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AB MCQ Set 120
43 Questions
Question 1 of 43
AB MCQ Set 120 0/43
Question 1 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 1
If \(y = x^2 e^x\), then \(\dfrac{d y}{d x} =\)
A
\(2 x e^x\)
✕
B
\(x(x + 2 e^x)\)
✕
C
\(x e^x (x + 2)\)
✕
D
\(2 x + e^x\)
✕
E
\(2 x + e\)
✕
Question 2 of 43   |  Limits and Continuity  · Level 2
What is the domain of \(f(x) = \dfrac{\sqrt{x^2 - 4}}{x - 3}\)?
A
\(\{x: x \neq 3\}\)
✕
B
\(\{x: |x| \leq 2\}\)
✕
C
\(\{x: |x| \geq 2\}\)
✕
D
\(\{x: |x| \geq 2\) and \(x \neq 3\}\)
✕
E
\(\{x: x \geq 2\) and \(x \neq 3\}\)
✕
Question 3 of 43   |  Applications of Integration  · Level 1
Particle velocity \(v(t) = e^t\). Distance from \(t = 0\) to \(t = 2\)?
A
\(e^2 - 1\)
✕
B
\(e - 1\)
✕
C
\(2 e\)
✕
D
\(e^2\)
✕
E
\(\dfrac{e^3}{3}\)
✕
Question 4 of 43   |  Analytical Applications of Differentiation  · Level 2
\(y = -5/(x - 2)\) is concave downward for what \(x\)?
A
\(x < 0\)
✕
B
\(x < 2\)
✕
C
\(x < 5\)
✕
D
\(x > 0\)
✕
E
\(x > 2\)
✕
Question 5 of 43   |  Integration and Accumulation of Change  · Level 1
\(\int \sec^2 x d x =\)
A
\(\tan x + C\)
✕
B
\(\csc^2 x + C\)
✕
C
\(\cos^2 x + C\)
✕
D
\(\dfrac{\sec^3 x}{3} + C\)
✕
E
\(2 \sec^2 x \tan x + C\)
✕
Question 6 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
If \(y = \ln \dfrac{x}{x}\), then \(\dfrac{d y}{d x} =\)
A
\(\dfrac{1}{x}\)
✕
B
\(\dfrac{1}{x^2}\)
✕
C
\(\dfrac{\ln x - 1}{x^2}\)
✕
D
\(\dfrac{1 - \ln x}{x^2}\)
✕
E
\(\dfrac{1 + \ln x}{x^2}\)
✕
Question 7 of 43   |  Integration and Accumulation of Change  · Level 2
\(\int \dfrac{x d x}{\sqrt{3 x^2 + 5}} =\)
A
\(\dfrac{1}{9}(3 x^2 + 5)^{\dfrac{3}{2}} + C\)
✕
B
\(\dfrac{1}{4}(3 x^2 + 5)^{\dfrac{3}{2}} + C\)
✕
C
\(\dfrac{1}{12}(3 x^2 + 5)^{\dfrac{1}{2}} + C\)
✕
D
\(\dfrac{1}{3}(3 x^2 + 5)^{\dfrac{1}{2}} + C\)
✕
E
\(\dfrac{3}{2}(3 x^2 + 5)^{\dfrac{1}{2}} + C\)
✕
Question 8 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
If \(x + 2 x y - y^2 = 2\), then at \((1, 1)\), \(\dfrac{d y}{d x} =\)
A
\(\dfrac{3}{2}\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(0\)
✕
D
\(-\dfrac{3}{2}\)
✕
E
nonexistent
✕
Question 9 of 43   |  Integration and Accumulation of Change  · Level 2
If \(\displaystyle\int_{0}^{k} (2 k x - x^2) d x = 18\), then \(k =\)
A
\(-9\)
✕
B
\(-3\)
✕
C
\(3\)
✕
D
\(9\)
✕
E
\(18\)
✕
Question 10 of 43   |  MCQ  · Level 2
