Covers Chapter 3

Sample Quiz Number 3, Math 1251, Scot Adams, Fall 1998

No calculators. 50 minutes. Every problem is worth one point.

%Answers: CHCEEF BFTDAEF CDEET AFFFB FCGYFAN

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1. {d\over dx}((\tan^{-1}(2x))^2) is equal to
(A) 2\tan^{-1}(2x)  \over  (1+4x^2)^2
(B) 4\tan^{-1}(2x)  \over  (1+4x^2)^2
(C) 4\tan^{-1}(2x)  \over  1+4x^2
(D) 2\tan^{-1}(2x)  \over  1+x^2
(E) 4\tan^{-1}(2x)\hfil
%section 3.6 Inverse trigonometric functions

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2. If f(x) is a one-to-one function such that
f(5)=2 and f'(5)=6, then (f^{-1})'(2) is equal to
(A) -7
(B) 7
(C) -1/7
(D) 1/7
(E) 5
(F) 6
(G) 2
(H) 1/6
(I) 1/2
%section 3.6 Inverse trigonometric functions

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3. A radioactive substance decays at a rate proportional to the amount
of the substance. If there are 90 grams of the substance initially,
and 35 grams remain after 5 years, how much is left after 7 years?
Express your answer in grams.
(A) 35\sqrt{6}/90
(B) 2.5
(C) 90(35/90)^{7/5}
(D) (35/7)e^{-90/5}
(E) 90e^{-7/5}
(F) none of these
%section 3.5 Exponential growth and decay

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4. \lim_{x\to0}{3x-\sin3x  \over   x^3} is equal to
(A) 4
(B) 9
(C) -4
(D) 9/6
(E) 9/2
(F) 2
(G) -2
(H) 0
(I) none of these
%section 3.8 Indeterminate forms and l'Hospital's Rule

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5. \lim_{x\to0}{[\cos2x][\ln(x+1)]  \over  \sin3x} is equal to
(A) 0/0
(B) 1/0
(C) 0
(D) 1
(E) 1/3
(F) 2/3
(G) \infty
(H) none of these
%section 3.8 Indeterminate forms and l'Hospital's Rule

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6. The function f(x)=\sin x is one-to-one.
(T) True
(F) False
%section 3.3 Logarithmic functions

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7. {d\over dx}[(x+e^x)^x] is equal to
(A) x(x+e^x)^{x-1}(1+e^x)
(B) [(x+e^x)^x][{x(1+e^x)  \over   x+e^x}+\ln(x+e^x)]
(C) x(x+e^x)^{x-1}
(D) (1+e^x)^x
(E) none of these
%section 3.1 Exponential functions and their derivatives

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8. The derivative (with respect to x) of
{\sec x  \over  \ln x} is
(A) \sec^2x  \over  1/x
(B) (\ln x)(\sec^2x)-(\sec x)(1/x)  \over  (\ln x)^2
(C) (\sec x)(1/x)-(\ln x)(\sec^2x)  \over  (\ln x)^2
(D) (\ln x)(\sec x)-(\sec x)(1/x)  \over  (\ln x)^2
(E) \sec x\tan x  \over  1/x
(F) (\ln x)(\sec x\tan x)-(\sec x)(1/x)  \over  (\ln x)^2
(G) (\sec x)(1/x)-(\ln x)(\sec x\tan x)  \over  (\ln x)^2
(H) (\ln x)(\sec x)-(\sec x)(1/x)  \over  (\ln x)^2
%section 3.4 Derivatives of logarithmic functions

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9. A function is one-to-one if and only if it has an inverse function.
(T) True
(F) False
%section 3.2 Inverse functions

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10. Sin^{-1}(\sin(1+4\pi)) is equal to
(A) 1+4\pi
(B) -1
(C) 0
(D) 1
(E) -6\pi
(F) none of these
%section 3.6 Inverse trigonometric functions

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11. Cos^{-1}(\cos(3\pi)) is equal to
(A) \pi
(B) 3\pi/2
(C) \pi/2
(D) 1
(E) -6\pi
%section 3.6 Inverse trigonometric functions

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12. {d\over dx}e^{x^3+x+1}} is equal to
(A) e^{x^3+x+1}
(B) 1  \over   x^2+x+1}
(C) (4x^3+2x+1)e^{x^3+x+1}}
(D) e^{3x^2+1}
(E) none of these
%section 3.1 Exponential functions and their derivatives

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13. For every real number x, e^{\ln x}=x.
(T) True
(F) False
%section 3.3 Logarithmic functions

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14. Using logarithmic differentiation, compute
dy/dx, where
y={e^x(x-2)^{3/4}\sqrt{x^3+1}  \over  (3x+1)^5(x-1)}.
(A)   [{e^x(x-2)^{3/4}\sqrt{x^3+1}  \over  (3x+1)^5(x-1)}]
      [1+(3/4){1  \over   x-2}+(1/2){3x^2  \over   x^3+1}
             +5{3  \over  3x+1}+{1  \over   x-1}]
(B)   [{e^x(x-2)^{3/4}\sqrt{x^3+1}  \over  (3x+1)^5(x-1)}]
      [e^x+(x-2)^{3/4}+\sqrt{x^3+1}-(3x+1)^5-(x-1)]
(C)   [{e^x(x-2)^{3/4}\sqrt{x^3+1}  \over  (3x+1)^5(x-1)}]
      [1+(3/4){1  \over   x-2}+(1/2){3x^2  \over   x^3+1}
             -5{3  \over  3x+1}-{1  \over   x-1}]
(D)   [{e^x(x-2)^{3/4}\sqrt{x^3+1}  \over  (3x+1)^5(x-1)}]
%section 3.4 Derivatives of logarithmic functions (Logarithmic differentiation)

