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AB MCQ Set 130
43 Questions
Question 1 of 43
AB MCQ Set 130 0/43
Question 1 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 1
If \(f(x) = x^{\dfrac{3}{2}}\), then \(f'(4) =\)
A
\(-6\)
✕
B
\(-3\)
✕
C
\(3\)
✕
D
\(6\)
✕
E
\(8\)
✕
Question 2 of 43   |  Limits and Continuity  · Level 2
\(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{3 n^3 - 5 n}{n^3 - 2 n^2 + 1} =\)
A
\(-5\)
✕
B
\(-2\)
✕
C
\(1\)
✕
D
\(3\)
✕
E
nonexistent
✕
Question 3 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
If \(x^3 + 3 x y + 2 y^3 = 17\), then \(\dfrac{d y}{d x} =\)
A
\(-\dfrac{x^2 + y}{x + 2 y^2}\)
✕
B
\(-\dfrac{x^2 + y}{x + y^2}\)
✕
C
\(-\dfrac{x^2 + y}{x + 2 y}\)
✕
D
\(-\dfrac{x^2 + y}{2 y^2}\)
✕
E
\(\dfrac{-x^2}{1 + 2 y^2}\)
✕
Question 4 of 43   |  Limits and Continuity  · Level 1
\(f\) continuous, \(f(x) = \dfrac{x^2 - 4}{x + 2}\) for \(x \neq -2\). \(f(-2) =\)
A
\(-4\)
✕
B
\(-2\)
✕
C
\(-1\)
✕
D
\(0\)
✕
E
\(2\)
✕
Question 5 of 43   |  Applications of Integration  · Level 2
Area enclosed by \(y = 1/(x-1)\), x-axis, \(x = 3\), \(x = 4\)
A
\(\dfrac{5}{36}\)
✕
B
\(\ln\left(\dfrac{2}{3}\right)\)
✕
C
\(\ln\left(\dfrac{4}{3}\right)\)
✕
D
\(\ln\left(\dfrac{3}{2}\right)\)
✕
E
\(\ln 6\)
✕
Question 6 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 3
Tangent to \(y = \dfrac{2x+3}{3x-2}\) at \((1, 5)\)
A
\(13 x - y = 8\)
✕
B
\(13 x + y = 18\)
✕
C
\(x - 13 y = 64\)
✕
D
\(x + 13 y = 66\)
✕
E
\(-2 x + 3 y = 13\)
✕
Question 7 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 1
If \(y = \tan x - \cot x\), then \(\dfrac{d y}{d x} =\)
A
\(\sec x \csc x\)
✕
B
\(\sec x - \csc x\)
✕
C
\(\sec x + \csc x\)
✕
D
\(\sec^2 x - \csc^2 x\)
✕
E
\(\sec^2 x + \csc^2 x\)
✕
Question 8 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
\(h(x) = f(g(x))\) where \(f(x) = 3 x^2 - 1\), \(g(x) = |x|\), then \(h(x) =\)
A
\(3 x^3 - |x|\)
✕
B
\(|3 x^2 - 1|\)
✕
C
\(3 x^2 |x| - 1\)
✕
D
\(3 |x| - 1\)
✕
E
\(3 x^2 - 1\)
✕
Question 9 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
\(f(x) = (x-1)^2 \sin x\), \(f'(0) =\)
A
\(-2\)
✕
B
\(-1\)
✕
C
\(0\)
✕
D
\(1\)
✕
E
\(2\)
✕
Question 10 of 43   |  Applications of Integration  · Level 3
Particle accel \(a(t) = 6t - 2\). \(v(3) = 25\), \(x(1) = 10\). Position \(x(t) =\)
A
\(9 t^2 + 1\)
✕
B
\(3 t^2 - 2 t + 4\)
✕
C
\(t^3 - t^2 + 4 t + 6\)
✕
D
\(t^3 - t^2 + 9 t - 20\)
✕
E
\(36 t^3 - 4 t^2 - 77 t + 55\)
✕
Question 11 of 43   |  Integration and Accumulation of Change  · Level 2
\(f, g\) continuous, \(f \geq 0\). Which must be true?
I. \(\int fg = (\int f)(\int g)\)
II. \(\int (f+g) = \int f + \int g\)
III. \(\int \sqrt{f} = \sqrt{\int f}\)
A
I only
✕
B
II only
✕
C
III only
✕
D
II and III only
✕
E
I, II, and III
✕
Question 12 of 43   |  MCQ  · Level 1
Period of \(2 \cos(3x)\)
A
\(2 \dfrac{\pi}{3}\)
✕
