Calculus

  1. Calculus check
    Can someone check my answers: 1) Use geometry to evaluate 6 int 2 (x) dx where f(x) = { |x|, -2 <= x <= 2} {2, 2 < x <= 4} {-x+4 4 < x <= 6} for my answer I got 0.512 2) The base of a solid is bounded by y = √x+2, the x-axis and the line x = 1. The cross ...
  2. Check Calculus answers please
    Can someone check my answers: 1) Use geometry to evaluate 6 int 2 (x) dx where f(x) = { |x|, -2 <= x <= 2} {2, 2 < x <= 4} {-x+4 4 < x <= 6} for my answer I got 0.512 2) The base of a solid is bounded by y = √x+2, the x-axis and the line x = 1. The cross ...
  3. Rate based Calculus Question help
    Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of...
  4. Calculus help please
    Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = -x^3 and y = -x
  5. Help me check my calculus questions
    Can someone check my answers: 1) Use geometry to evaluate 6 int 2 (x) dx where f(x) = { |x|, -2 <= x <= 2} {2, 2 < x <= 4} {-x+4 4 < x <= 6} for my answer I got 0.512 2) R is the first quadrant region enclosed by the x axis, the curve y = 2x+a and the line x...
  6. AP Calculus
    Can someone check my answers: 1) Use geometry to evaluate 6 int 2 (x) dx where f(x) = { |x|, -2 <= x <= 2} {2, 2 < x <= 4} {-x+4 4 < x <= 6} for my answer I got 0.512 2) R is the first quadrant region enclosed by the x axis, the curve y = 2x+a and the line x...
  7. Calculus
    Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of...
  8. AP Calculus
    Can someone help me solve these two questions? Thanks 1) Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = -x^3 and y = -x. 2) A rock is thrown upward with a speed of 48 feet per second from the edge of a cliff 400 feet above the ground. ...
  9. Advanced Placement Calculus
    Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of...
  10. Calculus
    Can someone help me solve these two questions? Thanks 1) Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = -x^3 and y = -x. 2) A rock is thrown upward with a speed of 48 feet per second from the edge of a cliff 400 feet above the ground. ...
  11. Calculus 2
    Find the power series representation about the center c=0 of (a) f(x)= 1/(1+x^2) (b) Use part (a) to find the power series of g(x)=ln(1+x^2) -Thank You
  12. Pre-Calculus Math
    Jack wants to install a swimming pool that is 12m long and 8m wide. He wants to install a rubberized safety border of uniform width around the pool. The width of the safety border is represented by x and the area of the safety border is 44 m^2. a) Write an expression in terms ...
  13. Help me check my calculus answers
    1. Which of the following functions grows the fastest as x goes to infinity? - 2^x - 3^x - e^x <-- my answer - x^20 2. Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity. - f(x) grows faster than g(x) as x goes to infinity. - g(x) grows ...
  14. Calculus
    A radar gun was used to record the speed of a car (in feet per minute) during selected times in the first 2 minutes of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the car covered during those 2 minutes. t: 0 0.3 1.0 1.6 2 v(t): 0 24.5 27.8 28.3 29.0
  15. Math - Calculus
    Problem solving with derivatives. A rectangular box has square base of edge length x cm. Its framework of 12 edges is constructed from wire of total length 36cm. Find: i. the height of the box in terms of x ? ii. the volume of the box in terms of x? ii. the value for x which ...
  16. Math - Calculus
    Problem solving with derivatives. Question: A piece of wire length of 30cm is cut into 2 sections. Each section is then bent into the shape of a square. Find the smallest possible value of the sum of the areas of the two squares.
  17. Math - Calculus
    Question: A box in the shape of a cuboid with a square base is to be made so that the sum of its dimensions (l + b + h) is 20cm. Find its maximum value.
  18. Math - Calculus
    Problem solving with derivatives. Question: A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares are cut out of each corner and the sides are turned up to form an open rectangular box. What is the maximum volume of the box?
  19. Math - Calculus
    A boat is pulled into a wharf by a rope at a speed of 20m/min. If the rope is attached to a point on the boat 7m vertically below the wharf, at what rate is the rope being drawn in when the boat is 24m from the wharf?
  20. Math - Calculus
    A circular oil slick floats on the surface of still water. Its area is increasing at a rate of 10m^2/min. At what rate is the radius (r metres) increasing?
  21. Pre-Calculus Math
    A rectangular field is to be enclosed by a fence and divided into two smaller plots by a fence parallel to one of the side. Find the dimensions of the largest such field if 1200 m of fencing material is available. What is the area of this field and what are the dimensions that...
