AP Calculus BC - AP Classroom Questions - Unit 5
The figure above represents a square sheet of cardboard with side length 40 inches. The sheet is cut and pieces are discarded. When the cardboard is folded, it becomes a rectangular box with a lid. The pattern for the rectangular box with a lid is shaded in the figure. Four squares with side length x and two rectangular regions are discarded from the cardboard. Which of the following statements is true? (The volume V of a rectangular box is given by V=lwℎ.) a) When x=20 inches, the box has a minimum possible volume. b) When x=20 inches, the box has a maximum possible volume. c) When x=20/3 inches, the box has a minimum possible volume. d) When x=20/3 inches, the box has a maximum possible volume.
d) When x=20/3 inches, the box has a maximum possible volume.
The figure above shows the graph of f on the interval [0,4]. Which of the following could be the graph of f′, the derivative of f, on the interval [0,4] ? a) \ b) /\/ c) /\/\/\ d) \/\
d) \/\
The point (1,1) is on the curve defined by x^2+x^3=2. Which of the following statements is true about the curve at the point (1,1) ? a) dy/dx>0 and d^2y/dx^2 > 0 b) dy/dx>0 and d^2y/dx^2 < 0 c) dy/dx<0 and d^2y/dx^2 > 0 d) dy/dx<0 and d^2y/dx^2 < 0
d) dy/dx<0 and d^2y/dx^2 < 0
Three graphs labeled I, II, and III are shown above. One is the graph of f, one is the graph of f′, and one is the graph of f″. Which of the following correctly identifies each of the three graphs? a) f f' f" I II III b) f f' f" II III I c) f f' f" III I II d) f f' f" II III I
d) f f' f" II III I
The graph of f′, the derivative of the continuous function f, is shown above on the interval −3<x<7. Which of the following statements is true about f on the interval −3<x<7 ? a) f has three relative extrema, and the graph of f has one point of inflection. b) f has three relative extrema, and the graph of f has four points of inflection. c) f has four relative extrema, and the graph of f has two points of inflection. d) f has four relative extrema, and the graph of f has four points of inflection.
d) f has four relative extrema, and the graph of f has four points of inflection.
Let f be the function defined by f(x)=3x^3−36x+6 for −4<x<4. Which of the following statements is true? a) f is decreasing on the interval (0,4) because f′(x)<0 on the interval (0,4). b) f is increasing on the interval (0,4) because f′(x)<0 on the interval (0,4). c) f is decreasing on the interval (−2,0) because f″(x)<0 on the interval (−2,0). d) f is decreasing on the interval (−2,2) because f′(x)<0 on the interval (−2,2).
d) f is decreasing on the interval (−2,2) because f′(x)<0 on the interval (−2,2).
The Second Derivative Test cannot be used to conclude that x=1 is the location of a relative minimum or relative maximum for which of the following functions? a) f(x)=cos(x^2−1), where f′(x)=−2xsin(x^2−1) and f"(x) = -2sin(x^2-1)-4x^2cos(x^2-1) b) f(x)=e^(x−1)^2, where f′(x)=2(x−1)e^(x−1)^2 and f"(x) = 4(x-1)^2e^(x-1)^2 + 2e^(x-1)^2 c) f(x)=x^3/3+x^2−3x+1, where f′(x)=x^2+2x−3 and f"(x) = 2x+2 d) f(x)=x^4−4x^3+6x^2−4x+1, where f′(x)=4x^3−12x^2+12x−4 and f"(x) = 12x^2-24x+12
d) f(x)=x^4−4x^3+6x^2−4x+1, where f'(x)=4x^3−12x^2+12x−4 and f"(x) = 12x^2-24x+12
At what values of x does the graph of y=e^−x+2xe^−x+x^2e^−x have a point of inflection? a) x = -1 only b) x = -1 and x = 1 c) x = -3-√2 and x = -3+√2 d) x = 1-√2 and x = 1+√2
d) x = 1-√2 and x = 1+√2
At what values of x does the graph of y=x^2e^−2x have a point of inflection? a) x = -2 and x = 0 b) x = 0 and x = 1 c) x = -2-√2 and x = -2+√2 d) x = 1-√2/2 and x = 1+√2/2
d) x = 1-√2/2 and x = 1+√2/2
The graph of f′, the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down? a) (−2,0) and (2, 4) b) (−3,2) and (0, 2) c) (−3,−1) only d) (0, 4)
a) (−2,0) and (2, 4)
Let g be the function given by g(x)=x^4−4/3x^3. At what value of x on the closed interval [−1,1] does g have an absolute maximum? a) -1 b) 0 c) 1 d) 4/3