B
\(2 \pi\)
✕
C
\(6 \pi\)
✕
D
\(2\)
✕
E
\(3\)
✕
Question 13 of 43   |  Integration and Accumulation of Change  · Level 2
\(\int \dfrac{3 x^2}{\sqrt{x^3 + 1}} d x =\)
A
\(2 \sqrt{x^3 + 1} + C\)
✕
B
\(\dfrac{3}{2} \sqrt{x^3 + 1} + C\)
✕
C
\(\sqrt{x^3 + 1} + C\)
✕
D
\(\ln \sqrt{x^3 + 1} + C\)
✕
E
\(\ln(x^3 + 1) + C\)
✕
Question 14 of 43   |  Analytical Applications of Differentiation  · Level 3
\(f(x) = (x-2)(x-3)^2\) has rel max at \(x =\)
A
\(-3\)
✕
B
\(-\dfrac{7}{3}\)
✕
C
\(-\dfrac{5}{2}\)
✕
D
\(\dfrac{7}{3}\)
✕
E
\(\dfrac{5}{2}\)
✕
Question 15 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
Slope of normal to \(y = 2 \ln(\sec x)\) at \(x = \dfrac{\pi}{4}\)
A
\(-2\)
✕
B
\(-\dfrac{1}{2}\)
✕
C
\(\dfrac{1}{2}\)
✕
D
\(2\)
✕
E
nonexistent
✕
Question 16 of 43   |  Integration and Accumulation of Change  · Level 2
\(\int (x^2 + 1)^2 d x =\)
A
\(\dfrac{(x^2+1)^3}{3} + C\)
✕
B
\(\dfrac{(x^2+1)^3}{6 x} + C\)
✕
C
\((x^3/3 + x)^2 + C\)
✕
D
\(\dfrac{2 x (x^2+1)^3}{3} + C\)
✕
E
\(\dfrac{x^5}{5} + \dfrac{2 x^3}{3} + x + C\)
✕
Question 17 of 43   |  Analytical Applications of Differentiation  · Level 3
MVT for \(f(x) = \sin\left(\dfrac{x}{2}\right)\) on \(\left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right)\)
A
\(2 \dfrac{\pi}{3}\)
✕
B
\(3 \dfrac{\pi}{4}\)
✕
C
\(5 \dfrac{\pi}{6}\)
✕
D
\(\pi\)
✕
E
\(3 \dfrac{\pi}{2}\)
✕
Question 18 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
\(f(x) = x^3\) for \(x \leq 0\), \(x\) for \(x > 0\). Which is true?
A
\(f\) odd
✕
B
\(f\) discontinuous at 0
✕
C
\(f\) has rel max
✕
D
\(f'(0) = 0\)
✕
E
\(f'(x) > 0\) for \(x \neq 0\)
✕
Question 19 of 43   |  Applications of Integration  · Level 3
Region in Q1 enclosed by \(y = (x+1)^{\dfrac{1}{3}}\), \(x = 7\), axes. Revolved about y-axis.
A
\(\pi \displaystyle\int_{0}^{7} (x+1)^{\dfrac{2}{3}} d x\)
✕
B
\(2 \pi \displaystyle\int_{0}^{7} x (x+1)^{\dfrac{1}{3}} d x\)
✕
C
\(\pi \displaystyle\int_{0}^{2} (x+1)^{\dfrac{2}{3}} d x\)
✕
D
\(2 \pi \displaystyle\int_{0}^{2} x (x+1)^{\dfrac{1}{3}} d x\)
✕
E
\(\pi \displaystyle\int_{0}^{7} (y^3 - 1)^2 d y\)
✕
Question 20 of 43   |  Analytical Applications of Differentiation  · Level 2
Inflection point of \(y = 1/x^2 - 1/x^3\)
A
\(0\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(3\)
✕
E
no value
✕
Question 21 of 43   |  Integration and Accumulation of Change  · Level 3
Antiderivative of \(1/(x^2 - 2 x + 2)\)
A
\(-(x^2 - 2x + 2)^{-2}\)
✕
B
\(\ln(x^2 - 2x + 2)\)
✕
C
\(\ln|\dfrac{x-2}{x+1}|\)
✕
D
\(arcsec(x-1)\)
✕
E
\(\arctan(x-1)\)
✕
Question 22 of 43   |  Analytical Applications of Differentiation  · Level 2
Critical points of \(f(x) = (x+2)^5 (x-3)^4\)
A
One
✕
B
Two
✕
C
Three
✕
D
Five
✕
E
Nine
✕
Question 23 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
\(f(x) = (x^2 - 2x - 1)^{\dfrac{2}{3}}\), \(f'(0) =\)
A
\(\dfrac{4}{3}\)
