Covers Chapter 2

Sample Quiz Number 2, Math 1251, Scot Adams, Fall 1998

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

%Answers: EAFH DGTFH EDAJ ETEFDG ABFCDF AAFFF

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1. Let f(x)=(2x^2-7x+8)^4. Then f'(1) is equal to
(A) 12
(B) 4(2-7+8)^3
(C) (2-7+8)^3
(D) (2-7+8)^5
(E) 4(2-7+8)^3(4-7)
(F) 5(2-7+8)^4(4-7)
(G) 4(4-7)^3
(H) 5(4-7)^4
(I) 3(4-7)^2
(J) 144
(K) none of these
%section 2.5 The chain rule

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2. Let f(x)={x^4-3   \over   3x^2+1}. Then f'(3) is equal to
(A) (3\cdot3^2+1)(4\cdot3^3)-(3^4-3)(6\cdot3)   \over   (3\cdot3^2+1)^2
(B) (3^4-3)(6\cdot3)   \over   (3\cdot3^2+1)^2-(3\cdot3^2+1)(4\cdot3^3)
(C) 3\cdot3^2+1   \over   3^4-3
(D) 3^4-3   \over   3\cdot3^2+1}
(E) 3
(F) \left[3^4-3   \over   3\cdot3^2+1\right]
    \left[(3\cdot3^2+1)(4\cdot3^3)-(3^4-3)(6\cdot3)   \over   (3\cdot3^2+1)^2\right]
(G) none of these
%section 2.2 Differentiation formulas 

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3. (d/dx)[\sin(2x)] is equal to
(A) \cos(2x)
(B) \cos(2)
(C) [\sin(2x)][2x]
(D) [\cos(2x)][2x]
(E) [\sin(2x)][2]
(F) [\cos(2x)][2]
(G) [\sin(2)][2x]
(H) [\cos(2)][2x]
(I) [\sin(2)][2]
(J) [\cos(2)][2]
(K) none of these
%section 2.5 The chain rule

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4. {d\over dx}(\tan^48x) is equal to
(A) \sec^2(8x)
(B) (\sec^2)^4(8x)
(C) \tan^4(8)
(D) 4[\tan^38x]
(E) 4[\tan^3x][8]
(F) 4[\tan^38x][8]
(G) 4[\tan^38x][\sec^2x][8]
(H) 4[\tan^38x][\sec^28x][8]
(I) 4[\tan^38x][\sec^28x][-8]
(J) none of these
%section 2.5 The chain rule

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5. The equation x^2y+xy^2-y=5 defines y implicitly as a function
of x. Then dy/dx is equal to
(A) x^2y+xy^2-y
(B) 2xy+x^2y+y^2+2xyy'-y'
(C) 2xy+y^2   \over    x^2+2xy-1
(D) -2xy+y^2   \over    x^2+2xy-1
(E) -5{2xy+y^2   \over    x^2+2xy-1}
(F) -2x^2y+y^3   \over    x^2+2x^2y-1
(G) 5
(H) none of these
%section 2.6 Implicit differentiation

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6. If we estimate \sqrt{64.6} by differentials, we get
(A) 8
(B) 8 - {0.6 \over 2\sqrt{8}}
(C) 8+\sqrt{64}
(D) 8 - {\sqrt{64} \over 64.6\root3\of{64}}
(E) 64 - {0.6 \over 2\sqrt{8}}
(F) 64 + {0.6 \over 2\sqrt{8}}
(G) 8 + {0.6 \over 16}
%section 2.9 Differentials; linear and quadratic approximations

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7. If f(t)=\cos t, then f''(t)=-f(t).
(T) True
(F) False
%section 2.7 Higher derivatives

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8. (Note: The area enclosed in a circle of radius r is \pi r^2.)
A circle is expanding.  The area enclosed in the circle is increasing
at the rate of 10 square inches per second.  When the radius is 8
inches, the radius of the circle is increasing at what rate?
Express your answer in inches per second.
(A) -4\pi   \over   \pi^2\sec^2(3\pi/4)
(B) -\pi^2\sec^2(3\pi/4)
(C) \pi(4^2)
(D) \pi(8^2)
(E) 16\pi
(F) 10   \over   16\pi
(G) 5   \over   16\pi
(H) none of these
%section 2.8 Related rates

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9. Assume that f(1)=3 and g(3)=5, so that (g\circ f)(1)=5.
Assume that f'(1)=3,000 and g'(3)=2,000. Then (g\circ f)'(1) is
(A) 1
(B) 3
(C) 5
(D) 6
(E) 2,000
(F) 3,000
(G) 6,000
(H) 6,000,000
(I) none of these
%section 2.5 The chain rule

