Calculus AB Multiple Choice (Morales)
An equation for a tangent to the graph of y=arcsin x/2 at the origin is a) x-2y=0 b) x-y=0
A
An equation of the line tangent to the graph of f(x) = x(1-2x)³ at the point (1,-1) is a) y=-7x+6 b) y= -6x+5 c) y= -2x+1 d) y= 2x-3 e) y= 7x-8
A
At what values of x does f(x)= 3x⁵ -5x³ +15 have a relative maximum? a) -1 only b) 0 only
A
GRAPH GRAPH GRAPH (Pg.44) The graph of f', the derivative of f, is shown in the figure above. Which of the following describes all relative extrema of f on the open interval (a,b)? a) one relative maximum and two relative minima b) one relative maximum and three relative minima
A
GRAPH GRAPH GRAPH (Pg.52) The graph of f is shown in the figure below. Which of the following could be the graph of the derivative of f? A
A
GRAPH GRAPH GRAPH (Pg.54) The graph of a function f is show above. Which of the following statements about f is false? A
A
GRAPH GRAPH GRAPH (Pg.55) The graph of y=h(x) is shown above. Which of the following could be the graph of y=h'(x)? A
A
GRAPH GRAPH GRAPH (Pg.55) The graphs of the derivates of the functions f,g, and h are shown above. Which of the functions f,g, or h have a relative maximum on the open interval a<x<b? a) f only b) g only
A
If f(x) = x + sin x. then f'(x) = a) 1+ cos x b) 1 -cos x c) cos x d) sin x- xcos x e) sin x+ xcos x
A
If f(x) = x+ 1/x , then the set of values for which f increases is a) (-∞, -1] U [1,∞) b) [-1,1]
A
If f(x) = x√2x-3, then f'(x)= a) 3x-3/ √2x-3 b) x/ √2x-3 c) 1/ √2x-3 d) -x+3/ √2x-3 e) 5x-6/ 2√2x-3
A
If f(x)= (x²-2x-1)^(2/3), then f'(0) is a) 4/3 b) 0 c) -2/3 d) -4/3 e) -2
A
If y= 3/ 4+x², then dy/dx = a) -6x/ (4+x²)² b) 3x/ (4+x²)² c) 6x/ (4+x²)² d) -3/ (4+x²)² e) 3/2x
A
If y= arctan (cos x), then dy/dx= a) -sin x/ 1+cos²x b) -(arcsec (cos x))² sin x c) (arcsec (cos x))² d) 1/(arccos x)² +1 e) 1/ 1+cos² x
A
If y= cos² 3x, the dy/dx= a) -6sin 3xcos 3x b) 2cos3x
A
Let f be a function defined for all real numbers x. If f'(x) = |4-x²|/x-2 , then f is decreasing on the interval a) (-∞,2) b) (-∞,∞)
A
The absolute maximum value of f(x) = x³-3x²+12 on the closed interval [-2,4] occurs at x= a) 4 b) 2
A
The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. At what value of x does the absolute minimum of occur? a) 0 b) 6
A
The value of the derivative of y = ³√x²+8 / ⁴√2x+1 at x=0 is a) -1 b) 0
A
lim f(x)=7, which of the following must be true? x→3 I. F is continuous at x=3 II. F is differentiable at x=3 III. F(3)=7 a) none b) II only c) III only d) I and III only e) I,II,III
A
A polynomial p(x) has a relative maximum at (-2,4) , a relative minimum at (1,1) , a relative maximum at (5,7) and no other critical points. How many zeros does p(x) have? a) One b) Two
B
Ab equation of the line tangent to y= x³+3x²+2 at its point of inflection is a) y=-6x-6 b) y= -3x+1
B
An equation of the line tangent to the graph of y= 2x+3/ 3x-2 at the point (1,5) is... a) 13x-y=8 b) 13x+y=18 c) x-13y=64 d) x+13y=66 e) -2x+3y=13
B
An equation of the line tangent to the graph of y= x+ cos x at the point (0,1) is a) y= 2x+1 b) y= x+1 c) y=x d) y=x-1 e) y=0
B
An equation of the line tangent to the graph of y=cos (2x) at x= π/4 is a) y-1=-(x-π/4) b) y=-2(x-π/4)
B
At what point on the graph of y=1/2x² is the tangent line parallel to the line 2x-4y=3? a) (1/2, -1/2) b) (1/2, 1/8)
B