a) -1
Let g be the function given by g(x)=3x^4−8x^3. At what value of x on the closed interval [−2,2] does g have an absolute maximum? a) -2 b) 0 c) 2 d) 8/3
a) -2
Let f be the function defined by f(x)=x^3/3−x^2/2−6x. On which open intervals is f decreasing? a) -2<x<3 only b) x<−2 and x>3 c) x<1/2 only d) There are no such intervals
a) -2<x<3 only
Let g be the function defined by g(x)=|x|+2 |1−x|. What is the absolute minimum value of g on the closed interval [−2,2] ? a) 1 b) 2 c) 4 d) 8
a) 1
Let g be the function defined by g(x)=|x|−3 |x+1|. What is the absolute maximum value of g on the closed interval [−2,2] ? a) 1 b) -1 c) -3 d) -7
a) 1
Let f be a function with first derivative given by f′(x)=(x+1)(x−2)(x−3). At what values of x does f have a relative maximum? a) 2 only b) -1 and 3 only c) 3 only d) -1, 2, and 3
a) 2 only
Let f be a function such that f(1)=2. At each point (x,y) on the graph of f, the slope is given by dy/dx = 5xy-x^2-y^2-5. Which of the following statements is true? a) f has a relative minimum at x=1. b) f has a relative maximum at x=1. c) f has neither a relative minimum nor a relative maximum at x=1. d) There is insufficient information to determine whether f has a relative minimum, a relative maximum, or neither at x=1.
a) f has a relative minimum at x=1.
Let f be a function such that f(−1)=1. At each point (x,y) on the graph of f, the slope is given by dy/dx = -x^2-xy+y^2-1. Which of the following statements is true? a) f has a relative minimum at x=−1. b) f has a relative maximum at x=−1. c) f has neither a relative minimum nor a relative maximum at x=−1. d) There is insufficient information to determine whether f has a relative minimum, a relative maximum, or neither at x=−1.
a) f has a relative minimum at x=−1.
Let f be the function with derivative given by f′(x)=x^2−(a+b)x+ab=(x−a)(x−b), where a and b are constants such that a<b. Which of the following statements is true? a) f is decreasing for a<x<b because f′(x)<0 for a<x<b. b) f is decreasing for x<a and x>b because f′(x)<0 for x<a and x>b. c) f is decreasing for x<(a+b)/2 because f′(x)<0 for x<(a+b)/2. d) f is decreasing for x<(a+b)/2 because f"(x)<0 for x<(a+b)/2.
a) f is decreasing for a<x<b because f′(x)<0 for a<x<b.
Let f be the function with derivative given by f'(x)=x^2−a^2=(x−a)(x+a), where a is a positive constant. Which of the following statements is true? a) f is decreasing for −a<x<a because f′(x)<0 for −a<x<a. b) f is decreasing for x<−a and x>a because f′(x)<0 for x<−a and x>a. c) f is decreasing for x<0 because f′(x)<0 for x<0. d) f is decreasing for x<0 because f″(x)<0 for x<0.
a) f is decreasing for −a<x<a because f′(x)<0 for −a<x<a.
Let f be a function with selected values given in the table above. Which of the following statements must be true? I. By the Intermediate Value Theorem, there is a value c in the interval (0,3) such that f(c)=10. II. By the Mean Value Theorem, there is a value c in the interval (0,3) such that f′(c)=−2. III. By the Extreme Value Theorem, there is a value c in the interval [0,3] such that f(c)≤f(x) for all x in the interval [0,3]. a) none b) I only c) II only d) I, II, and III
a) none
Let f be a function with selected values given in the table above. Which of the following statements must be true? I. By the Intermediate Value Theorem, there is a value c in the interval (0,3) such that f(c)=2. II. By the Mean Value Theorem, there is a value c in the interval (0,3) such that f′(c)=2. III. By the Extreme Value Theorem, there is a value c in the interval [0,3] such that f(c)≤f(x) for all x in the interval [0,3]. a) none b) I only c) II only d) I, II, and III
a) none
Consider the curve in the xy-plane defined by y√y^3+1=x for y>−1. For what value of y, y>−1, does the derivative of y with respect to x not exist? a) y = (-2/5)^1/3 b) y = (-1/4)^1/3 c) y = 0 d) The derivative of y with respect to x exists for all y>−1.
