Section 6.17 True Or False and Multiple Choice Problems
These answers correspond to the problems in Section 5.
True.
False.
False.
True.
True.
False.
True.
False
False.
False. Take \(f(x)=|x|\text{.}\)
False. Find \(\ds \lim _{x\to 2}f(x)\text{.}\)
False. Take \(f(x)=1\) and \(g(x)=2\text{.}\)
True.
False. Apply the Mean Value Theorem.
False. Apply the chain rule.
False.
False.
True.
True. The limit equals \(g'(2)\text{.}\)
False.
True. \(\tan ^2x-\sec ^2 x=-1\text{.}\)
False. \(\ds y=|x^2+x|\) is not differentiable for all real numbers.
False.
False.
False. Take \(\ds \lim _{x\to 5}\frac{x-5}{x-5}\text{.}\)
False. Take \(\ds f(x)=\frac{1}{x-1}\) if \(x>1\) and \(f(x)=0\) if \(x\leq 1\text{.}\)
False.
True.
False. \(c\) might be an isolated point.
False. Take \(f(x)=x^3\text{.}\)
True.
True. \(\frac{1}{\sqrt{x}}\cdot\ln 2^{\sqrt{x}}=\ln 2\text{,}\) \(x>0\text{.}\)
True.
False. \(f(x)\geq 1\text{.}\)
True.
False \(\ds f(x)=-\frac{x^4-256}{4}+3\text{.}\)
False. Take \(f(x)=x^2\) and \(c=1\text{.}\)
True \(\ds \frac{1}{\csc u} =\sin u\) with \(\sin u\not= 0\text{.}\)
True. Use the chain rule.
False. \(\sinh ^2x-\cosh ^2x=-1\text{.}\)
False. \(\ds \int \frac{dx}{x^2+1}=\arctan x+C\text{.}\)
False. \(\ds \int \frac{dx}{3-2x}=-\frac{\ln |3-2x|}{2}+C\text{.}\)
False.
False. Take \(\ds f(x)=\frac{x^2}{x}\) and \(a=0\text{.}\)
False. Take \(\ds f(x)=\frac{x^2}{x}\text{,}\) \(a=-1\text{,}\) and \(b=1\text{.}\)
False. Take \(f(x)=x|x|\text{.}\)
False. Take \(f(x)=1\) if \(x\) is rational and \(f(x)=0\) if \(x\) is irrational.
False. Take \(g(x)=0\text{.}\)
True. Take \(f(x)=\frac{1}{x}\) if \(x\not= 0\text{,}\) \(f(0)=0\text{,}\) and \(g(x)=-f(x)\text{.}\)
False. Take \(f(x)=\sin x\) and \(g(x)=-\sin x\text{.}\)
False. The numerator is an exponential function with a base greater than 1 and the denominator is a polynomial.
False. Take \(\ds f(x)=\tan \frac{\pi x}{2}\text{.}\)
False. Take \(f(x)=10x\) and \(g(x)=20x\) if \(x\in [0,0.5]\) and \(g(x)=10x\) if \(x\in (0.5,1]\text{.}\)
True. Take \(F(x)=f(x)-g(x)\) and apply Rolle's Theorem.
True.
False. The limit is missing.
False. The Squeeze Theorem.
True.
True.
True
False. For \(x\lt 3\) the function is decreasing.
True.
False. It should be \(L(x)=f(a)+f'(a)(x-a)\text{.}\)
False. The eccentricity of a circle is \(e=0\text{.}\)
True. Note that \(g'(x)=-0.5\) and \(f'(3)\approx 0.5\text{.}\)
True.
True.
False. \(f(g(x))=(x+1)^2\text{.}\)
True.
True.
False. \(f(3)=16\text{.}\)
False.
True.
False.
False. It is a quadratic polynomial.
False. The function should be also continuous on \([a,b]\text{.}\)
False. Take \(f(x)=-x\text{.}\)
False. Take \(f(x)=-|x|\text{.}\)
False. Use the Mean Value Theorem.
False. Take \(y=(x-5)^4\text{.}\)
False. Take \(f(x)=x^3\text{,}\) \(c=0\text{.}\)
True. Since \(f\) is differentiable, by Rolle's Theorem there is a local extremum between any two isolated solutions of \(f(x)=0\text{.}\)
False. Take \(f(x)=x-1\text{.}\)
False.
False. Take \(f(x)=\sin x\text{.}\)
False. \(g^\prime(2)=4\text{.}\)
False.
False.
