If F X is Not Defined at C Then F X Cannot Be Continuous on Any Interval True or False
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Chapter 5 True Or False and Multiple Choice Problems
Answer the following questions.
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For each of the following ten statements answer TRUE or FALSE as appropriate:
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If \(f\) is differentiable on \([-1,1]\) then \(f\) is continuous at \(x=0\text{.}\)
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If \(f'(x)\lt 0\) and \(f"(x)>0\) for all \(x\) then \(f\) is concave down.
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The general antiderivative of \(f(x)=3x^2\) is \(F(x)=x^3\text{.}\)
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\(\ln x\) exists for any \(x>1\text{.}\)
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\(\ln x=\pi\) has a unique solution.
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\(e^{-x}\) is negative for some values of \(x\text{.}\)
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\(\ln e^{x^2}=x^2\) for all \(x\text{.}\)
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\(f(x)=|x|\) is differentiable for all \(x\text{.}\)
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\(\tan x\) is defined for all \(x\text{.}\)
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All critical points of \(f(x)\) satisfy \(f'(x)=0\text{.}\)
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Answer each of the following either TRUE or FALSE.
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The function \(f(x)=\left\{ \begin{array}{lll} 3+\frac{\sin (x-2)}{x-2}\amp \mbox{if} \amp x\not=2 \\ 3\amp \mbox{if} \amp x=2 \end{array} \right.\) is continuous at all real numbers \(x\text{.}\)
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If \(f'(x)=g'(x)\) for \(0\lt x\lt 1\text{,}\) then \(f(x)=g(x)\) for \(0\lt x\lt 1\text{.}\)
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If \(f\) is increasing and \(f(x)>0\) on \(I\text{,}\) then \(\ds g(x)=\frac{1}{f(x)}\) is decreasing on \(I\text{.}\)
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There exists a function \(f\) such that \(f(1)=-2\text{,}\) \(f(3)=0\text{,}\) and \(f'(x)>1\) for all \(x\text{.}\)
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If \(f\) is differentiable, then \(\ds \frac{d}{dx}f(\sqrt{x})=\frac{f'(x)}{2\sqrt{x}}\text{.}\)
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\(\displaystyle \ds \frac{d}{dx}10^x=x10^{x-1}\)
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Let \(e=\exp (1)\) as usual. If \(y=e^2\) then \(y'=2e\text{.}\)
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If \(f(x)\) and \(g(x)\) are differentiable for all \(x\text{,}\) then \(\ds \frac{d}{dx}f(g(x))=f'(g(x))g'(x)\text{.}\)
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If \(g(x)=x^5\text{,}\) then \(\ds \lim _{x\to 2}\frac{g(x)-g(2)}{x-2}=80\text{.}\)
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An equation of the tangent line to the parabola \(y=x^2\) at \((-2,4)\) is \(y-4=2x(x+2)\text{.}\)
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\(\displaystyle \ds \frac{d}{dx}\tan ^2x=\frac{d}{dx}\sec ^2x\)
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For all real values of \(x\) we have that \(\ds \frac{d}{dx}|x^2+x|=|2x+1|\text{.}\)
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If \(f\) is one-to-one then \(\ds f^{-1}(x)=\frac{1}{f(x)}\text{.}\)
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If \(x>0\text{,}\) then \((\ln x)^6=6\ln x\text{.}\)
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If \(\ds \lim _{x\to 5}f(x)=0\) and \(\ds \lim _{x\to 5}g(x)=0\text{,}\) then \(\ds \lim _{x\to 5}\frac{f(x)}{g(x)}\) does not exist.
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If the line \(x=1\) is a vertical asymptote of \(y=f(x)\text{,}\) then \(f\) is not defined at 1.
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If \(f'(c)\) does not exist and \(f'(x)\) changes from positive to negative as \(x\) increases through \(c\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)
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\(\sqrt{a^2}=a\) for all \(a>0\text{.}\)
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If \(f(c)\) exists but \(f'(c)\) does not exist, then \(x=c\) is a critical point of \(f(x)\text{.}\)
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If \(f"(c)\) exists and \(f'''(c)>0\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)
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Are the following statements TRUE or FALSE.
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\(\ds \lim _{x\to 3}\sqrt{x-3}=\sqrt{\lim _{x\to 3}(x-3)}\text{.}\)
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\(\displaystyle \ds \frac{d}{dx}\left( \frac{\ln 2^{\sqrt{x}}}{\sqrt{x}}\right) =0\)
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If \(f(x)=(1+x)(1+x^2)(1+x^3)(1+x^4)\text{,}\) then \(f'(0)=1\text{.}\)
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If \(y=f(x)=2^{|x|}\text{,}\) then the range of \(f\) is the set of all non-negative real numbers.