a) y = (-2/5)^1/3
Let f be the function defined by f(x) = 1/3x^3 - 4x^2 - 9x + 5. On which of the following intervals is the graph of f both decreasing and concave down? a) (-∞, 4) b) (-1, 4) c) (4, 9) d) (9, ∞)
b) (-1, 4)
if x^2y - x/y = -2, then dy/dx = a) y/(x+x^2y^2) b) (y-2xy^3)/(x+x^2y^2) c) (y+2xy^3)/(x-x^2y^2) d) (y-2xy^3)/(x^2y^2-x)
b) (y-2xy^3)/(x+x^2y^2)
if xy^2 + x^2/y = 5, then dy/dx = a) (-2xy)/x(2y^3-x) b) -y(y^3+2x)/x(2y^3-x) c) -y^4/x(2y^3-x) d) (5y^2-y^4-2xy)/x(2y^3+x)
b) -y(y^3+2x)/x(2y^3-x)
Let f be the function defined by f(x)=12x−x^3. What is the absolute minimum value of f on the closed interval [0,3] ? a) -16 b) 0 c) 9 d) 16
b) 0
Consider the curve defined by y^2=x^3−3x+3 for x>0. At what value of x does the curve have a horizontal tangent? a) √3/3 b) 1 c) √3 d) There is no such value of x
b) 1
Let f be the function defined by f(x)=3x^2−x^3. What is the absolute minimum value of f on the closed interval [1,52] ? a) 0 b) 2 c) 25/8 d) 4
b) 2
Let f be the function defined by f(x)=xcosx−sinx. What is the absolute maximum value of f on the interval [−π/2,2π] ? a) -π b) 2π c) 0 d) 1
b) 2π
The total cost, in dollars, to order x units of a certain product is modeled by C(x)=7x^2+252. According to the model, for what size order is the cost per unit a minimum? a) An order of 1 unit has a minimum cost per unit. b) An order of 6 units has a minimum cost per unit. c) An order of 84 units has a minimum cost per unit. d) An order of 252 units has a minimum cost per unit.
b) An order of 6 units has a minimum cost per unit.
Selected values of a continuous function f are given in the table above. Which of the following statements could be false? a) By the Intermediate Value Theorem applied to f on the interval [0,4], there is a value c such that f(c)=5. b) By the Mean Value Theorem applied to f on the interval [0,4], there is a value c such that f′(c)=4. c) By the Extreme Value Theorem applied to f on the interval [0,4], there is a value c such that f(c)≤f(x) for all x in [0,4]. d) By the Extreme Value Theorem applied to f on the interval [0,4], there is a value c such that f(c)≥f(x) for all x in [0,4].
b) By the Mean Value Theorem applied to f on the interval [0,4], there is a value c such that f′(c)=4.
Let f be a twice-differentiable function. Selected values of f′ and f″ are shown in the table above. Which of the following statements are true? I. f has neither a relative minimum nor a relative maximum at x=1. II. f has a relative maximum at x=1. III. f has a relative maximum at x=4. a) I only b) II only c) III only d) I and III only
b) II only
Let f be a twice-differentiable function. Selected values of f′ and f″ are shown in the table above. Which of the following statements are true? I. f has neither a relative minimum nor a relative maximum at x=2. II. f has a relative maximum x=2. III. f has a relative maximum x=8. a) I only b) II only c) III only d) I and III only
b) II only
Let f be a differentiable function with f(−3)=7 and f(3)=8. Which of the following must be true for some c in the interval (−3,3) ? a) f′(c)=0, since the Extreme Value Theorem applies. b) f′(c)=8+7/3−(−3), since the Mean Value Theorem applies. c) f′(c)=8−7/3−(−3), since the Mean Value Theorem applies. d) f′(c)=7.5, since the Intermediate Value Theorem applies.
c) f′(c)=8−7/3−(−3), since the Mean Value Theorem applies.