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10. Let f(x) be defined by
f(x)=\cases{
x^2, &if x<0;\cr
0, &if 0\le x\le1;\cr
x^3, &if x>1.\cr}
Then f is differentiable
(A) everywhere;
(B) nowhere;
(C) except at x=-1;
(D) except at x=0;
(E) except at x=1;
(F) except at x=0 and x=1;
(G) except at x=-1 and x=1;
(H) except at x=-1, x=0 and x=1;
(I) none of these
%section 2.1 Derivatives

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11. Let f(x) = {2x+1 \over 3x+4}. Then f'(x) is equal to
(A) (3x+4)(2)-(2x+1)(3)   \over   3x+4
(B) (3)(2x)-(2x+1)(3x+4)   \over   (3x+4)^2
(C) -2   \over   3
(D) (3x+4)(2)-(2x+1)(3)   \over   (3x+4)^2
(E) 2   \over   3
(F) -(3x+4)(2)-(2x+1)(3)   \over   3x+4
(G) -(3)(2x)-(2x+1)(3x+4)   \over   (3x+4)^2
(H) -(2x+1)(3)-(3x+4)(2)   \over   (3x+4)^2
(I) -(3x+4)(2)-(2x+1)(3)   \over   (3x+4)^2
(J) (2x+1)(3)-(3x+4)(2)   \over   (3x+4)^2
(K) none of these
%section 2.2 Differentiation formulas

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12. The equation of the tangent line to y=x^2 at the point (2,4) is
(A) y-4=4(x-2)
(B) y+4=4(x+2)
(C) y-4=2x(x-2)
(D) y+4=2x(x+2)
(E) y-2=4(x-4)
(F) y+2=4(x+4)
(G) y-2=2x(x-4)
(H) y+2=2x(x+4)
(I) y=x
(J) none of these
%section 1.7 Tangents, velocities and other rates of change

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13. Find the derivative of f(x)=\sqrt{x^2+2x+3}
(A) \sqrt{2x+2}
(B) x/\sqrt{x^2+2x+3}
(C) (1/2)(x^2+2x+3)^{-1/2}
(D) \sqrt{x}
(E) (x^2+2x+3)^{-1/2}(2x+2)
(F) \sqrt{x^2+2x+3}
(G) \sqrt{x^2+2x+3}-\sqrt{x}
(H) 2x+2
(I) 1/(2\sqrt{x})
(J) none of these
%section 2.5 The chain rule

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14. Find the derivative of
f(x) = {\sin x \over x}
(A) \cos x   \over   1
(B) (\sin x)(x)-(x)(\sin x)   \over    x^2
(C) (\sin x)(1)-(x)(\cos x)   \over    x^2
(D) (x)(\sin x)-(\sin x)(x)   \over    x^2
(E) (x)(\cos x)-(\sin x)(1)   \over    x^2
(F) (x)(\sin x)-(\sin x)(x)   \over   \sin^2x
(G) (x)(\cos x)-(\sin x)(1)   \over   \sin^2x
(H) none of these
%section 2.5 The chain rule

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15. Two different functions can have the same derivative.
(T) True
(F) False
%section 2.2 Differentiation formulas

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16. The equation x^2+y^2=10 defines y implicitly as a function
of x. Then dy/dx is equal to
(A) 2x+2y
(B) -2x-2y
(C) 10-x^2
(D) -10+x^2
(E) -x/y
(F) x/y
(G) [2x]/[2y-1]
(H) -[2x]/[2y-1]
(I) none of these
%section 2.6 Implicit differentiation

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17. Let f be a differentiable function. Find a formula
for {d\over dx}\sin(f(x)).  Your answer should be in
terms of f(x) and f'(x).
(A) 2f(x)
(B) (f'(x))^2
(C) 2x
(D) 2f(x)f'(x)
(E) \cos(f(x))
(F) [\cos(f(x))][f'(x)]
(G) [\cos(f(x)f'(x))][f''(x)]
(H) \cos(f'(x))
(I) none of these
%section 2.5 The chain rule

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18. Compute \lim_{t\to(\pi/4)^-}\tan(t).
(A) -\infty
(B) \infty
(C) 0
(D) 1
(E) -1
(F) \sqrt{2}/2
(G) -\sqrt{2}/2
(H) \sqrt{3}/2
(I) -\sqrt{3}/2
(J) none of these
%section 2.5 The chain rule

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19. Let f(x)=x^3-9. Starting with the approximate
root x_1:=2, use Newton's method to find the second
approximation x_2.
(A) 2
(B) 2+(4/5)
(C) 2-(4/5)
(D) 2+(4/5)^2
(E) 2-(4/5)^2
(F) 2+(-1/12)
(G) 2-(-1/12)
(H) 2+(-1/12)^2
(I) 2-(-1/12)^2
(J) none of these
%section 2.10 Newton's method