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\(\ds \frac{d}{dx}\left( \frac{\log x^2}{\log x}\right) =0\text{.}\)
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If \(f'(x)=-x^3\) and \(f(4)=3\text{,}\) then \(f(3)=2\text{.}\)
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If \(f"(c)\) exists and if \(f"(c)>0\text{,}\) then \(f(x)\) has a local minimum at \(x=c\text{.}\)
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\(\ds \frac{d}{du}\left( \frac{1}{\csc u}\right) =\frac{1}{\sec u}\text{.}\)
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\(\ds \frac{d}{dx}(\sin ^{-1}(\cos x)=-1\) for \(0\lt x\lt \pi\text{.}\)
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\(\sinh ^2x-\cosh ^2x=1\text{.}\)
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\(\ds \int \frac{dx}{x^2+1}=\ln (x^2+1)+C\text{.}\)
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\(\ds \int \frac{dx}{3-2x}=\frac{1}{2}\ln |3-2x|+C\text{.}\)
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Answer each of the following either TRUE or FALSE.
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For all functions \(f\text{,}\) if \(f\) is continuous at a certain point \(x_0\text{,}\) then \(f\) is differentiable at \(x_0\text{.}\)
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For all functions \(f\text{,}\) if \(\ds \lim _{x\to a^-}f(x)\) exist, and \(\ds \lim _{x\to a^+}f(x)\) exist, then \(f\) is continuous at \(a\text{.}\)
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For all functions \(f\text{,}\) if \(a\lt b\text{,}\) \(f(a)\lt 0\text{,}\) \(f(b)>0\text{,}\) then there must be a number \(c\text{,}\) with \(a\lt c\lt b\) and \(f(c)=0\text{.}\)
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For all functions \(f\text{,}\) if \(f'(x)\) exists for all \(x\text{,}\) then \(f"(x)\) exists for all \(x\text{.}\)
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It is impossible for a function to be discontinues at every number \(x\text{.}\)
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If \(f\text{,}\) \(g\text{,}\) are any two functions which are continuous for all \(x\text{,}\) then \(\ds \frac{f}{g}\) is continuous for all \(x\text{.}\)
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It is possible that functions \(f\) and \(g\) are not continuous at a point \(x_0\text{,}\) but \(f+g\) is continuous at \(x_0\text{.}\)
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If \(\ds \lim _{x\to \infty }(f(x)+g(x))\) exists, then \(\ds \lim _{x\to \infty }f(x)\) exists and \(\ds \lim _{x\to \infty }g(x)\) exists.
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\(\displaystyle \ds \lim _{x\to \infty}\frac{(1.00001)^x}{x^{100000}}=0\)
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Every continuous function on the interval \((0,1)\) has a maximum value and a minimum value on \((0,1)\text{.}\)
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Answer each of the following either TRUE or FALSE.
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Let \(f\) and \(g\) be any two functions which are continuous on \([0,1]\text{,}\) with \(f(0)=g(0)=0\) and \(f(1)=g(1)=10\text{.}\) Then there must exist \(c,d\in [0,1]\) such that \(f'(c)=g'(d)\text{.}\)
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Let \(f\) and \(g\) be any two functions which are continuous on \([0,1]\) and differentiable on \((0,1)\text{,}\) with \(f(0)=g(0)=0\) and \(f(1)=g(1)=10\text{.}\) Then there must exist \(c\in [0,1]\) such that \(f'(c)=g'(c)\text{.}\)
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For all \(x\) in the domain of \(\sec ^{-1}x\text{,}\)
\begin{equation*} \sec (\sec ^{-1}(x))=x\text{.} \end{equation*}
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Answer each of the following either TRUE or FALSE.
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The slope of the tangent line of \(f(x)\) at the point \((a,f(a))\) is given by \(\ds \frac{f(a+h)-f(a)}{h}\text{.}\)
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Using the Intermediate Value Theorem it can be shown that \(\ds \lim _{x\to 0}x\sin \frac{1}{x}=0\text{.}\)
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The graph on Figure 5.0.1 exhibits three types of discontinuities.
