MATH 140 EXAM 1 Professor: Dr. J. Beyers NAME________________________ INSTRUCTIONS The exam is worth 100 points. There are 20 problems (each worth 5 points). This quiz is open book and open notes, unlimited time. This means that you may refer to your textbook, notes, and online classroom materials. You may take as much time as you wish, provided you turn in your exam no later than the due date posted in our course syllabus. You must show your work to receive full credit. If you do not show your work, you may earn only partial or no credit at the discretion of the professor. If you have any questions, please contact me by e-mail (jbeyers@umuc.edu). (1) _____ (11) _____ (2) _____ (12) _____ (3) see item #3 (13) _____ (4) see item #4 (14) _____ (5) _____ (15) _____ (6) _____ (16) _____ (7) _____ (17) _____ (8) _____ (18) see item #18 (9) _____ (19) _____ (10) _____ (20) _____ 1 1) Express the distance between the point (3, 0) and the point y=x 2 as a function of P ( x , y ) of the parabola x . 1) _____________ ( A ) D ( x )=x 2 ( B ) D ( x )= x2 6 x+ 9 ( C ) D ( x ) =|( x3 ) +x 2| ( D ) D ( x )= x 4 + x 26 x +9 of the above (E ) 2) If f ( x )=cos x and g ( x ) =x 4 , find f o g and gof . 2) _____________ ( A ) ( f o g ) ( x )=cos 4 x ; ( g o f )( x )=cos x 4 ( B )( f o g ) ( x )=cos x 4 ; ( g o f ) ( x )=cos 4 x ( C ) ( f o g )( x )=( cos x )4 ; ( g o f )( x )=x 4 cos x ( D )( f o g ) ( x ) =( cos 4 x ) ( x 4 ) ; ( g o f )( x )=( x 4 ) ( cos x ) of the above (E ) |x| 3) Sketch the graph of the function f ( x )= y = x . Indicate any points of discontinuity. Explain precisely why your choice of points is/are in fact discontinuities. 3) _____________ y x 4) Sketch the translated circle 2 2 4 x +4 y 4 x+12 y=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: (I) 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. (II) Estimate the instantaneous rate of change at the age of 40 and interpret your answer. Table 1 Age Number of (years) Arrests (per 100,000) 14 113.00 16 308.77 18 421.32 20 485.32 22 519.13 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 534.54 537.99 533.67 524.25 511.47 496.52 480.17 462.97 445.30 427.40 409.45 391.59 373.89 356.41 339.18 322.23 305.57 289.21 273.14 5) _____________ ( A ) ( I ) 14.58 arrests per 100,000 per 100,000 ; ( II )8.893, decrease of approximately 8.893 arrests year year ( B )( I ) 14.58 arrests per 100,000 per 100,000 ; ( II ) 8.835, decrease of approximately 8.835 arrests year year ( C ) ( I ) 14.58 arrests per 100,000 per 100,000 ; ( II ) 8.893,increase of approximately 8.893 arrests year year ( D )( I ) 30.35 arrests per 100,000 per 100,000 ; ( II )8.950, decrease of approximately 8.950 arrests year year of t h e above ( E) (Write Solution on next page:) Solution to #5: 6) Evaluate lim x 3 (A) 2/3 (B) 1 2x 6 . 6) _____________ x 2 4x 3 (C) 3 (D) does not exist (E) none of the above 7) Given G(x) = 2x 2 5 2x 10 , 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 1 sin x , 2x x 2 4, x 0 x 0 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: (A) 0 (B) 3/2 (C) 1 10) Given L(x) = (A) 0 (B) 1 (C) (- x sin x lim sin 3x x 0 2x . (D) does not exist 9) _____________ (E) none of the above , tell where L is continuous. (Give your answer in interval form.) 10) ____________ , 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 (A) dy dx given y=x 3 sin 3 ( x 3 ) dy =3 x 2 3 sin 2 ( 3 x2 ) dx (B ) dy =3 x 2 sin3 ( x3 ) + x 3 3 sin2 ( x 3 ) 3 x 2 dx (C) dy =3 x 2 sin 3 ( x 3 ) + x 3 3 sin2 ( x3 ) cos ( x 3 ) 3 x 2 dx ( D) dy 2 3 3 3 2 3 2 3 2 =3 x sin ( x )+ x 3 sin ( x ) cos ( x ) 3 x dx of the above (E ) 13) Find dx dt given . 13) ____________ x= 1+ cot 3 t 2 1 /2 ( A ) 3 csc 3 t ( B ) 1 (1+cot 3 t )1/ 2 ( C ) 1 ( 1+cot 3 t )1 /2 ( 3 ) ( D ) 1 ( 1+csc 2 3 t ) 2 2 2 2 1+ cot 3t of the above (E ) 14) Find dy /dx y= given 1 ( 3 x2 )3 (A) 1 2 3 ( 3 x2 ) (3 ) (B ) 3 ( 3 x2 )2 (C) 9 ( 3 x2 )2 ( D) 9 ( 3 x2 )4 of the above (E ) 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) = 7. 6t t 2 0. 2 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 cm 3/s, then what is the rate at which the depth of the coffee grounds is rising when they are 2 cm deep? 16) ____________ of the above ( A ) 1/ cm/s ( B ) 2/ cm/s ( C ) 4/ cm /s ( D ) 6 / cm/ s ( E ) 17) Differentiate the function h(z) = 1 3 (A) - 3 3 (B) - 5 3z 1 (5 3z) 4 / 3 (C) 1 (5 3z) 4 / 3 (D) 3 (5 3z) 2 / 3 (E) none of the above 5 3z . 17) ____________ 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.) (A) (0, 1) (B) (-1, 0) (C) (0, 0) (D) not enough information (E) none of the above 19) ____________ 20) 20) Find the limit if it exists lim ( x 2+ x x ) x (A)0 ( B ) 1/ 4 ( C ) 1/2 ( D)1 of the above (E ) 20) ____________