At how many points on the curve x^2/3+y^2/3=9 in the xy-plane does the curve have a tangent line that is horizontal? a) none b) one c) two d) three
c) two
At how many points on the curve x^2/5+y^2/5=1 in the xy-plane does the curve have a tangent line that is horizontal? a) none b) one c) two d) three
c) two
The function f is defined by f(x)=e^−x(x^2+2x). At what values of x does f have a relative maximum? a) x = -2 + √2 and x = -2 - √2 b) x = -√2 only c) x = -2 and x = 0 d) x = √2 only
c) x = -2 and x = 0
The acceleration, in centimeters per second per second, of a projectile is modeled by A(t)=2t^3−15t^2+36t, where t is measured in seconds. What is the projectile's maximum acceleration on the time interval 0≤t≤4 ? a) The maximum acceleration of the projectile is 4 centimeters per second per second and occurs at t=32 seconds. b) The maximum acceleration of the projectile is 27 centimeters per second per second and occurs at x=3 seconds. c) The maximum acceleration of the projectile is 28 centimeters per second per second and occurs at t=2 seconds. d) The maximum acceleration of the projectile is 32 centimeters per second per second and occurs at t=4 seconds.
d) The maximum acceleration of the projectile is 32 centimeters per second per second and occurs at t=4 seconds.
Which of the following describes the x-coordinates of the points on the curve cosx=e^y in the xy-plane where the curve has a horizontal tangent line? a) The values nπ for all integers n b) The values π/2+nπ for all integers n c) The values (2n+1)π for all integers n d) The values 2nπ for all integers n
d) The values 2nπ for all integers n
Which of the following describes the y-coordinates of the points on the curve e^x=siny in the xy-plane where the curve has a vertical tangent line? a) The values π/2+nπ for all nonnegative integers n b) The values π/2+2nπ for all nonnegative integers n c) The values π/2+nπ for all integers n d) The values π/2+2nπ for all integers n
d) The values π/2+2nπ for all integers n
The figure above represents a square sheet of cardboard with side length 20 inches. The sheet is cut and pieces are discarded. When the cardboard is folded, it becomes a rectangular box with a lid. The pattern for the rectangular box with a lid is shaded in the figure. Four squares with side length x and two rectangular regions are discarded from the cardboard. Which of the following statements is true? (The volume V of a rectangular box is given by V=lwℎ.) a) When x=10 inches, the box has a minimum possible volume. b) When x=10 inches, the box has a maximum possible volume. c) When x=10/3 inches, the box has a minimum possible volume. d) When x=10/3 inches, the box has a maximum possible volume.
d) When x=10/3 inches, the box has a maximum possible volume.
Let f be a differentiable function with a domain of (0,5). It is known that f'(x), the derivative of f(x), is negative on the intervals (0,1) and (2,3) and positive on the intervals (1,2) and (3,5). Which of the following statements is true? a) f has no relative minima and three relative maxima. b) f one relative minimum and two relative maxima. c) f has two relative minima and one relative maximum. d) f has three relative minima and no relative maxima.
c) f has two relative minima and one relative maximum.
The Mean Value Theorem can be applied to which of the following functions on the closed interval [−3,3] ? a) f(x) = x^2/3 b) f(x) = |x-1| c) f(x) = (x-2)/(x-5) d) f(x) = (x-5)/(x-2)
c) f(x) = (x-2)/(x-5)
The graph of f′, the derivative of the function f, is shown above. Which of the following could be the graph of f ? (concave down graph) a) b) c) d)
c)
The graph of f′, the derivative of the function f, is shown above. Which of the following could be the graph of f ? (concave up graph) a) b) c) d)
c)
The graph of f″, the second derivative of the function f, is shown above on the interval 0≤x≤6. Which of the following could be the graph of f ? a) b) c) d)
c)
The graph of f″, the second derivative of the function f, is shown above on the interval 0≤x≤8. Which of the following could be the graph of f ? a) b) c) d)
c)
The graph of y=f(x) is shown above. Which of the following could be the graph of y=f″(x)? a) / b) /\/ c) /\/\ d) \/\/
c) /\/\
The Mean Value Theorem can be applied to which of the following functions on the closed interval [−5,5] ? a) f(x) = 1/sinx b) f(x) = (x-1)/|x-1| c) f(x) = x^2/(x^2-36) d) f(x) = x^2/(x^2-4)
c) f(x) = x^2/(x^2-36)
Let f be a differentiable function with a domain of (0,10). It is known that f'(x), the derivative of f(x), is negative on the intervals (0,2) and (4,6) and positive on the intervals (2,4) and (6,10). Which of the following statements is true? a) f has no relative minima and three relative maxima. b) f has one relative minimum and two relative maxima. c) f has two relative minima and one relative maximum. d) f has three relative minima and no relative maxima.
c) f has two relative minima and one relative maximum.