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If \(w=f(x)\text{,}\) \(x=g(y)\text{,}\) \(y=h(z)\text{,}\) then \(\ds \frac{dw}{dz}=\frac{dw}{dx}\cdot \frac{dx}{dy}\cdot \frac{dy}{dz}\text{.}\)
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Suppose that on the open interval \(I\text{,}\) \(f\) is a differentiable function that has an inverse function \(f^{-1}\) and \(f'(x)\not= 0\text{.}\) Then \(f^{-1}\) is differentiable and \(\ds \left( f^{-1}(x)\right) '=\frac{1}{f'(f^{-1}(x))}\) for all \(x\) in the domain of \(f^{-1}\text{.}\)
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If the graph of \(f\) is on Figure 5.0.2 to the left, the graph to the right must be that of \(f^\prime\text{.}\)
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The conclusion of the Mean Value Theorem says that the graph of \(f\) has at least one tangent line in \((a,b)\text{,}\) whose slope is equal to the average slope on \([a,b]\text{.}\)
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The linear approximation \(L(x)\) of a function \(f(x)\) near the point \(x=a\) is given by \(L(x)=f'(a)+f(a)(x-a)\text{.}\)
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The graphs in Figure 5.0.3 are labeled correctly with possible eccentricities for the given conic sections:
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Given \(h(x)=g(f(x))\) and the graphs of \(f\) and \(g\) on Figure 5.0.4 then a good estimate for \(h'(3)\) is \(-\frac{1}{4}\text{.}\)
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Answer TRUE or FALSE to the following questions.
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If \(f(x)=7x+8\) then \(f'(2)=f'(17.38)\text{.}\)
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If \(f(x)\) is any function such that \(\ds \lim _{x\to 2}f(x)=6\) the \(\ds \lim _{x\to 2^+}f(x)=6\text{.}\)
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If \(f(x)=x^2\) and \(g(x)=x+1\) then \(f(g(x))=x^2+1\text{.}\)
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The average rate of change of \(f(x)\) from \(x=3\) to \(x=3.5\) is \(2(f(3.5)-f(3))\text{.}\)
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An equivalent precise definition of \(\ds \lim _{x\to a}f(x)=L\) is: For any \(0\lt \epsilon \lt 0.13\) there is \(\delta >0\) such that
\begin{equation*} \mbox{if } |x-a|\lt \delta \mbox{ then } |f(x)-L|\lt \epsilon\text{.} \end{equation*}
The last four True/False questions ALL pertain to the following function. Let
\begin{equation*} f(x)\left\{ \begin{array}{lll} x-4\amp \mbox{if} \amp x\lt 2\\ 23\amp \mbox{if} \amp x=2\\ x^2+7\amp \mbox{if} \amp x>2 \end{array} \right. \end{equation*}
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\(\displaystyle f(3)=-1\)
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\(\displaystyle f(2)=11\)
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\(f\) is continuous at \(x=3\text{.}\)
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\(f\) is continuous at \(x=2\text{.}\)
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Answer TRUE or FALSE to the following questions.
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If a particle has a constant acceleration, then its position function is a cubic polynomial.
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If \(f(x)\) is differentiable on the open interval \((a,b)\) then by the Mean Value Theorem there is a number \(c\) in \((a,b)\) such that \((b-a)f'(c)=f(b)-f(a)\text{.}\)
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If \(\ds \lim _{x\to \infty }\left( \frac{k}{f(x)}\right) =0\) for every number \(k\text{,}\) then \(\ds \lim _{x\to \infty }f(x)=\infty\text{.}\)
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If \(f(x)\) has an absolute minimum at \(x=c\text{,}\) then \(f'(c)=0\text{.}\)
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True or False. Give a brief justification for each answer.
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There is a differentiable function \(f(x)\) with the property that \(f(1)=-2\) and \(f(5)=14\) and \(f^\prime (x)\lt 3\) for every real number \(x\text{.}\)
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If \(f"(5)=0\) then \((5,f(5))\) is an inflection point of the curve \(y=f(x)\text{.}\)
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If \(f^\prime (c)=0\) then \(f(x)\) has a local maximum or a local minimum at \(x=c\text{.}\)
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If \(f(x)\) is a differentiable function and the equation \(f^\prime (x)=0\) has 2 solutions, then the equation \(f(x)=0\) has no more than 3 solutions.
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If \(f(x)\) is increasing on \([0,1]\) then \([f(x)]^2\) is increasing on \([0,1]\text{.}\)
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Answer the following questions TRUE or False.
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If \(f\) has a vertical asymptote at \(x=1\) then \(\ds \lim _{x\to 1}f(x)=L\text{,}\) where \(L\) is a finite value.
