x)=x$.
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Prove that \[\lim_{h\to 0}\,\frac{0}{h}=0.\]
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The Derivative of a Constant
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Prove that if \[\lim_{x\to a}\,g(x)=0\] then the limit \[\lim_{x\to a}\,\frac{1}{g(x)}\] does not exist.
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