Equation of tangent to \(f(x) = x(1 - 2 x)^3\) at \((1, -1)\)
A
\(y = -7 x + 6\)
✕
B
\(y = -6 x + 5\)
✕
C
\(y = -2 x + 1\)
✕
D
\(y = 2 x - 3\)
✕
E
\(y = 7 x - 8\)
✕
Question 11 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 1
If \(f(x) = \sin x\), then \(f'\left(\dfrac{\pi}{3}\right) =\)
A
\(-\dfrac{1}{2}\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(\dfrac{\sqrt{2}}{2}\)
✕
D
\(\dfrac{\sqrt{3}}{2}\)
✕
E
\(\sqrt{3}\)
✕
Question 12 of 43   |  Integration and Accumulation of Change  · Level 1
\(\displaystyle\int_{0}^{c} f'(x) d x\) where \(f\) has continuous derivative
A
\(f(c) - f(0)\)
✕
B
\(|f(c) - f(0)|\)
✕
C
\(f(c)\)
✕
D
\(f(x) + c\)
✕
E
\(f''(c) - f''(0)\)
✕
Question 13 of 43   |  Integration and Accumulation of Change  · Level 3
\(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{\cos \theta}{\sqrt{1 + \sin \theta}} d \theta =\)
A
\(-2(\sqrt{2} - 1)\)
✕
B
\(-2 \sqrt{2}\)
✕
C
\(2 \sqrt{2}\)
✕
D
\(2(\sqrt{2} - 1)\)
✕
E
\(2(\sqrt{2} + 1)\)
✕
Question 14 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 1
If \(f(x) = \sqrt{2 x}\), then \(f'(2) =\)
A
\(\dfrac{1}{4}\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(\dfrac{\sqrt{2}}{2}\)
✕
D
\(1\)
✕
E
\(\sqrt{2}\)
✕
Question 15 of 43   |  Contextual Applications of Differentiation  · Level 2
\(x(t) = t^3 - 3 t^2 - 9 t + 1\). Particle at rest when?
A
No values
✕
B
\(1\) only
✕
C
\(3\) only
✕
D
\(5\) only
✕
E
\(1\) and \(3\)
✕
Question 16 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{0}^{1} (3 x - 2)^2 d x =\)
A
\(-\dfrac{7}{3}\)
✕
B
\(-\dfrac{7}{9}\)
✕
C
\(\dfrac{1}{9}\)
✕
D
\(1\)
✕
E
\(3\)
✕
Question 17 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
If \(y = 2 \cos\left(\dfrac{x}{2}\right)\), then \(\dfrac{d^2 y}{d x^2} =\)
A
\(-8 \cos\left(\dfrac{x}{2}\right)\)
✕
B
\(-2 \cos\left(\dfrac{x}{2}\right)\)
✕
C
\(-\sin\left(\dfrac{x}{2}\right)\)
✕
D
\(-\cos\left(\dfrac{x}{2}\right)\)
✕
E
\(-\left(\dfrac{1}{2}\right) \cos\left(\dfrac{x}{2}\right)\)
✕
Question 18 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{2}^{3} \dfrac{x}{x^2 + 1} d x =\)
A
\(\dfrac{1}{2} \ln \dfrac{3}{2}\)
✕
B
\(\dfrac{1}{2} \ln 2\)
✕
C
\(\ln 2\)
✕
D
\(2 \ln 2\)
✕
E
\(\dfrac{1}{2} \ln 5\)
✕
Question 19 of 43   |  Analytical Applications of Differentiation  · Level 3
Polynomial degree > 2, \(a \neq b\), \(f(a)=f(b)=1\). Which must be true between \(a\) and \(b\)?