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If has domain \([0,\infty )\) and has no horizontal asymptotes, then \(\lim _{x\to \infty }f(x)=\pm \infty\text{.}\)
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If \(g(x)=x^2\) then \(\ds \lim _{x\to 2}\frac{g(x)-g(2)}{x-2}=0\text{.}\)
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If \(f"(2)=0\) then \((2,f(2))\) is an inflection point of \(f(x)\text{.}\)
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If \(f^\prime(c)=0\) then \(f\) has a local extremum at \(c\text{.}\)
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If \(f\) has an absolute minimum at \(c\) then \(f^\prime (c)=0\text{.}\)
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If \(f^\prime (c)\) exists, then \(\ds \lim _{x\to c}f(x)=f(c)\text{.}\)
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If \(f(1)\lt 0\) and \(f(3)>0\text{,}\) then there exists a number \(c\in (1,3)\) such that \(f(c)=0\text{.}\)
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If \(\ds f^\prime (g)=\frac{1}{(3-g)^2}\text{,}\) then \(f(g)\) is differentiable on \((-\infty ,3)\cup (3,\infty )\text{.}\)
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If \(\ds f^\prime (g)=\frac{1}{(3-g)^2}\text{,}\) the equation of the tangent line to \(f(g)\) at \((0,1/3)\) is \(y=\frac{1}{9}g+\frac{1}{3}\text{.}\)
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Are the following statements true or false?
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The points described by the polar coordinates \((2,\pi /4)\) and \((-2,5\pi /4)\) are the same.
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If the limit \(\displaystyle \lim _{x\to \infty }\frac{f^\prime (x)}{g^\prime (x)}\) does not exist, then the limit \(\displaystyle \lim _{x\to \infty }\frac{f(x)}{g(x)}\) does not exist.
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If \(f\) is a function for which \(f"(x)=0\text{,}\) then \(f\) has an inflection point at \(x\text{.}\)
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If \(f\) is continuous at the number \(x\text{,}\) then it is differentiable at \(x\text{.}\)
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Let \(f\) be a function and \(c\) a number in its domain. The graph of the linear approximation of \(f\) at \(c\) is the tangent line to the curve \(y=f(x)\) at the point \((c,f(c))\text{.}\)
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Every function is either an odd function or an even function.
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A function that is continuous on a closed interval attains an absolute maximum value and an absolute minimum value at numbers in that interval.
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An ellipse is the set of all points in the plane the sum of whose distances from two fixed points is a constant.
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For each statement indicate whether is True or False.
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There exists a function \(g\) such that \(g(1)=-2\text{,}\) \(g(3)=6\) and \(g^\prime(x)>4\) for all \(x\text{.}\)
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If \(f(x)\) is continuous and \(f^\prime(2)=0\) then \(f\) has either a local maximum to minimum at \(x=2\text{.}\)
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If \(f(x)\) does not have an absolute maximum on the interval \([a,b]\) then \(f\) is not continuous on \([a,b]\text{.}\)
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If a function \(f(x)\) has a zero at \(x=r\text{,}\) then Newton's method will find \(r\) given an initial guess \(x_0\not= r\) when \(x_0\) is close enough to \(r\text{.}\)
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If \(f(3)=g(3)\) and \(f^\prime(x)=g^\prime(x)\) for all \(x\text{,}\) then \(f(x)=g(x)\text{.}\)
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The function \(\ds g(x)=\frac{7x^4-x^3+5x^2+3}{x^2+1}\) has a slant asymptote.
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For each statement indicate whether is True or False.
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If \(\ds \lim_{x\to a}f(x)\) exists then \(\ds \lim_{x\to a}\sqrt{f(x)}\) exists.
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If \(\ds \lim_{x\to 1}f(x)=0\) and \(\ds \lim_{x\to 1}g(x)=0\) then \(\ds \lim_{x\to 1}\frac{f(x)}{g(x)}\) does not exist.
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\(\ds \sin^{-1}\left(\sin \left(\frac{7\pi}{3}\right)\right)=\frac{7\pi}{3}\text{.}\)
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If \(h(3)=2\) then \(\ds \lim_{x\to 3}h(x)=2\text{.}\)
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The equation \(\ds e^{-x^2}=x\) has a solution on the interval \((0,1)\text{.}\)
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If \((4,1)\) is a point on the graph of \(h\) then \((4,0)\) is a point on the graph \(f\circ h\) where \(f(x)=3^x+x-4\text{.}\)
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If \(-x^3+3x^2+1\leq g(x)\leq (x-2)^2+5\) for \(x\geq 0\) then \(\ds \lim _{x\to 2}g(x)=5\text{.}\)
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If \(g(x)=\ln x\text{,}\) then \(g(g^{-1}(0))=0\text{.}\)
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For each of the following, circle only one answer.