  22. Pre-Calculus Math
    An amusement park charges $8 admission and averages 2000 visitors per day. A survey shows that, for each $1 increase in admission cost, 100 fewer people would visit the park. Find the admission cost and number of visitors that gives the maximum revenue.
  23. Pre-Calculus Math
    The Summer Theatre charges $4 per ticket, and it sells an average of 400 tickets nightly. The manager estimates that the ticket sales would decrease by 50 for each $1 increase in the ticket cost. What is the most profitable price to charge?
  24. Calculus
    A radar gun was used to record the speed of a car (in feet per minute) during selected times in the first 2 minutes of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the car covered during those 2 minutes. Give a 2 decimal place answer and include ...
  25. Calculus check please
    1. Which of the following functions grows the fastest as x goes to infinity? - 2^x - 3^x - e^x (my answer) - x^20 2. Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity. - f(x) grows faster than g(x) as x goes to infinity. - g(x) grows faster ...
  26. Calculus check my answers
    1. The general solution of the differential equation dy - 0.2x dx = 0 is a family of curves. These curves are all: a. lines b. hyperbolas c. parabolas (my answer) d. ellipses 2. The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 ...
  27. Calculus check
    Help me check my calculus 1. The graph of f '(x) is continuous, positive, and has a relative maximum at x = 0. Which of the following statements must be true? a. The graph of f is always concave down. b.The graph of f is always increasing. c. The graph of f has a relative ...
  28. Calculus
    Can someone help me check my answers 1. Find the range of the function f(x) = x∫-6 √36-t^2 dt - [-6, 0] - [0, 6] - [0, 9π] (my answer) - [0, 18π] 2. Use the graph of f(t) = 2t + 2 on the interval [-1, 4] to write the function F(x), where f(x) = x∫-1 f(t) dt a. F(x) = ...
  29. Calculus
    The differential equation dy/dx = (y-2)/(y-1) I.produces a slope field with horizontal tangents at y = 2 II.produces a slope field with vertical tangents at y = -1 III.produces a slope field with columns of parallel segments A. I only B. II only C. I and II D. III only
  30. Calculus
    For an object whose velocity in ft/sec is given by v(t) = -2t^2 + 4, what is its distance travelled, in feet, on the interval t = 0 to t = 2 secs?
  31. Calculus 2
    Consider the polar curve: r=3cos(3ϴ) a) Find the area of the first leaf of the graph. -Thank you.
  32. Calculus
    A radar gun was used to record the speed of a car (in feet per minute) during selected times in the first 2 minutes of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the car covered during those 2 minutes. t 0 0.3 1.0 1.6 2 v(t) 0 24.5 27.8 28.3 29.0
  33. Calculus
    The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of 2∫1 f(x) dx . x 1 1.1 1.3 1.6 1.7 1.8 2.0 f(x) 1 3 5 8 10 11 14
  34. Calculus
    An object has a constant acceleration of 40 ft/sec2, an initial velocity of -20 ft/sec, and an initial position of 10 ft. Find the position function, s(t), describing the motion of the object.
  35. Calculus
    An oceanographer measured an ocean wave during a storm. The vertical displacement, h, of the wave, in metres, can be modelled by h(t) = 0.8cost + 0.5sin2t, where t is the time in seconds. a) Determine the vertical displacement of the wave when the velocity is 0.8 m/s. b) ...
  36. math, calculus
    Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by f1(t) = C sin ωt and f2(t) = C sin(ωt + α). The two sound waves interfere to produce a single sound modeled by the sum of these functions f(t...
  37. Calculus
    Function f(x) is positive, increasing and concave up on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the upper sum, lower sum, and trapezoidal rule approximations for the value of (b)∫(a) f(x)dx Which ...
  38. AP Calculus
    2) Given the table below for selected values of f(x), use 6 trapezoids to estimate the value of (10)∫(1) f(x)dx x 1 3 4 6 7 9 10 f(x) 4 8 6 10 10 12 16
  39. Calculus
    1) Which are true about the differential equation dy/dx = (x)/(x-4) I.will have a slope field with negative slopes in quadrant I II.will have a slope field with positive slopes in all quadrants III.will produce a slope field with columns of parallel tangents
  40. Calculus
    Which are true about the differential equation dy/dx = 2x(4-y) I.will have a slope field with negative slopes in quadrant I II.will have a slope field with positive slopes in all quadrants III.will produce a slope field with columns of parallel tangents
  41. Calculus
    Given a curve, say y^3 = x^2, and a value, say 4. How can I find a point in which the curve will have slope 4?