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15. Compute {d\over dx}\arctan(x^2)}.
(A) 2x  \over  \sqrt{1-x^2}
(B) 2x  \over  \sqrt{1-x^4}
(C) 2x  \over  1+x^2
(D) 2x  \over  1+x^4
(E) 1  \over  \sqrt{1-x^4}
(F) \arcsin(2x)
%section 3.6 Inverse trigonometric functions

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16. \lim_{x\to-\infty}x+e^x} is equal to
(A) 0
(B) 1
(C) -1
(D) \infty
(E) -\infty
%section 3.8 Indeterminate forms and l'Hospital's Rule

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17. The range of \arccos is
(A) (-\infty,\infty)
(B) (-\pi/2,\pi/2)
(C) [-\pi/2,\pi/2]
(D) (0,\pi)}
(E) [0,\pi]
(F) (-1,1)
(G) [-1,1]}
(H) (0,\infty)
(I) [0,\infty)
(J) none of these
%section 3.6 Inverse trigonometric functions

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18. The function f(x)=-2x+3 is one-to-one
(T) True
(F) False
%section 3.2 Inverse functions

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Note: \root3\of{...} means the cube root of ...
19. {d\over dx}  \ln\root3\of{x-1  \over   x+1} is equal to
(A) (1/3)  [{1  \over   x-1}  -  {1  \over   x+1}]
(B) \root3\of{x+1  \over   x-1}
(C) \root3\of{x-1  \over   x+1}
(D) x+1  \over   x-1
(E) x-1  \over   x+1
(F) (4/3)[{1  \over   x-1}-{1  \over   x+1}]
(G) \ln({x-1  \over   x+1})
%section 3.4 Derivatives of logarithmic functions

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20. {d\over dx}\cos(e^x) is equal to
(A) \sin(e^x)
(B) e^{\sin x}
(C) e\sin(e^x)
(D) -2\sin(e^x)
(E) e^x\sin(e^x)
(F) -e^x\sin(e^x)
(G) 2\sin(e^x)\cos(e^x)
(H) -2\sin(e^x)\cos(e^x)
(I) none of these
%section 3.1 Exponential functions and their derivatives

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21. Compute \lim_{x\to\infty}(1+x^{-3})^{x^2}.
(A) 0
(B) \infty
(C) e
(D) e^{2/3}
(E) e^{3/2}
(F) 1
(G) 2
(H) none of the above
%section 3.8 Indeterminate forms and l'Hospital's rule

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22. [d/dx][(\cos x)^{\sin x}] is equal to
(A) (\cos x)(\ln(\sin x))
(B) [(\cos x)^{\sin x}]
    [{\cos^2x  \over  \sin x}+(\sin x)(\ln(\sin x))]
(C) [(\cos x)^{\sin x}]
    [{\cos^2x  \over  \sin x}-(\sin x)(\ln(\sin x))]
(D) (\cos x)^{-\sin x}
(E) (\sin x)(\ln(\cos x))
(F) none of these
%section 3.4 Derivatives of logarithmic functions (Logarithmic differentiation)

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23. [d/dx][\sin^{-1}(x^2+2x+1)] is equal to
(A) \sin^{-1}(2x+2)
(B) 2x+2  \over  \sqrt{1-(x^2+2x+1)^2}
(C) -2x-2  \over  \sqrt{1-(x^2+2x+1)^2}
(D) 2x+2  \over  1+(x^2+2x+1)^2
%section 3.6 Inverse trigonometric functions

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24. The function f(x)=x^2 has an inverse
(T) True
(F) False
%section 3.2 Inverse functions

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25. Let f(x)=3x and let g be the inverse of f. Then g(x)
is equal to
(A) -3x
(B) 3x 
(C) x/3
(D) -x/3
(E) (x-3)/3
(F) none of these
%section 3.2 Inverse functions

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26. Suppose that f(1)=10, that f'(1)=5
and that f is one-to-one. Let g=f^{-1}.
Answer the following three questions:
Q1: Does this give enough information to compute g'(1)?
Q2: Does this give enough information to compute g'(5)?
Q3: Does this give enough information to compute g'(10)?
(A) YYY
(B) YYN
(C) YNY
(D) YNN
(E) NYY
(F) NYN
(G) NNY
(H) NNN
%section 3.2 Inverse functions

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27. Suppose that f is a one-to-one function such that
the domain of f is (-\infty,\infty), such that the range of f
is (-\infty,\infty) and such that f'(x)>0, for all x\in(-\infty,\infty).
Let g=f^{-1}.
Does it necessarily follow that g'(x)>0, for all x\in(-\infty,\infty).
(Y) Yes
(N) No
%section 3.2 Inverse functions


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28. Compute [d/dx][\log_5(x)]
(A) \log_5(x)
(B) 1  \over   x
(C) 1  \over   x\log_5(x)
(D) 5^x
(E) 5^x\ln 5
(F) none of these
%section 3.4 Derivatives of logarithmic functions

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29. Compute \lim_{x\to\infty}
5x^7+\log_3(x)+\sin(e^x)  \over  8x^7+3x^4+4x^3+7-\cos(e^{x^2})
(A) 5/8
(B) -6
(C) \infty
(D) -\infty
(E) 0
(F) does not exist
(G) none of these
%section 3.8 Indeterminate forms and l'Hospital's rule

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30. Suppose f and g are functions, and suppose that
\lim_{x\to1}f(x)=1} and \lim_{x\to1}g(x)=0.
Does it follow that \lim_{x\to1}  {f(x)  \over   g(x)}  =  \infty?
(Y) Yes
(N) No
%section 3.8 Indeterminate forms and l'Hospital's rule

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