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If \(h(x)=\ln (1-x^2)\) where \(-1\lt x\lt 1\text{,}\) then \(h^\prime(x)=\)
A. \(\ds \frac{1}{1-x^2}\text{,}\) B. \(\ds \frac{1}{1+x}-\frac{1}{1-x}\text{,}\) C. \(\ds \frac{2}{1-x^2}\text{,}\) D. None of these.
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The derivative of \(f(x)=x^2\tan x\) is
A. \(2x\sec^2x\text{,}\) B. \(2x\tan x+x^2\cot x\text{,}\) C. \(2x\tan x+(x\sec x)^2\text{,}\) D. None of these.
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If \(\cosh y=x+x^3y\text{,}\) then at the point \((1,0)\) we have \(y^\prime\)
A. \(0\text{,}\) B. \(3\text{,}\) C. \(-1\text{,}\) D. Does not exist.
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The derivative of \(\ds g(x)=e^{\sqrt{x}}\) is
A. \(e^{\sqrt{x}}\text{,}\) B. \(\sqrt{x}e^{\sqrt{x}-1}\text{,}\) C. \(\frac{0.5e^{\sqrt{x}}}{\sqrt{x}}\text{,}\) D. None of these.
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For each of the following, circle only one answer.
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Suppose \(y^{\prime\prime}+y=0\text{.}\) Which of the following is a possibility for \(y=f(x)\text{.}\)
A. \(y=\tan x\text{,}\) B. \(y=\sin x\text{,}\) C. \(y=\sec x\text{,}\) D. \(y=1/x\text{,}\) E. \(y=e^x\)
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Which of the following is \(\ds \arcsin \left( \sin \left( \frac{3\pi }{4}\right) \right)\text{?}\)
A. \(0\text{,}\) B. \(\ds \frac{\pi }{4}\text{,}\) C. \(\ds -\frac{\pi }{4}\text{,}\) D. \(\ds \frac{5\pi }{4}\text{,}\) E. \(\ds \frac{3\pi }{4}\)
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Let \(f(x)\) be a continuous function on \([a,b]\) and differentiable on \((a,b)\) such that \(f(b)=10\text{,}\) \(f(a)=2\text{.}\) On which of the following intervals \([a,b]\) would the Mean Value Theorem guarantee a \(c\in (a,b)\) such that \(f'(c)=4\text{.}\)
A. \([0,4]\text{,}\) B. \([0,3]\text{,}\) C. \([2,4]\text{,}\) D. \([1,10]\text{,}\) E. \((0,\infty )\)
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Let \(P(t)\) be the function which gives the population as a function of time. Assuming that \(P(t)\) satisfies the natural growth equation, and that at some point in time \(t_0\text{,}\) \(P(t_0)=500\text{,}\) \(P'(t_0)=1000\text{,}\) find the growth rate constant \(k\text{.}\)
A. \(\ds -\frac{1}{2}\text{,}\) B. \(\ds \ln \left( \frac{1}{2}\right)\text{,}\) C. \(\ds \frac{1}{2}\text{,}\) D. \(2\text{,}\) E. \(\ln 2\)
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Suppose that \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) If \(f^\prime(x)>0\) on \((a,b)\text{.}\) Which of the following is necessarily true?
A. \(f\) is decreasing on \([a,b]\text{,}\)
B. \(f\) has no local extrema on \((a,b)\text{,}\)
C. \(f\) is a constant function on \((a,b)\text{,}\)
D. \(f\) is concave up on \((a,b)\text{,}\)
E. \(f\) has no zero on \((a,b)\)
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For each of the following, circle only one answer.
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The equation \(x^5+10x+3=0\) has
A. no real roots, B. exactly one real root, C. exactly two real roots, D. exactly three real roots, E. exactly five real roots
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The value of \(\cosh (\ln 3)\) is
A. \(\ds \frac{1}{3}\text{,}\) B. \(\ds \frac{1}{2}\text{,}\) C. \(\ds \frac{2}{3}\text{,}\) D. \(\ds \frac{4}{3}\text{,}\) E. \(\ds \frac{5}{3}\)
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The function \(f\) has the property that \(f(3)=2\) and \(f'(3)=4\text{.}\) Using a linear approximation to \(f\) near \(x=3\text{,}\) an approximation to \(f(2.9)\) is
A. \(1.4\text{,}\) B. \(1.6\text{,}\) C. \(1.8\text{,}\) D. \(1.9\text{,}\) E. \(2.4\)
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Suppose \(F\) is an antiderivative of \(f(x)=\sqrt[3]{x}\text{.}\) If \(\ds F(0)=\frac{1}{4}\text{,}\) then \(F(1)\) is
A. \(-1\text{,}\) B. \(\ds -\frac{3}{4}\text{,}\) C. \(0\text{,}\) D. \(\ds \frac{3}{4}\text{,}\) E. \(1\)
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Suppose \(f\) is a function such that \(f'(x)=4x^3\) and \(f"(x)=12x^2\text{.}\) Which of the following is true?