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In problems 20 through 24, use the following information:
Two particles are traveling in the plane. The first
one travels along the x-axis, and its position at time t is
given by (2t^2,0). The second one travels along the y-axis,
and its position at time t is given by (0,4t^3).
20. Find the location of the first particle at time t=1.
(A) (2,0)
(B) (0,1)
(C) (1,0)
(D) (0,2)
(E) (4,0)
(F) (0,-2)
(G) (-2,0)
(H) (0,-4)

%*****************

21. Find the location of the second particle at time t=1.
(A) (8,0)
(B) (0,4)
(C) (4,0)
(D) (0,8)
(E) (2,0)
(F) (0,1)
(G) (1,0)
(H) (0,2)

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22. Find the distance from the first particle to the
second at time t=1.
(A) 2
(B) \sqrt{3}
(C) \sqrt{t^2+t^3}
(D) \sqrt{2}
(E) \sqrt{2^2+3^2}
(F) \sqrt{2^2+4^2}
(G) \root3\of{2^2+4^2}
(H) \sqrt{2t^2+4t^3}
(I) none of these

%*****************

23. Find the distance from the first particle to the
second at an arbitrary time t.
(A) \sqrt{2t^2+4t^3}
(B) \sqrt{2t^4+4t^6}
(C) \sqrt{4t^4+16t^6}
(D) 4t^4+16t^6
(E) t^2+t^3
(F) \sqrt{t^4+t^6}
(G) \sqrt{t^2+t^3}
(H) t^2-t^3

%*****************

24. Find how fast the distance (between the two
particles) is increasing at an arbitrary time t.
(A) 1   \over   2\sqrt{4t^4+16t^6}
(B) 4t^4+16t^6   \over   \sqrt{4t^4+16t^6}
(C) 4(4t^3)+16(6t^5)   \over   \sqrt{4t^4+16t^6}
(D) 4(4t^3)+16(6t^5)   \over   2\sqrt{4t^4+16t^6}
(E) 2t^3+3t^5   \over   \sqrt{t^4+t^6}
(F) (1/2)(t^4+t^6)^{-1/2}
(G) \sqrt{4t^3+6t^5}
(H) 2t+3t^2
%section 2.5 The chain rule

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25. Let f(x)=e^{-2x}. Then f'''(x) is equal to
(A) e^{-2x}
(B) -2e^{-2x}
(C) -2e^{27x}
(D) 27e^{-2x}
(E) 4e^{-2x}
(F) -8e^{-2x}
(G) 16e^{27x}
(H) -27e^{-2x}
(I) -27e^{-27x}
(J) e^{-8x}
%section 2.7 Higher derivatives

%*****************

26. Compute \lim_{h\to0} { \sin(\pi+h)-\sin\pi \over h}.
(A) -1
(B) -\sqrt{3}/2
(C) -\sqrt{2}/2
(D) -1/2
(E) 0
(F) 1/2
(G) \sqrt{2}/2
(H) \sqrt{3}/2
(I) 1
(J) none of these
%section 2.1 Derivatives

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27. [d/dx]^{100}[\sin x] is equal to
(A) \sin x
(B) -\sin x
(C) \cos x
(D) -\cos x
(E) \tan x
(F) -\tan x
(G) \sec^2x
(H) -\sec^2x
%section 2.7 Higher derivatives

%*****************

28. Find the 1-jet of f(x)=x^2 at 4.
(A) x^2
(B) 2x
(C) 16
(D) 8
(E) 8 and then 16
(F) 16 and then 8
(G) x^2 and then 2x
(H) none of these
%section 2.9 Differentials; Linear and Quadratic Approximations

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29. Find the solution to the differential equation
f''=-f satisfying f(0)=1 and f'(0)=3.
(A) f(x)=e^x
(B) f(x)=e^{-x}
(C) f(x)=e^x+3e^{-x}
(D) f(x)=\sin x
(E) f(x)=\cos x
(F) f(x)=\cos x+3\sin x
(G) f(x)=\sin x+3\cos x
(H) f(x)=e^x+3e^{-1}+\sin x+3\cos x
(I) none of these
%section Spring motion 

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30. Find [d/dx][\ln(\cos15)]
(A) 1/\cos15
(B) \ln(-\sin15)
(C) \ln(\cos15)
(D) 1/(-\sin15)
(E) [1/(\cos15)][-\sin15]
(F) none of these
%section 2.2 Differentiation formulas

%*****************

END OF TEST

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