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.