A. \(f\) has a local maximum at \(x=0\) by the first derivative test
B. \(f\) has a local minimum at \(x=0\) by the first derivative test.
C. \(f\) has a local maximum at \(x=0\) by the second derivative test.
D. \(f\) has a local minimum at \(x=0\) by the second derivative test.
E. \(f\) has an inflection point at \(x=0\)
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Circle clearly your answer to the following 10 multiple choice question.
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Evaluate \(\ds \frac{d}{dx}\sin (x^2)\)
A. \(2x\cos (x^2)\text{,}\) B. \(2x\sin (x^2)\text{,}\) C. \(2x\cos (x)\text{,}\) D. \(2x\cos (2x)\text{,}\) E. \(2x\cos (2x)\)
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Evaluate \(\ds \lim _{x\to 0^+}\frac{\ln x}{x}\)
A. \(0\text{,}\) B. \(\infty\text{,}\) C. \(1\text{,}\) D. \(-1\text{,}\) E. none of the above
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Evaluate \(\ds \lim _{x\to 0^+}\frac{1-e^x}{\sin x}\)
A. \(1\text{,}\) B. \(-1\text{,}\) C. \(0\text{,}\) D. \(\infty\text{,}\) E. \(\sin e\)
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The circle described by the equation \(x^2+y^2-2x-4=0\) has center \((h,k)\) and radius \(r\text{.}\) The values of \(h\text{,}\) \(k\text{,}\) and \(r\) are
A. \(0\text{,}\) \(1\text{,}\) and \(\sqrt{5}\text{,}\) B. \(1\text{,}\) \(0\text{,}\) and \(5\text{,}\) C. \(1\text{,}\) \(0\text{,}\) and \(\sqrt{5}\text{,}\) D. \(-1\text{,}\) \(0\text{,}\) and \(5\text{,}\) E. \(-1\text{,}\) \(0\text{,}\) and \(\sqrt{5}\)
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The edge of the cube is increasing at a rate of 2 cm/hr. How fast is the cube's volume changing when its edge is \(\sqrt{x}\) cm in length?
A. 6 cm\(^3\)/hr, B. 12 cm\(^3\)/hr, C. \(3\sqrt{2}\) cm\(^3\)/hr, D. \(6\sqrt{2}\) cm\(^3\)/hr, E. none of the above
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Given the polar equation \(r=1\text{,}\) find \(\ds \frac{dy}{dx}\)
A. \(\cot \theta\text{,}\) B. \(-\tan \theta\text{,}\) C. \(0\text{,}\) D. \(1\text{,}\) E. \(-\cot \theta\)
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Let \(A(t)\) denote the amount of a certain radioactive material left after time \(t\text{.}\) Assume that \(A(0)=16\) and \(A(1)=12\text{.}\) How much time is left after time \(t=3\text{?}\)
A. \(\ds \frac{16}{9}\text{,}\) B. \(8\text{,}\) C. \(\ds \frac{9}{4}\text{,}\) D. \(\ds \frac{27}{4}\text{,}\) E. \(4\)
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Which of the following statements is always true for a function \(f(x)\text{?}\)
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If \(f(x)\) is concave up on the interval \((a,b)\text{,}\) then \(f(x)\) has a local minimum \((a,b)\text{.}\)
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It is possible for \(y=f(x)\) to have an inflection point at \((a,f(a))\) even if \(f'(x)\) does not exist at\(x=a\text{.}\)
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It is possible for \((a,f(a))\) to be both a critical point and an inflection point of \(f(x)\text{.}\)