An athlete is planning for the "Land and Lake Race," the path of which is shown in the figure above. The contestants will start at the dot, run up to 4 miles along the beach, enter the water at any time, and swim to the island. The athlete estimates that they can run along the beach at a constant rate of 8 miles per hour and swim at a constant rate of 2 miles per hour. Let T be a function that represents the time to complete the race, where x is the distance in miles that the athlete runs and T(x) is measured in hours. Which of the following methods best explains how to determine the minimum time, in hours, the athlete should run along the beach and then swim to the island? a) Let T(x)=8x+2√1^2+(4−x)^2. Solve T′(x)=0 and find the values of x where T′(x) changes sign from negative to positive. Evaluate T for those values of x to determine the minimum time. b) Let T(x)=x/8+√1^2+(4−x)^2/2. Solve T′(0)=
b) Let T(x)=x/8+√1^2+(4−x)^2/2. Solve T′(0)=0 and find the values of x where T′(x) changes sign from negative to positive. Evaluate T for those values of x to determine the minimum time.
The graph of f″, the second derivative of the continuous function f, is shown above on the interval [0,π/2]. On this interval f has only one critical point, which occurs at x=π/16. Which of the following statements is true about the function f on the interval [0,π/2]? a) f has a relative minimum at x=π/16 but not an absolute minimum. b) The absolute minimum of f is at x=π/16. c) f has a relative maximum at x=π/16 but not an absolute maximum. d) The absolute maximum of f is at x=π/16.
b) The absolute minimum of f is at x=π/16.
Let f be the function defined by f(x)=x^2+1/x+1 with domain [0,∞). The function f has no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem? a) The domain of f is not an open interval. b) The domain of f is not a closed and bounded interval. c) The function f is not continuous on its domain. d) The function f is not differentiable on its domain.
b) The domain of f is not a closed and bounded interval.
Consider all lines in the xy-plane that pass through both the origin and a point (x,y) on the graph of y=x^2−x+16 for 1≤x≤8. The figure above shows one such line and the graph of y=x^2−x+16. Which of the following statements is true? a) The line with minimum slope passes through the graph of y=x^2−x+16 at x=1. b) The line with minimum slope passes through the graph of y=x^2−x+16 at x=4. c) The line with minimum slope passes through the graph of y=x^2−x+16 at x=7. d) The line with minimum slope passes through the graph of y=x^2−x+16 at x=8.
b) The line with minimum slope passes through the graph of y=x^2−x+16 at x=4.
Let f be the function with derivative f′(x)=x^3−3x+2 . Which of the following statements is true? a) f has no relative minima and one relative maximum. b) f has one relative minimum and no relative maxima. c) f has one relative minimum and one relative maximum. d) f has two relative minima and one relative maximum.
b) f has one relative minimum and no relative maxima.
Let f be the function with derivative f′(x)=x^3−3x−2. Which of the following statements is true? a) f has no relative minima and one relative maximum. b) f has one relative minimum and no relative maxima. c) f has one relative minimum and one relative maximum. d) f has two relative minima and one relative maximum.
b) f has one relative minimum and no relative maxima.
The graph of f′, the derivative of the function f, is shown above for −1<x<5. Which of the following statements is true for −1<x<5 ? a) f has one relative minimum and two relative maxima. b) f has two relative minima and one relative maximum. c) f has two relative minima and three relative maxima. d) f has three relative minima and three relative maxima
b) f has two relative minima and one relative maximum.