At what value of x does the graph of y= 1/x² - 1/x³ have a point of inflection? a) 0 b) 2
B
At x=0 which of the following is true of the function f defined by f(x) = x²+e⁻²ⁿ? a) f is increasing b) f is decreasing
B
At x=0, which of the following is true of the function f defined by f(x) = x² +e⁻²ⁿ ? a) f is increasing b) f is decreasing
B
For what value of k will x+ k/x have a relative maximum at x=-2? a) 2 b) 4
B
For what value of k will x+ k/x have a relative minimum at x=-2? a) -4 b) 4
B
For what value of x does the function f(x) = (x-2) (x-3)² have a relative maximum? a) -3 b) 7/3
B
GRAPH GRAPH GRAPH (Pg.49) If y is a function x such that y'>0 for all y"<0 for all x, which of the following could be part of the graph of y=f(x)? B
B
GRAPH GRAPH GRAPH (Pg.49) The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f? B
B
GRAPH GRAPH GRAPH (Pg.50) The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? a) f(1)< f'(1) < f"(1) b) f"(1)< f(1) <f'(1)
B
GRAPH GRAPH GRAPH (Pg.50) The graph of y=f(x) on the closed interval [2,7] is shown above. How many points of inflection does this graph have on this interval? a) One b) Three
B
GRAPH GRAPH GRAPH (Pg.51) The graph of y= f(x) is shown in the figure above. On which of the following intervals are dy/dx >0 and d²y/dx² <0? I. a<x<b II. b<x<c III. c<x<d a) I only b) II only
B
GRAPH GRAPH GRAPH (Pg.51) Which of the following pairs of graphs could represent the graph of a function and the graph of its derivative? I. Graph II. Graph III. Graph a) III only b) I and III
B
GRAPH GRAPH GRAPH (Pg.53) If y is a function of x such that y'>0 for all x and y"<0 for all x, which of the following could be part of the graph of y=f(x)? B
B
GRAPH GRAPH GRAPH (Pg.53) The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1,-1) and (3,1). For what values of x, -2<x<4, is not differentiable? a) 0 only b) 0 and 2 only
B
Given the function defined by f(x) =3x⁵ -20x³ , find all values of x for which the graph of f is concave up. a) x>5 b) -√2 <x< 0 or x> √2
B
How many critical points does the function f(x) = (x+2)⁵ (x-3)⁴ have ? a) One b) Three
B
If a function f is continuous for all x and if f has a relative minimum at (-1,4) and a relative minimum at (3,-2), which of the following statements must be true? a) The graph of f has a horizontal asymptote b) The graph of f intersects both axes
B
If f and g are twice differentiable and if h(x) =f(g(x)), the h"(x)= a) f"(g(x)) b) f"(g(x)) [g'(x)]² + f'(g(x)) g"(x)
B
If f is a continuous function defined for all real numbers x and if the maximum value of f(x) is 5 and the minimum value of f(x) is 7, then which of the following must be true? I. The maximum value of f(|x|) is 5. II. The maximum value of |f(x)| is 7. III. The minimum value of f(|x|) is 0. a) I only b) II only
B
If f is continuous for a≤x≤b and differentiable for a<x<b , which of the following could be false? a) f has a minimum value on a≤x≤b b) f'(c)=0 for some c such that a<c<b.
B
If f is the function defined by f(x) = 3x⁵ -5x⁴ , what are all the x-coordinates of points of inflection for the graph of f? a) 1 b) -1,0, and 1
B
If f" (x) = x(x+1) (x-2)² , then the graph of f has inflection points when x= a) -1 only b) -1 and 0 only
B
If f(x) = (x+1)^3/2 + e^x-2/2, then f'(2)= a) 1 b) 2
B
If f(x) = 1/3x³ -4x²+12x-5 and the domain is the set of all x such that 0≤x≤9 , then the absolute maximum value of the function f occurs when x is a) 2 b) 0