  42. Calculus
    A girl throws a tennis ball straight into the air with a velocity of 64 feet/sec. If acceleration due to gravity is -32 ft/sec2, how many seconds after it leaves the girl's hand will it take the ball to reach its highest point? Assume the position at time t = 0 is 0 feet. 4 ...
  43. Calculus
    For an object whose velocity in ft/sec is given by v(t) = -3t2 + 5, what is its displacement, in feet, on the interval t = 0 to t = 2 secs? 6.607 2 -2.303 2.303
  44. AP Calculus
    Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = cos(t) - sin(t) and v(0) = 3 v(t) = sin(t) + cos(t) + 3 v(t) = sin(t) + cos(t) + 2 v(t) = sin(t) - cos(t) + 3 v(t) = sin(t) - cos(t) + 4
  45. Calculus
    find the approximate volume in cubic units of the solid created when the region under the curve y = sec(x) on the interval [0, pi / 3 ] is rotated around the x-axis.
  46. Calculus
    Find the points on the curve y=x^3-4x^2+1 where the tangent is parallel to y+4x=0. y'=3x^2-8x y+4x=3x^2-8x y=3x^2-12x y=3x(x-4) Ya... I need some help The answers are (2, -7) (2/3, -67/27)
  47. Calculus
    If the graph of the function y-(ax^2+b)/(x^2+cx+4) has vertical asymptotes x=1,x=4, a horizontal asymptote of y=2 and x-intercepts at x=+-2, find a, b and c. Answers: a=2, b=-8 and c=-5 I found a and b but I'm having trouble finding c. I made y=0 and x=2 0=(4a+b)/(4+2c+4) Then...
  48. calculus
    Use the instantaneous rate of change of f(x) = e^(5x) to find the equation of the tangent line to f(x) at x = 0.
  49. calculus
    If a ball is thrown at the height of 10m then it bounces by 3\5 of the 10 meter and continues to resting condition, what is the total distance covered by the ball until rest?
  50. Calculus
    1. use the definition mtan=(f(x)-f(x))/(x-a) to find the SLOPE of the line tangent to the graph of f at P. 2. Determine an equation of the tangent line at P. 3. Given 1 & 2, how would I plot the graph of f and the tangent line at P if : f(x)=x^2 +4, P(4,20)
  51. Calculus
    Let f(x)= (x^2 -4)/(x-2) and note that lim as x approaches 2 , f(x)=4. For each value of E, use a graphing utility to find all values whenever 0<|x-2|< delta . looks like an S. a) e=2 b) e=1 c) For any e>0, make a conjecture about the value of delta S that satisfies ...
  52. calculus
    d/dx (16-3x- 1/2x^2)= ? I got dy/dx= -3 - x
  53. Calculus
    How do you take the limit of a composite piecewise function f(f(x))? Only the graph is given. lim f(f(x)) x->2 I figured the way to do it was to find the lim f(x) = C then find lim f(x). x->2 x-> C The simple questions on Khan academy's "Limits of composite functions...
  54. Pre-Calculus
    Which is the polar form of the parametric equations x=5cos(theta) and y=5sin(theta) ? a. r= 5(theta) b. r= 5 c. r= 25 cos (theta) sin (theta)***** d. r= 25cos^2 (theta) + 25sin^2 (theta) Which is the polar form of the parametric equations x=4t and y=t^2? a. r= 16 tan (theta) ...
  55. Calculus
    What is the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.
  56. Calculus
    Use geometry to evaluate the integral from 0 to 6 of the function f of x, dx for f of x equals 3 for x less than or equal to 3 and equals the quantity 6 minus x for x greater than 3 - 27 - 13.5 -12 - 10.5
  57. calculus
    If 'a' and 'b' are unit vectors that make an angle of 60 degrees with each other, calculate l 3a - 5b l and l 8a + 3b l *the 'a' and 'b' have a carat of top of them* How do i answer this without using components?
  58. calculus
    If 'a' and 'b' are unit vectors that make an angle of 60 degrees with each other, calculate l 3a - 5b l and l 8a + 3b l *the 'a' and 'b' have a carat of top of them* How do i answer this without using components?
  59. calculus
    If 'a' and 'b' are unit vectors that make an angle of 60 degrees with each other, calculate l 3a - 5b l and l 8a + 3b l the 'a' and 'b' have a carat of top of them
  60. Pre-Calculus
    1.) Which of the following polar equations is equivalent to the parametric equations below? x=t^2 y=2t A.) r=4cot(theta)csc(theta) B.) r=4tan(theta)sec(theta) C.) r=tan(theta)sec(theta)/4 D.) r=16cot(theta)csc(theta) 2.) Which polar equation is equivalent to the parametric ...