The function f is continuous on the interval (0,16), and f is twice differentiable except at x=3, where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection? a) x = 9 only b) x = 3 and x = 9 c) x = 3 and x = 11 d) x = 9 and x = 11
b) x = 3 and x = 9
The function f is continuous on the interval (0,16), and f is twice differentiable except at x=5 where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection? a) x = 8 only b) x = 5 and x = 8 c) x = 5 and x = 12 d) x = 8 and x = 5
b) x = 5 and x = 8
Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0,2π] ? a) y = 1/(1+sinx) b) y = 1/(x^2+π) c) y = (x^2 - 2πx + x^2)/(x-π) d) y = |x-π|/(x-π)
b) y = 1/(x^2+π)
Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0,4] ? a) y = tanx b) y = tan^-1x c) y = (x^2-16)/(x^2+x+20) d) y = 1/(e^x-1)
b) y = tan^-1x
Let f be the function defined by f(x)=x^5/20−x^4/12−x^3/3. The graph of f′, the derivative of f, is shown above. On which of the following intervals is the graph of f concave up? a) x<−1 and 0<x<2 b) −1<x<0 and x>2 c) x<2/3 - 2√10/3 and x>2/3 + 2√10/3 d) 2/3 - 2√10/3 < x < 2/3 + 2√10/3
b) −1<x<0 and x>2
The function f is differentiable and decreasing on the interval 0≤x≤6, and the graph of f has exactly two points of inflection on this interval. Which of the following could be the graph of f′, the derivative of f ? a) b) c) d)
c)
The function f is differentiable and increasing on the interval 0≤x≤6, and the graph of f has exactly two points of inflection on this interval. Which of the following could be the graph of f′, the derivative of f ? a) b) c) d)
c)
The function f shown in the figure above is continuous on the closed interval [0,12] and differentiable on the open interval (0,12). Based on the graph, what are all values of x that satisfy the conclusion of the Mean Value Theorem applied to f on the closed interval [0,12] ? a) 4.5 only because this is the value where f(x) equals the average rate of change of f on [0,12]. b) 3 and 8 because these are the values where f′(x)=0 on [0,12]. c) 2 and 9 only because these are the values where the instantaneous rate of change of f at those values is equal to the average rate of change of f on [0,12]. d) 2, 4.5, and 9 because these are the values where either the instantaneous rate of change of f at the value is equal to the average rate of change of f on [0,12] or the value of f(x) is equal to the average rate of change of f on [0,12].
c) 2 and 9 only because these are the values where the instantaneous rate of change of f at those values is equal to the average rate of change of f on [0,12].
Let f be the function defined by f(x)=x^3−32x^2−6x. What is the absolute maximum value of f on the interval [−2,3] ? a) -10 b) -9/2 c) 7/2 d) 3
c) 7/2
Let f be the function given by f(x)=x^2−9/sinx on the closed interval [0,5]. Of the following intervals, on which can the Mean Value Theorem be applied to f? I. [1,3], because f is continuous on [1,3] and differentiable on (1,3). II. [4,5], because f is continuous on [4,5] and differentiable on (4,5). III. [1,4], because f is continuous on [1,4] and differentiable on (1,4). a) none b) I only c) I and II only d) I, II, and III
c) I and II only
Let C be the curve defined by xy=−4. Which of the following statements is true of curve C at the point (−2,2) ? a) It has a relative minimum because y′=0 and y″>0. b) It has a relative maximum because y′=0 and y″<0. c) It is increasing and concave up because y′>0 and y″>0. d) It is increasing and concave down because y′>0 and y″<0.
c) It is increasing and concave up because y′>0 and y″>0.
Let C be the curve defined by x^2−y^2=1. Consider all points (x,y) on curve C where x>1 and y>0. Which of the following statements provides a justification for the concavity of the curve? a) The curve is concave down because y″=−x/y^2<0. b) The curve is concave up because y″=1/y^2>0. c) The curve is concave down because y″=−1/y^3<0. d) The curve is concave up because y″=1/y^3>0.
c) The curve is concave down because y″=−1/y^3<0.
Consider the curve defined by x^2/16−y^2/9=1. It is known that dy/dx=9x/16y and d^2y/dx^2=−81/16y^3. Which of the following statements is true about the curve in Quadrant IV ? a) The curve is concave up because dy/dx>0. b) The curve is concave down because dy/dx<0. c) The curve is concave up because d^2y/dx^2>0. d) The curve is concave down because d^2y/dx^2<0.
c) The curve is concave up because d^2y/dx^2>0.
Consider the curve in the xy-plane defined by x^2−y^2/5=1. It is known that dy/dx=5x/y and d^2ydx^2=−25/y^3. Which of the following statements is true about the curve in Quadrant IV ? a) The curve is concave up because dy/dx>0. b) The curve is concave down because dy/dx<0. c) The curve is concave up because d^2y/dy^2>0. d) The curve is concave down because d^2y/dy^2<0.
c) The curve is concave up because d^2y/dy^2>0.
The figure above shows a rectangle inscribed in a semicircle with a radius of 20. The area of such a rectangle is given by A(x)=2x√400−x^2, where the width of the rectangle is 2x. It can be shown that A'(x)=((−2x^2)/(√400−x^2))+2√400−x^2 and A has critical values of −20, −10√2, 10√2, and 20. It can also be shown that A′(x) changes from positive to negative at x=10√2. Which of the following statements is true? a) The inscribed rectangle with maximum area has dimensions 10√2 by 10√2. b) The inscribed rectangle with minimum area has dimensions 10√2 by 10√2. c) The inscribed rectangle with maximum area has dimensions 20√2 by 10√2. d) The inscribed rectangle with minimum area has dimensions 20√2 by 10√2.
c) The inscribed rectangle with maximum area has dimensions 20√2 by 10√2.