B
If f(x) = e^1/x, then f'(x)= a) -e^1/x b) - e^1/x / x²
B
If f(x) = e^tan²x, then f'(x) = a) 2tan xe b)2tan xsec² x e^tan²x
B
If f(x) = eⁿ , then ln (f'(2))= a) 0 b) 2
B
If f(x) = eⁿ sin x, then the number of zeros of f on the closed interval [0,2π] is a) 1 b) 3
B
If f(x) = ln (ln x), then f'(x)= a) x b) 1/ x ln x
B
If f(x) = ln (√x), then f"(x)= a) 2/x² b) 1/2x²
B
If f(x) = sin (x/2) , then there exists a number c in the interval π/2 <x< 3π/2 that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c? a) 2π/3 b) π
B
If f(x) = sin x, then f' (π/3)= a) -1/2 b) 1/2 c) √2/2 d) √3/2 e) √3
B
If f(x) = x/tan x, then f' (π/4)= a)2 b) 1-π/2
B
If f(x) = x²eⁿ , then the graph of f is decreasing for all x such that a) x<-2 b) -2<x<0
B
If f(x) = √2x, then f'(2)= a) 1/4 b) 1/2 c) √2/2 d) 1 e)√2
B
If f(x)= ln(e^2x), the f'(x) - a) 1 b) 2
B
If the graph of y= x³ +ax² +bx -4 has a point of inflection at (1, -6) , what is the value of b? a) 0 b) It cannot be determined from the information given
B
If the line 3x-4y=0 is tangent in the first quadrant to the curve y=x³+ k, the k is a) 1/2 b) 1/4
B
If u, v, and w, are nonzero differentiable functions, then the derivative of uv/w is a) uv' +u'v/w' b) uv'w+ u'vw - uvw' / w²
B
Let f and g be differentiable functions such that f(1) =2, f'(1)=3, f'(2)=-4 g(1) =2, g'(1) =-3 g'(2)=5 If h(x)= f(g(x)), the h'(1)= a)15 b) 12
B
Let f be a continuous function on the closed interval [-3,6]. If f(-3) =-1 and f(6) =3, then the Intermediate Value Theorem guarantees that a) f(0)= 0 b) f(c) =1 for at least one c between -3 and 6
B
Let f be a function defined and continuous on the closed interval [a,b]. If f has a relative maximum at c and a<c<b , which of the following statements must be true? I. f'(c) exists II. If f'(c) exists, then f'(c)=0 III. If f'(c) exists, then f"(c)≤0 a) II only b) II and III only
B
Let f be a function that is differentiable on the open interval (1,10). If f(2) =-5 , f(5)=5, and f(9)=-5, which of the following must be true? I. f has at least 2 zeros II. The graph of f has at least one horizontal tangent. III. For some c, 2<c<5 , f(c) =3. a) I and II only b) I, II, III
B
Let f be the function defined by f(x)= {x³ for x≤0 {x for x>0 Which of the following statements about f is true? a) f is an odd function b) f'(x)>0 for x≠0
B
Let f be the function defined by the following {sin x, x<0 {x², 0≤x<1 {2-x, 1≤x<2 {x-3, x≥2 For what values of x is f NOT continuous? a) 1 only b) 2 only
B
Let f be the function given by f(x) = x³ -3x². What are all values of c that satisfy the conclusion of the Mean Value Theorem of differential calculus on the closed interval [0,3]? a) 0 only b) 2 only
B
Let f be the function given by f(x) = |x|. Which of the following statements about f are true? I. F is continuous at x=0 II. F is differentiable at x=0. III. f has an absolute minimum at x=0. a) I only b) I and III only
B
Let f(x) = |sin x-1/2|. The maximum value attained by f is a) 1 b) 3/2
B
TABLE TABLE TABLE TABLE ( Pg.30) The function f is continuous on the closed interval [0,2] and has values that are given in the table above. The equation f(x) =1/2 must have at least two solutions in the interval [0,2] if k= a) 1/2 b) 0
B
The Mean Value Theorem guarantees the existence of a special point on the graph of y=√x between (0,0) and (4,2). What are the coordinates of this point? a) (2,1) b) (1,1)