I. \(f=0\)
II. \(f'=0\)
III. \(f''=0\)
A
None
✕
B
I only
✕
C
II only
✕
D
I and II only
✕
E
I, II, and III
✕
Question 20 of 43   |  Applications of Integration  · Level 2
Area enclosed by \(y = x\) and \(y = x^2 - 3 x + 3\)
A
\(\dfrac{2}{3}\)
✕
B
\(1\)
✕
C
\(\dfrac{4}{3}\)
✕
D
\(2\)
✕
E
\(\dfrac{14}{3}\)
✕
Question 21 of 43   |  MCQ  · Level 2
If \(\ln x - \ln\left(\dfrac{1}{x}\right) = 2\), then \(x =\)
A
\(1/e^2\)
✕
B
\(\dfrac{1}{e}\)
✕
C
\(e\)
✕
D
\(2 e\)
✕
E
\(e^2\)
✕
Question 22 of 43   |  MCQ  · Level 3
\(f'(x) = \cos x\), \(g'(x) = 1\), \(f(0) = g(0) = 0\). \(\operatorname*{lim}\limits_{x\rightarrow 0} f(x)/g(x) =\)
A
\(\dfrac{\pi}{2}\)
✕
B
\(1\)
✕
C
\(0\)
✕
D
\(-1\)
✕
E
nonexistent
✕
Question 23 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 4
\(\dfrac{d}{d x} (x^{\ln x}) =\)
A
\(x^{\ln x}\)
✕
B
\((\ln x)^x\)
✕
C
\(\left(\dfrac{2}{x}\right)(\ln x)(x^{\ln x})\)
✕
D
\((\ln x)(x^{\ln x - 1})\)
✕
E
\(2(\ln x)(x^{\ln x})\)
✕
Question 24 of 43   |  Integration and Accumulation of Change  · Level 1
For \(x > 1\), \(f(x) = \displaystyle\int_{1}^{x} d \dfrac{t}{t}\), then \(f'(x) =\)
A
\(1\)
✕
B
\(\dfrac{1}{x}\)
✕
C
\(\ln x - 1\)
✕
D
\(\ln x\)
✕
E
\(e^x\)
✕
Question 25 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{0}^{\dfrac{\pi}{2}} x \cos x d x =\)
A
\(-\dfrac{\pi}{2}\)
✕
B
\(-1\)
✕
C
\(1 - \dfrac{\pi}{2}\)
✕
D
\(1\)
✕
E
\(\dfrac{\pi}{2} - 1\)
✕
Question 26 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
At \(x = 3\), \(f(x) = x^2\) for \(x<3\), \(f(x) = 6x - 9\) for \(x \geq 3\)
A
undefined
✕
B
continuous but not differentiable
✕
C
differentiable but not continuous
✕
D
neither continuous nor differentiable
✕
E
both continuous and differentiable
✕
Question 27 of 43   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{1}^{4} |x - 3| d x =\)
A
\(-\dfrac{3}{2}\)
✕
B
\(\dfrac{3}{2}\)
✕
C
\(\dfrac{5}{2}\)
✕
D
\(\dfrac{9}{2}\)
✕
E
\(5\)
✕
Question 28 of 43   |  Differentiation: Definition and Fundamental Properties  · Level 2
\(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\tan(3(x+h)) - \tan(3x)}{h}\)
A
\(0\)
✕
B
\(3 \sec^2(3x)\)
✕
C
\(\sec^2(3 x)\)
✕
D
\(3 \cot(3 x)\)
✕
E
nonexistent
✕
Question 29 of 43   |  Applications of Integration  · Level 3
Region in Q1 enclosed by \(y = e^{2x}\), \(x = 1\), axes. Rotated about y-axis. Volume?
A
\(2 \pi \displaystyle\int_{0}^{1} x e^{2x} d x\)
✕
B
\(2 \pi \displaystyle\int_{0}^{1} e^{2x} d x\)
✕
C
\(\pi \displaystyle\int_{0}^{1} e^{4x} d x\)
✕
D
\(\pi \displaystyle\int_{0}^{e} y \ln y d y\)
✕
E
\(\left(\dfrac{\pi}{4}\right) \displaystyle\int_{0}^{e} \ln^2 y d y\)
✕
Question 30 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
If \(f(x) = x/(x+1)\), find \(f^{-1}(x)\)
A
\(\dfrac{x - 1}{x}\)
✕
B
\(\dfrac{x + 1}{x}\)
✕
C
\(\dfrac{x}{1 - x}\)
✕
D
\(\dfrac{x}{x + 1}\)
✕
E
\(x\)
✕
Question 31 of 43   |  MCQ  · Level 2
Which does NOT have period \(\pi\)?