False. Take \(f(x)=|x|\text{.}\)
True. If if is differentiable at \(c\) then \(f\) is continuous at \(c\text{.}\)
False. It is not given that \(f\) is continuous.
True.
True.
True.
False. Take functions \(\displaystyle f(x)=xe^{-1/x^2}\sin (x^{-4})\) and \(\displaystyle g(x)=e^{-1/x^2}\text{.}\)
False. Take \(f(x)=x^4\text{.}\)
False. Take \(f(x)=|x|\) and \(x=0\text{.}\)
True.
False. Take \(f(x)=e^x\text{.}\)
True.
True.
False. Use the Mean Value Theorem.
False. Take \(\displaystyle f(x)=(x-2)^2\text{.}\)
True.
False. Take \(f(x)=\sqrt[3]{x}\text{.}\)
True.
False.
False. Take \(f(x)=x^2-2\) and \(a=0\text{.}\)
False. Take \(\displaystyle f(x)=g(x)=x\text{.}\)
False. Recall, \(\ds \sin^{-1}x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) for all \(x\in[-1,1]\text{.}\)
False. Take \(\displaystyle f(x)=\frac{1}{x-3}\) if \(x\not= 3\) and \(f(3)=2\text{..}\)
True.
True.
True.
True.
B.
C.
C.
C.
B.
C. The range of \(y=\arcsin x\) is \(\ds\left[ -\frac{\pi }{2}, \frac{\pi }{2}\right]\text{.}\)
C. \(\ds \frac{10-2}{4-2}=4\text{.}\)
C. Use \(\ds \frac{dP}{dt}=kP\text{.}\)
B. \(f\) is increasing.
B. Consider \(f(x)=x^5+10x+3\) and its first derivative.
E. \(\ds \cosh (\ln 3)=\frac{3+\frac{1}{3}}{2}\text{.}\)
B. \(f(2.9)\approx 2+4(2.9-3)\text{.}\)
E. \(F(x)=\frac{3}{4}x^{\frac{4}{3}}+\frac{1}{4}\text{.}\)
B.
A.
E. \(\ds \lim _{x\to 0^+}\frac{\ln x}{x}=-\infty\text{.}\)
B. Use L'Hospital's rule.
C. \((x-1)^2+y^2=5\text{.}\)
B. \(\ds \frac{dV}{dt}=3x^2\frac{dx}{dt}\text{.}\)
E. \(\ds \frac{dy}{dt}=\frac{\frac{dy}{d\theta }}{\frac{dy}{d\theta }}\text{.}\)
D. \(\ds A(t)=16\left( \frac{3}{4}\right) ^t\text{.}\)
D. For (1) take \(f(x)=x^3\) on \((0,1)\text{.}\) For (2) take \(f(x)=\sqrt[3]{x}\text{.}\) For (3) take \(f(x)=x^4\text{.}\)
B. For (1) take \(g(x)=0\text{.}\) For (3) take \(f(x)=|x|\text{,}\) \(g(x)=-|x|\text{,}\) and \(a=0\text{.}\)
A.
C.
B. Note \(\ds y^\prime \sinh y=1+3x^2y+x^3y^\prime\text{.}\)
D.
D.
B.
C.
B.
E.
E.
B.
A
C.
\(\pi\text{.}\)
\(f'(x)=\ln |x|\text{.}\)
0. \(\ds \frac{dy}{dt}=2\sin x\cdot \cos x\cdot \frac{dx}{dt}\text{.}\)
\(F(x)=e^{x^2}+C\text{.}\)
\(0\text{.}\) \(\ds \lim _{t\to \infty }\ln \frac{t+1}{t}\text{.}\)
Yes.
\(r=5\text{.}\)
1.
\(\ds F=x\cdot \sin \frac{1}{x}\) and \(a=0\text{.}\)
\(f(x)=|x|\text{.}\)
\(f(x)=x^3\text{.}\)
\(f(x)=x^4\text{.}\)
The derivative of function \(f\) at a number \(a\text{,}\) denoted by \(f^\prime(a)\text{,}\) is \(\ds f'(a)= \lim _{h \to 0} \frac{f (a+h)- f (a)}{h}\) if this limit exists.
A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that \(f'(c)=0\) or \(f'(c)\) does not exist.
If \(f\) is continuous on a closed interval \([a, b]\text{,}\) the \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c\) and \(d\) in \([a, b]\text{.}\)
ii
viii
v
iv
no match
no match
vii
vi
iii
no match
i
no match
ix.