  61. Pre Calculus
    Identify the parametric equations that represent the same path as the following parametric equations. x(t)=2cos2t y(t)=sin3t a. x(t)=2cos2t y(t)=sin6t b. x(t)=4cos4t y(t)=sin6t c. x(t)=2cos4t y(t)=sin6t d. x(t)=4cos2t y(t)=2sin3t
  62. Pre-Calculus Help?
    Jimmy wants to rewrite the set of parametric equations x = 1/2 T + 3 and y = 2T - 1 in rectangular form by eliminating T. Which of the following equations would help him to eliminate T. A) t = 2(x-3) B) t = 2(x+3) C) t = y-1 / 2 D) t = 2(y+1)
  63. Pre-Calculus Help?
    A class is to eliminate t from the parametric equations x = t^2 + 3 and y = 4t. Beth says that she can write t = sqrt (x-3) to eliminate the parameter. Why is this wrong? A) She should have added 3 to x, not subtracted B) She should always solve for t as a function of y C) She...
  64. Pre-Calculus Help?
    A biologist determines that the path of a bee from its hive to its foraging site can be described by the parametric equations x- 4t-1 and y-22+3t-4. Which of the following equations is the curve described by these parametric equations? y=20x-9 y=20x+16 x^2+8x-25/8 x^2+4x-37/8
  65. Pre-Calculus Help?
    What conic section is drawn by the parametric equations x=csc t and y cot? A. Parabola B. Circle C. Ellipse D. Hyperbola
  66. Pre-Calculus Help?
    the vertical and horizontal positions of a zip line cable are represented by the following parametric equations. Rewrite the parametric equations by elimination the the parameter. x(t)= 3t + 4 y(t) = 5-t a) y = -1/3x+19/3 b) y= 19 - 3x c) y= 2x + 9 d) y= 5-x
  67. Pre-Calculus Help?
    Rewrite the following parametric equations by solving for y x(t) = e^-t y(t) = 3e^2t a) y=1/3x^2,x>0 b)3/x^2, x>0 c) y= 3e^t, x>0 d) y=4e^t, x>0
  68. Pre-Calculus Help?
    A 5-lb force acting in the direction of (5, -3) moves and object just left over 12 ft. from point (0, 6) to (7, -4). Find the work done to move the object to the nearest foot-pound. a. 11 ft. * lbs b. 34 ft. * lbs c. 56 ft. * lbs d. 61 ft. * lbs
  69. Pre-Calculus Help? Vectors?
    How does multiplying a vector by a scalar value of 2pi change the vector? A.) The vector will change direction and increase in magnitude. B.) The vector will change direction and decrease in magnitude. C.) The vector will not change direction but will increase in magnitude. D...
  70. Pre-Calculus Help? Vectors?
    How does multiplying a vector by a scalar value of -pi/4 change the vector? A.) The vector will change direction and increase in magnitude. B.) The vector will change direction and decrease in magnitude. C.) The vector will not change direction but will increase in magnitude. ...
  71. Pre Calculus
    Find the magnitude and direction angle for the vector v=3 cos 123 degrees i + 3 sin 123 degrees j.
  72. Calculus
    Use geometry to evaluate integral[-3,3] for: f(x)= { √(4-(x+1)^2), -3≤x≤1 { |x-2| - 1, 1<x≤3 A. 2pi + 1 B. 2pi - 1 C. pi - 1 D. pi + 1
  73. Pre Calculus
    A travel agent is trying to schedule a client's trip from City A to City B . There are 3 direct flights,3 flights from A to a connecting city C , and 4 flights from this connecting City C to City B .How many trips are possible?
  74. Calculus
    Show that the function F(x) = integral[x to 3x](1/t)dt is constant on the interval (0, +∞).
  75. Calculus
    Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of...
  76. Calculus
    Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 5.
  77. Calculus
    Find d/dt integral[2 to x^2](e^x^3)dx
  78. Calculus
    The function f is continuous on the interval [4, 15], with some of its values given in the table above. Estimate the average value of the function with a Trapezoidal Approximation, using the 4 intervals between those given points. x 4 9 11 14 15 f(x) -6 -11 -18 -21 -25
  79. Calculus
    Find F '(x) for F(x) = integral[x^3 to 1](cos(t^4)dt) a. cos(x^7) b. -cos(x^12) c. -3x^2cos(x^12) d. cos(1) - cos(x^12)
  80. calculus
    Let F(x)= the integral from 0 to 2x of tan(t^2) dt. Use your calculator to find F″(1) By applying the fundamental theorem of calculus, I got the derivative of the integral (F'(x)) to be 2tan(2x^2) When I take the derivative to find F''(x) I get 8x sec^2(2x^2). When I plug 1 ...