✕
B
\(0\)
✕
C
\(-\dfrac{2}{3}\)
✕
D
\(-\dfrac{4}{3}\)
✕
E
\(-2\)
✕
Question 24 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 1
\(\dfrac{d}{d x} (2^x) =\)
A
\(2^{x-1}\)
✕
B
\((2^{x-1}) x\)
✕
C
\((2^x) \ln 2\)
✕
D
\((2^{x-1}) \ln 2\)
✕
E
\(\dfrac{2x}{\ln} 2\)
✕
Question 25 of 43   |  Contextual Applications of Differentiation  · Level 3
[Calc] \(s(t) = -4 \cos t - t^2/2 + 10\). \(v\) when \(a = 0\)?
A
\(-5.19\)
✕
B
\(0.74\)
✕
C
\(1.32\)
✕
D
\(2.55\)
✕
E
\(8.13\)
✕
Question 26 of 43   |  Analytical Applications of Differentiation  · Level 2
\(f(x) = x^3 + 12 x - 24\) is
A
incr/decr/incr at \(-2, 2\)
✕
B
decr then incr at 0
✕
C
increasing for all \(x\)
✕
D
decreasing for all \(x\)
✕
E
decr/incr/decr at \(-2, 2\)
✕
Question 27 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{1}^{500} (13^x - 11^x) d x + \displaystyle\int_{2}^{500} (11^x - 13^x) d x =\)
A
\(0.000\)
✕
B
\(14.946\)
✕
C
\(34.415\)
✕
D
\(46.000\)
✕
E
\(136.364\)
✕
Question 28 of 43   |  Limits and Continuity  · Level 3
\(\operatorname*{lim}\limits_{\theta \rightarrow 0} \dfrac{1 - \cos \theta}{2 \sin^2 \theta}\)
A
\(0\)
✕
B
\(\dfrac{1}{8}\)
✕
C
\(\dfrac{1}{4}\)
✕
D
\(1\)
✕
E
nonexistent
✕
Question 29 of 43   |  Applications of Integration  · Level 3
Region under \(y = \sqrt{x}\), x-axis, \(x=3\) revolved about x-axis
A
\(3 \pi\)
✕
B
\(2 \sqrt{3} \pi\)
✕
C
\(9 \dfrac{\pi}{2}\)
✕
D
\(9 \pi\)
✕
E
\(36 \sqrt{3} \dfrac{\pi}{5}\)
✕
Question 30 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
\(f(x) = e^{3 \ln(x^2)}\), \(f'(x) =\)
A
\(e^{3 \ln(x^2)}\)
✕
B
\((3/x^2) e^{3 \ln(x^2)}\)
✕
C
\(6(\ln x) e^{3 \ln(x^2)}\)
✕
D
\(5 x^4\)
✕
E
\(6 x^5\)
✕
Question 31 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{0}^{\sqrt{3}} d \dfrac{x}{\sqrt{4 - x^2}}\)
A
\(\dfrac{\pi}{3}\)
✕
B
\(\dfrac{\pi}{4}\)
✕
C
\(\dfrac{\pi}{6}\)
✕
D
\(\left(\dfrac{1}{2}\right) \ln 2\)
✕
E
\(-\ln 2\)
✕
Question 32 of 43   |  Differential Equations  · Level 2
\(\dfrac{d y}{d x} = 2 y^2\), \(y(1) = -1\), find \(y(2)\)
A
\(-\dfrac{2}{3}\)
✕
B
\(-\dfrac{1}{3}\)
✕
C
\(0\)
✕
D
\(\dfrac{1}{3}\)
✕
E
\(\dfrac{2}{3}\)
✕
Question 33 of 43   |  Contextual Applications of Differentiation  · Level 3
25-ft ladder, top sliding down at 3 ft/min. When top is 7 ft up, \(\dfrac{d x}{d t}\)?
A
\(-\dfrac{7}{8}\)
✕
B
\(-\dfrac{7}{24}\)
✕
C
\(\dfrac{7}{24}\)
✕
D
\(\dfrac{7}{8}\)
✕
E
\(\dfrac{21}{25}\)
✕
Question 34 of 43   |  Limits and Continuity  · Level 2
\(y = \dfrac{a x + b}{x + c}\) has horizontal asymptote \(y = 2\) and vertical asymptote \(x = -3\). Find \(a + c\).
A
\(-5\)
✕
B
\(-1\)
✕
C
\(0\)
✕
D
\(1\)
✕
E
\(5\)
✕
Question 35 of 43   |  Integration and Accumulation of Change  · Level 3
[Calc] \(\displaystyle\int_{0}^{2} e^{x^2} d x\) approx by 2 inscribed rectangles vs trapezoidal \(n=2\). Difference?
A
\(53.60\)
✕
B