The figure above shows a rectangle inscribed in a semicircle with a radius of 2. The area of such a rectangle is given by A(x)=2x√4−x^2, where the width of the rectangle is 2x. It can be shown that A'(x)=((−2x^2)/(√4−x^2))+2√4−x^2 and A has critical values of −2, √−2, √2, and 2. It can also be shown that A′(x) changes from positive to negative at x=√2. Which of the following statements is true? a) The inscribed rectangle with maximum area has dimensions √2 by √2. b) The inscribed rectangle with minimum area has dimensions √2 by √2. c) The inscribed rectangle with maximum area has dimensions 2√2 by √2. d) The inscribed rectangle with minimum area has dimensions 2√2 by √2.
c) The inscribed rectangle with maximum area has dimensions 2√2 by √2.
Consider all lines in the xy-plane that pass through both the origin and a point (x,y) on the graph of y=x^2−4x+9 for 1≤x≤4. The figure above shows one such line and the graph of y=x^2−4x+9. Which of the following statements is true? a) The line with minimum slope passes through the graph of y=x^2−4x+9 at x=1 . b) The line with minimum slope passes through the graph of y=x^2−4x+9 at x=2. c) The line with minimum slope passes through the graph of y=x^2−4x+9 at x=3. d) The line with minimum slope passes through the graph of y=x^2−4x+9 at x=4.
c) The line with minimum slope passes through the graph of y=x^2−4x+9 at x=3.
Selected values of a differentiable function f are given in the table above. What is the fewest possible number of values of c in the interval [0,11] for which the Mean Value Theorem guarantees that f′(c)=4 ? a) Zero b) One c) Two d) Three
c) Two
Selected values of a differentiable function f are given in the table above. What is the fewest possible number of values of c in the interval [1,9] for which the Mean Value Theorem guarantees that f′(c)=6 ? a) Zero b) One c) Two d) Three
c) Two
In the xy-plane, how many points on the curve y^2+x^2=3−xy have horizontal or vertical tangent lines? a) No points have vertical tangent lines, and two points have horizontal tangent lines. b) One point has a vertical tangent line, and one point has a horizontal tangent line. c) Two points have vertical tangent lines, and two points have horizontal tangent lines. d) No points have vertical tangent lines, and no points have horizontal tangent lines.
c) Two points have vertical tangent lines, and two points have horizontal tangent lines.
Let f be the function given by f(x)=sinxcosx/(x^2−4) on the closed interval [−2π,2π]. On which of the following closed intervals is the function f guaranteed by the Extreme Value Theorem to have an absolute maximum and an absolute minimum? a) [-2π, 2π] b) [-2, 2] c) [-1, 1] d) [π/2, π]
c) [-1, 1]
The function g shown in the figure above is continuous on the closed interval [0,x6] and differentiable on the open interval (0,x6), where x1, x2, x3, x4, x5, and x6 are points on the x-axis. Based on the graph, what are all values of x that satisfy the conclusion of the Mean Value Theorem applied to g on the closed interval [0,x6] ? a) x3 only, because this is the value where g(x) equals the average rate of change of g on [0,x6]. b) x2 and x4 only, because these are the values where g′(x)=0 on [0,x6]. c) x1 and x5 only, because these are the values where the instantaneous rate of change of g at those values is equal to the average rate of change of g on [0,x6]. d) x1, x3, and x5 only, because these are the values where either the instantaneous rate of change of g at the value is equal to the average rate of change of g on [0,x6] or the value of g(x) is equal to the average rate of change of g on [0,x6].
c) x1 and x5 only, because these are the values where the instantaneous rate of change of g at those values is equal to the average rate of change of g on [0,x6].