A
\(\sin\left(\dfrac{x}{2}\right)\)
✕
B
\(|\sin x|\)
✕
C
\(\sin^2 x\)
✕
D
\(\tan x\)
✕
E
\(\tan^2 x\)
✕
Question 32 of 43   |  Analytical Applications of Differentiation  · Level 3
Absolute max of \(f(x) = x^3 - 3 x^2 + 12\) on \([-2, 4]\) at \(x =\)
A
\(4\)
✕
B
\(2\)
✕
C
\(1\)
✕
D
\(0\)
✕
E
\(-2\)
✕
Question 33 of 43   |  MCQ  · Level 2
\(4 \cos\left(x + \dfrac{\pi}{3}\right) =\)
A
\(2 \sqrt{3} \cos x - 2 \sin x\)
✕
B
\(2 \cos x - 2 \sqrt{3} \sin x\)
✕
C
\(2 \cos x + 2 \sqrt{3} \sin x\)
✕
D
\(2 \sqrt{3} \cos x + 2 \sin x\)
✕
E
\(4 \cos x + 2\)
✕
Question 34 of 43   |  Applications of Integration  · Level 3
Average value of \(y\) for \(y = 3 x - x^2\) in first quadrant
A
\(-6\)
✕
B
\(-2\)
✕
C
\(\dfrac{3}{2}\)
✕
D
\(\dfrac{9}{4}\)
✕
E
\(\dfrac{9}{2}\)
✕
Question 35 of 43   |  MCQ  · Level 3
If \(f(x) = e^x \sin x\), number of zeros on \([0, 2\pi]\) is
A
\(0\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(3\)
✕
E
\(4\)
✕
Question 36 of 43   |  Integration and Accumulation of Change  · Level 3
For \(x > 0\), \(\int \left(\dfrac{1}{x}\right) \displaystyle\int_{1}^{x} d \dfrac{u}{u} d x =\)
A
\(1/x^3 + C\)
✕
B
\(8/x^4 - 2/x^2 + C\)
✕
C
\(\ln(\ln x) + C\)
✕
D
\(\ln(x^2)/2 + C\)
✕
E
\((\ln x)^2/2 + C\)
✕
Question 37 of 43   |  Integration and Accumulation of Change  · Level 1
\(\displaystyle\int_{1}^{10} f d x = 4\), \(\displaystyle\int_{10}^3 f d x = 7\), then \(\displaystyle\int_{1}^{3} f d x =\)
A
\(-3\)
✕
B
\(0\)
✕
C
\(3\)
✕
D
\(10\)
✕
E
\(11\)
✕
Question 38 of 43   |  Contextual Applications of Differentiation  · Level 3
Rectangle: \(\dfrac{d z}{d t} = 1\), \(\dfrac{d x}{d t} = 3 \dfrac{d y}{d t}\). At \(x=4, y=3\), find \(\dfrac{d x}{d t}\).
A
\(\dfrac{1}{3}\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(\sqrt{5}\)
✕
E
\(5\)
✕
Question 39 of 43   |  Limits and Continuity  · Level 2
\(\operatorname*{lim}\limits_{x \rightarrow 3} f(x) = 7\), which must be true?
I. continuous at 3
II. differentiable at 3
III. \(f(3) = 7\)
A
None
✕
B
II only
✕
C
III only
✕
D
I and III only
✕
E
I, II, and III
✕
Question 40 of 43   |  Limits and Continuity  · Level 2
Which has \(y = 1\) as asymptote?
A
\(y = \ln x\)
✕
B
\(y = \sin x\)
✕
C
\(y = x/(x+1)\)
✕
D
\(y = x^2/(x-1)\)
✕
E
\(y = e^{-x}\)
✕
Question 41 of 43   |  Applications of Integration  · Level 3
Ellipse \(x^2 + 9 y^2 = 9\) revolved about x-axis. Volume?
A
\(2 \pi\)
✕
B
\(4 \pi\)
✕
C
\(6 \pi\)
✕
D
\(9 \pi\)
✕
E
\(12 \pi\)
✕
Question 42 of 43   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
\(f, g\) odd. Which must be odd?
I. \(f(g(x))\)
II. \(f+g\)
III. \(fg\)
A
I only
✕
B
II only
✕
C
I and II only
✕
D
II and III only
✕
E
I, II, and III
✕
Question 43 of 43   |  Analytical Applications of Differentiation  · Level 3
Cylinder volume \(16 \pi\), minimize tin (surface area). Height?
A
\(2 \sqrt[3]{2}\)
✕
B
\(2 \sqrt{2}\)
✕
C
\(2 \sqrt[3]{4}\)
✕
D
\(4\)
✕
E
\(8\)
✕

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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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