\(30.51\)
✕
C
\(27.80\)
✕
D
\(26.80\)
✕
E
\(12.78\)
✕
Question 36 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
\(f'(a)\) definition:
I. \(\operatorname*{lim}\limits_{h\rightarrow 0} (f(a+h)-f(a))/h\)
II. \(\operatorname*{lim}\limits_{x\rightarrow a} \dfrac{f(x)-f(a)}{x-a}\)
III. \(\operatorname*{lim}\limits_{x\rightarrow a} (f(x+h)-f(x))/h\)
A
I only
✕
B
II only
✕
C
I and II only
✕
D
I and III only
✕
E
I, II, and III
✕
Question 37 of 43   |  Integration and Accumulation of Change  · Level 3
\(f''(x) = 2 x - \cos x\), find \(f\)
A
\(x^3/3 + \cos x - x + 1\)
✕
B
\(x^3/3 - \cos x - x + 1\)
✕
C
\(x^3 + \cos x - x + 1\)
✕
D
\(x^2 - \sin x + 1\)
✕
E
\(x^2 + \sin x + 1\)
✕
Question 38 of 43   |  Contextual Applications of Differentiation  · Level 3
Radius increasing, area rate equals circumference rate. Radius?
A
\(\dfrac{1}{\pi}\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(\dfrac{2}{\pi}\)
✕
D
\(1\)
✕
E
\(2\)
✕
Question 39 of 43   |  Integration and Accumulation of Change  · Level 2
\(\dfrac{d}{d x} \displaystyle\int_{0}^{x} \cos(2 \pi u) d u =\)
A
\(0\)
✕
B
\((1/(2 \pi)) \sin x\)
✕
C
\((1/(2 \pi)) \cos(2 \pi x)\)
✕
D
\(\cos(2 \pi x)\)
✕
E
\(2 \pi \cos(2 \pi x)\)
✕
Question 40 of 43   |  Differential Equations  · Level 3
[Calc] Puppy 2 lb at birth, 3.5 lb at 2 months. Exponential growth. Weight at 3 months?
A
\(4.2\)
✕
B
\(4.6\)
✕
C
\(4.8\)
✕
D
\(5.6\)
✕
E
\(6.5\)
✕
Question 41 of 43   |  Integration and Accumulation of Change  · Level 2
\(\int x f(x) d x =\)
A
\(x f(x) - \int x f'(x) d x\)
✕
B
\((x^2/2) f(x) - \int (x^2/2) f'(x) d x\)
✕
C
\(x f(x) - (x^2/2) f(x) + C\)
✕
D
\(x f(x) - \int f'(x) d x\)
✕
E
\((x^2/2) \int f(x) d x\)
✕
Question 42 of 43   |  Analytical Applications of Differentiation  · Level 2
[Calc] Min of \(f(x) = x \ln x\) on \((0, \infty)\)
A
\(-e\)
✕
B
\(-1\)
✕
C
\(-\dfrac{1}{e}\)
✕
D
\(0\)
✕
E
no min
✕
Question 43 of 43   |  Contextual Applications of Differentiation  · Level 3
[Calc] Newton's method on \(x^3 + x - 1 = 0\), \(x_1 = 1\), find \(x_3\)
A
\(0.682\)
✕
B
\(0.686\)
✕
C
\(0.694\)
✕
D
\(0.750\)
✕
E
\(1.637\)
✕

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Reference Sheet

Area & Circumference

Circle$A = \pi r^2$,  $C = 2\pi r$
Rectangle$A = lw$
Triangle$A = \tfrac{1}{2}bh$
Trapezoid$A = \tfrac{1}{2}(b_1+b_2)h$

Volume

Box$V = lwh$
Cylinder$V = \pi r^2 h$
Sphere$V = \tfrac{4}{3}\pi r^3$
Cone$V = \tfrac{1}{3}\pi r^2 h$
Pyramid$V = \tfrac{1}{3}lwh$

Triangles

Pythagorean Thm$a^2 + b^2 = c^2$
30-60-90sides: $1,\, \sqrt{3},\, 2$
45-45-90sides: $1,\, 1,\, \sqrt{2}$
Triangle Anglessum $= 180°$

Other Facts

Circle Degrees$360° = 2\pi \text{ rad}$
Exterior Angle= sum of non-adjacent interior angles

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is $2\pi$.

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