A. i. and ii.
B. only iii.
C. i., ii., and iii.
D. ii. and iii.
E. only i.
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Which of the following statements is always true for a function \(f(x)\text{?}\)
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If \(f(x)\) and \(g(x)\) are continuous at \(x=a\text{,}\) then \(\ds \frac{f(x)}{g(x)}\) is continuous at \(x=a\text{.}\)
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If \(f(x)+g(x)\) is continuous at \(x=a\) and \(f'(a)=0\text{,}\) then \(g(x)\) is continuous ta \(x=a\text{.}\)
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If \(f(x)+g(x)\) is differentiable at \(x=a\text{,}\) then \(f(x)\) and \(g(x)\) are both differentiable at \(x=a\text{.}\)
A. only i.
B. only ii.
C. only iii.
D. i. and ii.
E. ii. and iii.
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The slant asymptote of the function \(\ds f(x)=\frac{x^2+3x-1}{x-1}\) is
A. \(y=x+4\text{,}\) B. \(y=x+2\text{,}\) C. \(y=x-2\text{,}\) D. \(y=x-4\text{,}\) E. none of the above
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This is a multiple choice question. No explanation is required
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The derivative of \(\ds g(x)=e^{\sqrt{x}}\) is
A. \(\sqrt{x}e^{\sqrt{x}-1}\text{,}\)
B. \(2e^{\sqrt{x}}x^{-0.5}\text{,}\)
C. \(\ds \frac{0.5e^{\sqrt{x}}}{\sqrt{x}}\text{,}\)
D. \(e^{\sqrt{x}}\text{,}\)
E. None of these
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If \(\cosh y=x+x^3y\text{,}\) then at the point \((1,0)\) \(y^\prime =\)
A. \(0\text{,}\)
B. \(-1\text{,}\)
C. \(1\text{,}\)
D. \(3\text{,}\)
E. Does not exist
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An antiderivative of \(f(x)=x-\sin x+e^x\) is
A. \(1-\cos x +e^x\text{,}\)
B. \(x^2+\ln x-\cos x\text{,}\)
C. \(\ds 0.5x^2+e^x-\cos x\text{,}\)
D. \(\cos x +e^x+0.5x^2\text{,}\)
E. None of these
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If \(h(x)=\ln (1-x^2)\) where \(-1\lt x\lt 1\text{,}\) then \(h^\prime (x)=\)
A. \(\ds \frac{1}{1-x^2}\text{,}\)
B. \(\ds \frac{1}{1+x}+\frac{1}{1-x}\text{,}\)
C. \(\ds \frac{2}{1-x^2}\text{,}\)
D. \(\ds \frac{1}{1+x}-\frac{1}{1-x}\text{,}\)
E. None of these
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The linear approximation to \(f(x)=\sqrt[3]{x}\) at \(x=8\) is given by
A. \(2\text{,}\)
B. \(\ds \frac{x+16}{12}\text{,}\)
C. \(\ds \frac{1}{3x^{2/3}}\text{,}\)
D. \(\ds \frac{x-2}{3}\text{,}\)
E. \(\ds \sqrt[3]{x}-2\)
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This is a multiple choice question. No explanation is required
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If a function \(f\) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\text{,}\) then there exists \(c\in (a,b)\) such that \(\ds f(b)-f(a)=f^\prime(c)(b-a)\) is:
A. The Extreme Value Theorem,
B. The Intermediate Value Theorem,
C. The Mean Value Theorem,
D. Rolle's Theorem,
E. None of these
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If \(f\) is continuous function on the closed interval \([a,b]\text{,}\) and \(N\) is a number between \(f(a)\) and \(f(b)\text{,}\) then there is \(c\in [a,b]\) such that \(f(c)=N\) is:
A. Fermat's Theorem
B. The Intermediate Value Theorem
C. The Mean Value Theorem
D. Rolle's Theorem
E. The Extreme Value Theorem
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If \(f\) is continuous function on the open interval \((a,b)\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\in (a,b)\) is:
A. The Extreme Value Theorem,
B. The Intermediate Value Theorem,
C. The Mean Value Theorem,
D. Rolle's Theorem,
E. None of these
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A function \(f\) is continuous at a number \(a\) …
A. … if \(f\) is defined at \(a\text{,}\)
B. … if \(\ds \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\) exists,
C. … if \(\ds \lim_{x\to a} f(x)\) exists,
D. … if \(f\) is anti-differentiable at \(a\text{,}\)
E. … if \(\ds \lim_{x\to a} f(x)=f(a)\)
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A function \(f\) is differentiable at a number \(a\) …
A. … if \(\ds \lim_{x\to a} f(x)=f(a)\text{,}\)
B. … if \(\ds \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\) exists,
C. … if \(f\) is defined at \(a\text{,}\)
D. … if \(f\) is continuous at \(a\text{,}\)
E. … if we can apply the Intermediate Value Theorem
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An antiderivative of a function \(f\) …
A. … is a function \(\ds F\) such that \(F^\prime(x)=f(x)\text{,}\)
B. … is a function \(\ds F\) such that \(F(x)=f^\prime(x)\text{,}\)
C.… is a function \(\ds F\) such that \(F^\prime(x)=f^\prime(x)\text{,}\)
D. … is a function \(\ds F\) such that \(F(x)=f(x)\text{,}\)
E. … is a function \(\ds F\) such that \(F"(x)=f(x)\)
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A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that …
A. … \(\ds f^\prime(c)=0\text{,}\)
B. … \(\ds f(c)\) is a local extremum,
C. … either \(\ds f^\prime(c)=0\) or \(f^\prime(x)\) is not defined,
D.… \(\ds (c,f(c))\) is an inflection point,
E. … we can apply the Extreme Value Theorem in the neighbourhood of the point \(\ds (c,f(c))\)
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Answer the following questions. You need not show work for this section.