Given the curve x^2y=x in the xy-plane, for what value of y, if any, does the derivative of y with respect to x not exist? a) y = 0 b) y = -1 c) y = -1/ln2 d) for no value of y
c) y = -1/ln2
Let f be the function defined by f(x)=(sinx)e^−x on the interval [−π/2,π/2]. On which of the following open intervals is f increasing? a) (-π/4, π/2) b) (0, π/2) only c) (π/4, π/2) only d) (-π/2, π/4)
d) (-π/2, π/4)
Let f be the function defined by f(x)=(x+x^2)e^−2x. On which of the following open intervals is f increasing?
d) (-√2/2, √2/2)
The function f is defined by f(x)=x^2e^−x^2. At what values of x does f have a relative maximum? a) -2 b) 0 c) 1 only d) -1 and 1
d) -1 and 1
Let f be the function given by f(x)=−x^3+3x^2+24x. What is the absolute maximum value of f on the closed interval [−6,6] ? a) -6 b) 36 c) 80 d) 180
d) 180
In the xy-plane, the point (0,2) is on the curve C. If dy/dx=−4x/3y for the curve, which of the following statements is true? a) At the point (0,2), the curve C has a relative minimum because dy/dx=0 and d^2y/dx^2>0. b) At the point (0,2), the curve C has a relative minimum because dy/dx=0 and d^2y/dx^2<0. c) At the point (0,2), the curve C has a relative maximum because dy/dx=0 and d^2y/dx^2>0. d) At the point (0,2), the curve C has a relative maximum because dy/dx=0 and d^2y/dx^2<0.
d) At the point (0,2), the curve C has a relative maximum because dy/dx=0 and d^2y/dx^2<0.
Let g be the function given by g(x)=√1+cosx. Which of the following statements could be false on the interval (π/2)≤x≤(7π/4) ? a) By the Extreme Value Theorem, there is a value c such that g(c)≤g(x) for (π/2)≤x≤(7π/4.) b) By the Extreme Value Theorem, there is a value c such that g(c)≥g(x) for (π/2)≤x≤(7π/4). c) By the Intermediate Value Theorem, there is a value c such that g'(c) = (g(7π/4)+g(π/2))/2 d) By the Mean Value Theorem, there is a value c such that g'(c)=(g(7π/4)−g(π/2)/7π/4−π/2.
d) By the Mean Value Theorem, there is a value c such that g'(c)=(g(7π/4)−g(π/2)/7π/4−π/2.
Let g be the function given by g(x)=√1−sin^2x. Which of the following statements could be false on the interval 0≤x≤π ? a) By the Extreme Value Theorem, there is a value c such that g(c)≤g(x) for 0≤x≤π. b) By the Extreme Value Theorem, there is a value c such that g(c)≥g(x) for 0≤x≤π. c) By the Intermediate Value Theorem, there is a value c such that g(c)=(g(0)+g(π))/2. d) By the Mean Value Theorem, there is a value c such that g′(c)=(g(π)−g(0))/(π−0.)
d) By the Mean Value Theorem, there is a value c such that g′(c)=(g(π)−g(0))/(π−0.)
A curve in the xy-plane is defined by the equation x^3/3+y^2/2−3x+2y=−1/6. Which of the following statements are true? I. At points where x=√3, the lines tangent to the curve are horizontal. II. At points where x=−2, the lines tangent to the curve are vertical. III. The line tangent to the curve at the point (1,1) has slope 2/3. a) I and II only b) II and III only c) I and III only d) I, II, and III
d) I, II, and III
Let f be a twice-differentiable function. Which of the following statements are individually sufficient to conclude that x=2 is the location of the absolute maximum of f on the interval [−5,5] ? I. f′(2)=0 II. x=2 is the only critical point of f on the interval [−5,5], and f″(2)<0. III. x=2 is the only critical point of f on the interval [−5,5], and f(−5)<f(5)<f(2). a) II only b) III only c) I and II only d) II and III only
d) II and III only
Let f be a twice-differentiable function. Which of the following statements are individually sufficient to conclude that x=4 is the location of the absolute maximum of f on the interval [0,10] ? I. f′(4)=0 II. x=4 is the only critical point of f on the interval [0,10], and f″(4)<0. III. x=4 is the only critical point of f on the interval [0,10], and f(10)<f(0)<f(4). a) II only b) III only c) I and II only d) II and III only
d) II and III only
The function f is continuous on the interval (0,5) and is twice differentiable except at x=2, where the derivatives do not exist (DNE). Information about the first and second derivatives of f for some values of x in the interval (0,5) is given in the table above. Which of the following statements could be false? a) The function f has a relative maximum at x=1. b) The function f has a relative minimum at x=2. c) The graph of f has a point of inflection at x=3. d) The graph of f has a point of inflection at x=4.
d) The graph of f has a point of inflection at x=4.