B
The function defined by f(x) = x³-3x² for all real numbers x has a relative minimum at x= a) -2 b) 0
B
The function f given by f(x) = 3x⁵ -4x³ -3x has a relative maximum at x= a) 0 b) -1
B
The function f given by f(x) = x³+12x-24 is a) decreasing for all x b) increasing for all x
B
The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. How many points of inflection does the graph of f have? a) two b) six
B
The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. The point (3,5) is on the graph of y=f(x). An equation of the line tangent to the graph of f at (3,5) is a) y=2 b) y-5=2(x-3)
B
The function f is given by f(x) = x⁴ +x² -2. On which of the intervals is f increasing? a) (-∞,0) b) (0,∞)
B
The graph of the function ƒ is shown in the figure above. Which of the following statements about ƒ is true? a) lim f(x) = lim f(x) x→a x→b b) lim f(x)=2 x→a c) lim f(x)=2 x→b d) lim f(x)=1 x→b e) lim f(x) does not exist x→a
B
The graph of y =3x⁴ -16x³+24x²+48 is concave down for a) x<0 b) 2/3 <x< 2
B
The graph of y= -5/ x-2 is concave downward for all values of x such that a) x<0 b) x>2
B
The graph of y=5x⁴ -x⁵ has a point of inflection at a) (0,0) only b) (3,162) only
B
The lim tan 3(x+h)-tan3x/h h→0 a) 0 b) 3sec² (3x) c) sec²(3x) d) 3cot (3x) e) nonexistent
B
The slope of the line tangent to the graph of y= ln (x/2) at x=4 is.... a) 1/8 b) 1/4
B
The slope of the line tangent to the graph of y= ln (x²) at x= e² is.... a) 1/e² b) 2/e² c) 4/e² d) 1/e⁴ e) 4/e⁴
B
What are all values of x for which the function f defined by f(x) =x³+3x²-9x+7 is increasing? a) -1<x<1 b) x<-3 or x>1
B
What are all values of x for which the function f defined by f(x)= (x²-3)e⁻ⁿ is increasing? a) -3<x<1 b) -1<x<3
B
What are the coordinates of the inflection point on the graph of y= (x+1)arctan x? a) (0,0) b) (1, π/2)
B
What is lim 8(½+h)⁸- 8(½)⁸/ h h→0 a) 0 b) ½ c) 1 d) The limit does not exist e) It cannot be determined from the given information
B
What is the minimum value of f(x) = xln x? a) 0 b) -1/e
B
What is the x-coordinate of the point of inflection on the graph of y= 1/3x³ +5x² +24? a) 5 b) -5
B
Which of the following functions shows that the statement "If a function is continuous at x=0, then it is differentiable at x=0", is false? a) f(x) =x³ b) f(x) = x^1/3
B
Which of the following is an equation of the line tangent to the graph of f(x) = x⁴+ 2x² at the point where f'(x) =1? a) y=8x-5 b) y=x-0.122
B
Which of the following is true about the graph of y=ln|x²-1| in the interval (-1,1)? a) It is increasing. b) It is concave down
B
d (1/x³- 1/x +x²) at x=-1 is dx a) -6 b) -4 c) 0 d) 2 e) 6
B
d/dx ln (1/1-x)= a) 1/x-1 b) 1/1-x
B
lim sin(x+h) - sin x/h is h→0 a) 1 b) cos x
B
If a ≠ 0, then lim (x²-a²/x⁴-a⁴) x→a a) 1/a² b) 1/2a² c) 1/6a² d) 0 e) nonexistent
C
If f is a continuous function on [a,b], which of the following is necessarily true? a) f' exists on (a,b). b) If f(x₀) is a maximum of f, then f'(x₀)=0 c) lim f(x) = f[lim x] for x₀ ∈ (a,b) x→x₀ x→x₀ d) f'(x)=0 for some x∈ [a,b]. e) The graph of f' is a straight line.
C
If f(x) = x^(3/2), then f'(4)= a) -6 b) -3 c) 3 d) 6 e) 8
C
If f(x) =x, then f'(5)= a)0 b) 1/5 c) 1 d) 5 e) 25/2
C
If y= cos² x- sin² x, y' = a) -1 b) 0 c) -2sin (2x) d) -2 (cos x+sin x) e) 2 (cos x- sin x)
C
