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1 ) Express the distance between the point (3, 0) and the point P(x,y) of the parabola
y=x^2 as a function of x.
1) _____________
(A) D(x)= x^2
(B) D(x)= √(x^2-6x+9)
(C) D(x)=|(x-3)+x^2 |
(D) D(x)= √(x^4+x^2-6x+9)
(E) None of the above
2) If f(x)=cosx and g(x)= x^4, find fog and gof.
2) _____________
(A) (fog)(x)=cos^4〖x; (gof)(x)=cos〖x^4 〗 〗
(B) (fog)(x)=cos〖x^4 〗; (gof)(x)=cos^4x
(C) (fog)(x)=(cosx )^4; (gof)(x)=x^4 cosx
(D) (fog)(x)= (cos^4x )(x^4 ) ; (gof)(x)=(x^4 )(cosx )
(E) None of the above
3) Sketch the graph of the functionf(x)= y= |x|/x. Indicate any points of discontinuity. Explain precisely why your choice of points is/are in fact discontinuities.
3) _____________
4) Sketch the translated circle 4x^2+4y^2-4x+12y=15. Indicate the center and radius.
4) _____________
5) According to the National Highway Traffic Safety Administration, the number of individuals arrested A(t) for driving under the influence of alcohol as a function of their age t is indicated in Table 1. Use the data in Table 1 to:
Estimate the average rate of change in the number of arrests (per 100,000 drivers) for driving under the influence of alcohol between the ages of 18 and 26.
Estimate the instantaneous rate of change at the age of 40 and interpret your answer.
Table 1
Age
(years) Number of Arrests
(per 100,000)
14 113.00
16 308.77
18 421.32
20 485.32
22 519.13
24 534.54
26 537.99
28 533.67
30 524.25
32 511.47
34 496.52
36 480.17
38 462.97
40 445.30
42 427.40
44 409.45
46 391.59
48 373.89
50 356.41
52 339.18
54 322.23
56 305.57
58 289.21
60 273.14
5) _____________
(A) (I) 14.58 arrests (per 100,000)/year; (II)-8.893,decrease of approximately 8.893 arrests (per 100,000)/year
(B) (I) 14.58 arrests (per 100,000)/year; (II)-8.835,decrease of approximately 8.835 arrests (per 100,000)/year
(C) (I) 14.58 arrests (per 100,000)/year; (II) 8.893,increase of approximately 8.893 arrests (per 100,000)/year
(D) (I) 30.35 arrests (per 100,000)/year; (II)-8.950,decrease of approximately 8.950 arrests (per 100,000)/year
(E) None of the above
(Write Solution on next page:)
Solution to #5:
6) Evaluate . 6) _____________
(A) 2/3 (B) 1 (C) 3 (D) does not exist (E) none of the above
7) Given G(x) = , tell where G is continuous. (Give your answer in interval form.)
7) _____________
(A) (- , )
(B) (-5, 5)
(C) (- , 5) (5, )
(D) (- , 0) (0, )
(E) none of the above
8) Given f(x) = , find all points where f is not continuous. For each such point, tell whether or not the discontinuity is removable or not removable. Explain why the point is a discontinuity and then explain why it is a removable or non-removable discontinuity.
8) _____________
(A) 0; removable discontinuity
(B) 0; not removable discontinuity
(C) -2, 2, 0; removable discontinuity
(D) -2, 2, 0; not removable discontinuity
(E) none of the above
9) Find the trigonometric limit: . 9) _____________
(A) 0 (B) 3/2 (C) 1 (D) does not exist (E) none of the above
10) Given L(x) = , tell where L is continuous. (Give your answer in interval form.)
10) ____________
(A) 0
(B) 1
(C) (- , 0) (0, )
(D) (- , )
(E) none of the above
11) State whether the function f(x) = 3 – x2 attains a maximum value or a minimum value (or both) on the interval [-2, 2).
11) ____________
(A) max at 0; min at -2
(B) max at 0
(C) min at -2
(D) min at 0, max at -2
(E) none of the above
12) Find given y =x^3 sin^3(x^3 )
(A) dy/dx=3x^2∙3 sin^2(3x^2 )
(B) dy/dx=3x^2∙sin^3〖(x^3 )+x^3∙3 sin^2〖(x^3 )∙3x^2 〗 〗
(C) dy/dx=3x^2∙sin^3〖(x^3 )+x^3∙3 sin^2〖(x^3 )∙cos〖(x^3 )∙3x^2 〗 〗 〗
(D) dy/dx=3x^2∙sin^3〖(x^3 )+x^3∙3 sin^2〖(x^3 )∙cos^2〖(x^3 )∙3x^2 〗 〗 〗
(E) None of the above
13) Find given x=√(1+cot3t ). 13) ____________
(A) (-3 csc^23t)/(2√(1+cot3t )) (B) 1/2 (1+cot3t )^(-1/2) (C) 1/2 (1+cot3t )^(-1/2) (3) (D) 1/2 (1+csc^23t )^(-1/2)
(E) None of the above
14) Find dy/dx given
y=1/(3x-2)^3
(A) 1/(3(3x-2)^2 (3) )
(B) (-3)/(3x-2)^2
(C) (-9)/(3x-2)^2
(D) (-9)/(3x-2)^4
(E) None of the above
15) When a person gets a single flu shot, the concentration of the drug in milligrams per liter after t hours in the bloodstream is modeled by the following equation. Find the horizontal asymptote of the function F(t) and interpret what the horizontal asymptote represents with respect to the concentration of flu medication in the bloodstream as time passes.
F(t) =
15) ____________
16) After studying calculus for a while, you decide to make a pot of coffee. You pour the coffee grounds into a cone-shaped filter whose radius is 3 cm and height 6 cm. If the coffee grounds are being poured into the filter at a rate of 4 cm3/s, then what is the rate at which the depth of the coffee grounds is rising when they are 2 cm deep?
16) ____________
(A) 1/π cm/s (B) 2/π cm/s (C) 4/π cm/s (D) 6/π cm/s (E) none of the above
17) Differentiate the function h(z) = . 17) ____________
(A) -
(B) -
(C)
(D)
(E) none of the above
18) Use the definition of the derivative to find f '(x) given f(x) = x2 + x + 1.
(You must use the definition and show all your work.)
18) ____________
19) Find the coordinates of the point or point(s) on the curve 2y2 = 5x + 5 which is (are) closest to the origin (0, 0). (You must show all your work in order to receive full credit.)
19) ____________
(A) (0, 1)
(B) (-1, 0)
(C) (0, 0)
(D) not enough information
(E) none of the above
20)
20) Find the limit if it exists
lim┬(x→∞)(√(x^2+x)-x)
(A) 0
(B) 1/4
(C) 1/2
(D) 1
(E) None of the above
20) ____________

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