  81. Calculus
    Consider a hemispherical bowl with a radius of 12 cm. The bowl contains water to a depth of 9 cm. The density of the water at any point x cm below the surface of the water is given by 2e^(0.15x) g/cm^3. A. Find the volume of the water in the bowl. B. Find the mass of the water...
  82. calculus
    Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x^3 and y = x.? I know how to use the midpoint rule to get the area under a curve but I'm confused on how to get the area between the two curves. Do I subtract them somehow? How do I set it up?
  83. Calculus
    Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x^3 and y = x. I just need to know how to use the midpoint rule when the area is between two curves instead of under a curve. Help please.
  84. calculus
    A rock is thrown upward with a speed of 48 feet per second from the edge of a cliff 400 feet above the ground. What is the speed of the rock when it hits the ground? Use acceleration due to gravity as -32 feet per second squared and approximate your answer to 3 decimal places...
  85. Calculus
    Find the area of the region bounded by the curves y = sin^-1(x/4) , y = 0, and x = 4 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative.
  86. Calculus
    Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = sqrt(x) − (1/9)x, [0, 81] c=?
  87. Calculus
    A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 69 cells. (a) Find the relative growth rate. (Assume t is measured in ...
  88. Calculus
    Consider the following. cos(x) + sqrt(y)= 1 (a) Find y' by implicit differentiation. y' = Correct: Your answer is correct. (b) Solve the equation explicitly for y and differentiate to get y' in terms of x. y' = ? (c) Check that your solutions to parts (a) and (b) are ...
  89. Calculus
    Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x^2 and y = x about the line x = 7. I have to use the volume by disks method, but I'm confused about how to set it up and evaluate
  90. Calculus
    Suppose that f(5) = 3, f '(5) = 4, g(5) = −6, and g'(5) = 1. Find the following values. (a) (fg)'(5) (b) (f/g)'(5) (c) (g/f)'(5)
  91. Calculus
    The table gives the values of a function obtained from an experiment. Use them to estimate 9 f(x) dx 3 using three equal subintervals with right endpoints, left endpoints, and midpoints. x 3 4 5 6 7 8 9 f(x) −3.5 −2.1 −0.5 0.2 0.8 1.3 1.8 (c) Estimate 9 f(x) dx 3 using ...
  92. Calculus
    A tank contains 2000 L of pure water. Brine that contains 15 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. The concentration of salt after t minutes (in grams per liter) is C(t) = 15t/80 + t. As t → ∞, what does the concentration approach?
  93. Calculus
    Find all vertical and horizontal asymptotes of the graph of the function. (Enter your answers as a comma-separated list.) f(x) = 5 − 3x/5 + 8x
  94. Calculus
    A bacteria population is 3000 at time t = 0 and its rate of growth is 1000 · 6t bacteria per hour after t hours. What is the population after one hour? (Round your answer to the nearest whole number.)
  95. Calculus
    Evaluate the integral by interpreting it in terms of areas. 0 (3 +sqrt(1 − x2)) dx −1
  96. Calculus
    The base of a solid is bounded by the curve y=sqrt(x+1) , the x-axis and the line x = 1. The cross sections, taken perpendicular to the x-axis, are squares. Find the volume of the solid a. 1 b. 2 c. 2.333 d. none of the above I got a little confused, but this is what I have so...
  97. Calculus
    An object has a constant acceleration of 40 ft/sec^2, an initial velocity of −20 ft/sec, and an initial position of 10 ft. Find the position function, s(t), describing the motion of the object.
  98. Calculus 2
    Convert r = 1 / (6cos(θ)+8sin(θ)) to an equation in rectangular coordinates (i.e., in terms of x and y). A similar example I found, where r = 1 / (cos (θ) − sin (θ)), found the rectangular equation to be y = x - 1. How would this answer change for my problem now that sin...
  99. Calculus
    Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim x→∞ sqrt(25x^2 + x) − 5x
  100. Calculus
    For the function f whose graph is given, state the following (a) lim x → ∞ f(x) (b) lim x → −∞ f(x) (c) lim x → 1 f(x) (d) lim x → 3 f(x) (e) the equations of the asymptotes (Enter your answers as a comma-separated list of equations.) vertical: horizontal:
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