If y= x²eⁿ , then dy/dx= a) 2xeⁿ b) x(x+ 2eⁿ) c) xeⁿ (x+2) d) 2x+ eⁿ e) 2x+ e
C
d/dx (2ⁿ) = a) 2ⁿ⁻¹ b) (2ⁿ⁻¹) x c) (2ⁿ) ln 2 d) (2ⁿ⁻¹) ln 2 e) 2x/ ln 2
C
lim θ→0 (1-cosθ/2sin²θ) a)0 b) 1/8 c)1/4 d)1 e) nonexistent
C
If f(x) = (x-1)/(x+1) for all x ≠ -1, then f'(1)= a) -1 b) -1/2 c) 0 d) 1/2 e) 1
D
If f(x) = (x-1)² sin x, then f'(0) = a) -2 b) -1 c) 0 d) 1 e) 2
D
If f(x) = e²ⁿ/2x , then f'(x)= a) 1 b) e²ⁿ(1-2x)/ 2x² c) e²ⁿ d) e²ⁿ(2x+1)/ x² e) e²ⁿ(2x-1)/ 2x²
D
If f(x)= -x³+x+1/x, then f'(-1)= a) 3 b) 1 c) -1 d) -3 e) -5
D
If y= ln x/x, then dy/dx= a) 1/x b) 1/x² c) ln x-1/ x² d) 1- ln x/ x² e) 1+ ln x/ x²
D
If y= tan u, u=v- 1/v, and v= ln x, what is the value of dy/dx at x=e? a) 0 b) 1/e c) 1 d) 2/e e) sec² e
D
If y=10^(ⁿ²⁻¹), then dy/dx = a) (ln 10) 10^(ⁿ²⁻¹) b) (2x) 10^(ⁿ²⁻¹) c) (ₙ²-1) 10^(ⁿ²⁻²) d) 2x (ln 10) 10^(ⁿ²⁻¹) e) x² (ln 10) 10^(ⁿ²⁻¹)
D
Let f be a function such that lim (f(2+h)-f(2))/h=5. Which of h→0 the following must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2. a) I only b) II only c) I and II only d) I and III only e) II and III only
D
d/dx (arcsin 2x)= a) -1/ 2√1-4x² b) -2/ √4x²-1 c) 1/ √1-4x² d) 2/ √1-4x² e) 2/ √4x²-1
D
d/dx cos²(x³)= a) 6x²sin (x³) cos (x³) b) 6x² cos (x³) c) sin² (x³) d) -6x² sin (x³) cos (x³) e) -2sin (x³) cos (x³)
D
lim n→∞ (3n³-5n/n³-2n²+1) a)-5 b)-2 c)1 d)3 e)nonexistent
D
lim n→∞ (4n²/n²+10,000n) a) 0 b) 1/2500 c) 1 d) 4 e) nonexistent
D
lim x→0 (xcsc x) a) -∞ b)-1 c) 0 d) 1 e) ∞
D
At x=3, the function given by f(x)= {x² , x<3 is {6x-9 , x≥3 a) undefined b) continuous but not differentiable c) differentiable but not continuous d) neither continuous nor differentiable e) both continuous and differentiable
E
GRAPH GRAPH GRAPH (Pg.52) Let f be a function that is continuous on the closed interval [-2,3] such that f'(0) does not exist, f'(2)=0 and f"(x) <0 for all x except x=0. Which of the following could be the graph of f? E
E
GRAPH GRAPH GRAPH (Pg.54) The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f? E
E
If f(x) = (2x+1)⁴ , then the 4th derivative of f(x) at x=0 is a) 0 b) 24 c) 48 d) 240 e) 384
E
If f(x) = e^(3ln(x²), then f'(x) = a) e^(3ln(x²) b) 3/x²(e^(3ln(x²) c) 6(ln x)e^(3ln(x²) d) 5x⁴ e) 6x⁵
E
If f(x) = sin (e⁻ⁿ), then f'(x)= a) -cos (e⁻ⁿ) b) cos (e⁻ⁿ)+ e⁻ⁿ c) cos (e⁻ⁿ) - e⁻ⁿ d) e⁻ⁿ cos (e⁻ⁿ) e) -e⁻ⁿ cos (e⁻ⁿ)
E
If f(x) = tan (2x), then f' (π/6)= a) √3 b) 2√3 c) 4 d) 4√3 e) 8
E
If f(x)= eⁿ, which of the following is equal to f'(e)? a) lim e^(x+h)/ h h→0 b) lim e^(x+h)-e^(e)/ h h→0 c) lim e^(e+h)-e/ h h→0 d) lim e^(x+h)-1/ h h→0 e) lim e^(e+h)-e^(e)/ h h→0
E
If f(x)= {ln x for 0<x≤2 then lim f(x) is {x²ln2 for 2<x≤4 x→2 a) ln2 b) ln8 c) ln16 d) 4 e) nonexistent
E
If lim f(x)=L, where L is a real number, which of the following x→a must be true? a) f'(a) exists. b) f(x) is continuous at x=a. c) f(x) is defined at x=a d) f(a) = L e) none of the above
E
If y= 2cos (x/2) , then d²y/dx² = a) -8cos (x/2) b) -2cos (x/2) c) -sin (x/2) d) -cos (x/2) e) -1/2cos (x/2)
E
If y=tan x- cot x, then dy/dx = a) sec x csc x b) sec x- csc x c) sec x+ csc x d) sec²x - csc²x e) sec²x+ csc²x
E
Let f and g be differentiable functions with the following properties: (i) g(x) >0 for all x (ii) f(0)=1 If h(x) = f(x)g(x) and h'(x)= f(x)g'(x), the f(x) = a) f'(x) b) g(x) c) eⁿ d) 0 e) 1
E
d/dx (ln e²ⁿ )= a) 1/e²ⁿ b) 2/ e²ⁿ c) 2x d) 1 e) 2
E
lim x→1 (x/ln x) a) 0 b) 1/e c) 1 d) e e) nonexistent
E