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What is the period of \(f(x)=\tan x\text{?}\)
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What is the derivative of \(f(x)=x\ln |x| -x\text{?}\)
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If \(y=\sin ^2 x\) and \(\ds \frac{dx}{dt}=4\text{,}\) find \(\ds \frac{dy}{dt}\) when \(x=\pi\text{.}\)
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What is the most general antiderivative of \(f(x)=2xe^{x^2}\text{?}\)
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Evaluate \(\ds \lim _{t\to \infty }(\ln (t+1)-\ln t)\text{?}\)
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Does differentiability imply continuity?
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Convert the Cartesian equation \(x^2+y^2=25\) into a polar equation.
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Simplify \(\cosh ^2x-\sinh ^2x\text{.}\)
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Give an example for the each of the following:
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Function \(F=f\cdot g\) so that the limits of \(F\) and \(f\) at \(a\) exist and the limit of \(g\) at \(a\) does not exist.
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Function that is continuous but not differentiable at a point.
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Function with a critical number but no local maximum or minimum.
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Function with a local minimum at which its second derivative equals 0.
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State the definition of the derivative of function \(f\) at a number \(a\text{.}\)
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State the definition of a critical number of a function.
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State the Extreme Value Theorem.
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Match the start of each definition/theorem with its conclusion.
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The Mean Value Theorem states that …
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The chain rule states that …
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A critical number is a number that …
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The Extreme Value Theorem states that …
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Fermat's Theorem states that
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An antiderivative of a function \(f\) is …
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The natural number \(e\) is …
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An inflection point is a point …
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The derivative of a function \(f\) at a number \(a\) is …
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L'Hospital's Rule states that …
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The Intermediate Value Theorem states that …
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A function \(f\) is continuous at a number \(a\) …
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The Squeeze Theorem states that …
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… if \(f\) is continuous on the closed interval \([a,b]\) and let \(N\) be any number between \(f(a)\) and \(f(b)\text{,}\) \(f(a)\not= f(b)\text{.}\) Then there exists a number \(c\) in \((a,b)\) such that \(f(c)=N\text{.}\)
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… if \(f\) is a function that satisfies the following hypotheses:
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\(f\) is continuous on the closed interval \([a,b]\text{.}\)
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\(f\) is differentiable on the open interval \((a,b)\text{.}\)
Then there is a number \(c\) in \((a,b)\) such that \(\displaystyle f^\prime (c)=\frac{f(b)-f(a)}{b-a}\text{.}\)
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… \(\displaystyle f^\prime (a)=\lim _{h\to 0}\frac{f(a+h)-f(a)}{h}\) if this limit exists.
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… If \(f\) is continuous on a closed interval \([a,b]\text{,}\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\in [a,b]\text{.}\)
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… is in the domain of \(f\) such that either \(f^\prime (c)=0\) or \(f^\prime (c)\) does not exist.
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… on a continuous curve where the curve changes from concave upward to concave downward or from concave downward to concave upward.
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… the base of the exponential function which has a tangent line of slope \(1\) at \((0,1)\text{.}\)
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… If \(f\) and \(g\) are both differentiable then \(\displaystyle \frac{d}{dx}\left[ f(g(x))\right] =f^\prime (g(x))\cdot g^\prime (x)\text{.}\)
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… If \(f(x)\leq g(x)\leq h(x)\) and \(\ds \lim_{x\to a}f(x)=\lim_{x\to a}h(x)=L\) then \(\lim_{x\to a}g(x)=L\text{.}\)
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Source: http://www.sfu.ca/~vjungic/Zbornik2/chap_True_Or_False.html
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