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BC MCQ Set 110
34 Questions
Question 1 of 34
BC MCQ Set 110 0/34
Question 1 of 34   |  Integration and Accumulation of Change  · Level 1
\(\displaystyle\int_{0}^{1} \sqrt{x}(x + 1) d x =\)
A
\(0\)
✕
B
\(1\)
✕
C
\(\dfrac{16}{15}\)
✕
D
\(\dfrac{7}{5}\)
✕
E
\(2\)
✕
Question 2 of 34   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 2
\(x = e^{2t}\), \(y = \sin(2t)\), \(\dfrac{d y}{d x} =\)
A
\(4 e^{2t} \cos(2t)\)
✕
B
\(\dfrac{e^{2t}}{\cos(2t)}\)
✕
C
\(\dfrac{\sin(2t)}{2 e^{2t}}\)
✕
D
\(\dfrac{\cos(2t)}{2 e^{2t}}\)
✕
E
\(\dfrac{\cos(2t)}{e^{2t}}\)
✕
Question 3 of 34   |  Analytical Applications of Differentiation  · Level 2
\(f(x) = 3 x^5 - 4 x^3 - 3 x\) has rel max at
A
\(-1\)
✕
B
\(-\dfrac{\sqrt{5}}{5}\)
✕
C
\(0\)
✕
D
\(\dfrac{\sqrt{5}}{5}\)
✕
E
\(1\)
✕
Question 4 of 34   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
\(\dfrac{d}{d x} (x e^{\ln x^2}) =\)
A
\(1 + 2 x\)
✕
B
\(x + x^2\)
✕
C
\(3 x^2\)
✕
D
\(x^3\)
✕
E
\(x^2 + x^3\)
✕
Question 5 of 34   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
\(f(x) = (x-1)^{\dfrac{3}{2}} + e^{x-2}/2\), \(f'(2) =\)
A
\(1\)
✕
B
\(\dfrac{3}{2}\)
✕
C
\(2\)
✕
D
\(\dfrac{7}{2}\)
✕
E
\((3+e)/2\)
✕
Question 6 of 34   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
Slope of normal to \(y = \sqrt{16 - x}\) at \((0, 4)\)
A
\(8\)
✕
B
\(4\)
✕
C
\(\dfrac{1}{8}\)
✕
D
\(-\dfrac{1}{8}\)
✕
E
\(-8\)
✕
Question 7 of 34   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
\(y = x y + x^2 + 1\), find \(y'\) at \(x = -1\)
A
\(\dfrac{1}{2}\)
✕
B
\(-\dfrac{1}{2}\)
✕
C
\(-1\)
✕
D
\(-2\)
✕
E
nonexistent
✕
Question 8 of 34   |  Integration and Accumulation of Change  · Level 2
\(\displaystyle\int_{1}^{\infty} x/(1+x^2)^2 d x\)
A
\(-\dfrac{1}{2}\)
✕
B
\(-\dfrac{1}{4}\)
✕
C
\(\dfrac{1}{4}\)
✕
D
\(\dfrac{1}{2}\)
✕
E
divergent
✕
Question 9 of 34   |  Contextual Applications of Differentiation  · Level 3
\(a(t) = 2t - 7\), \(v(0) = 6\). Particle farthest right at \(t =\)
A
\(0\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(3\)
✕
E
\(4\)
✕
Question 10 of 34   |  Infinite Sequences and Series  · Level 2
Sum of geometric \(\dfrac{3}{2} + \dfrac{9}{16} + \dfrac{27}{128} + ...\)
A
\(1.60\)
✕
B
\(2.35\)
✕
C
\(2.40\)
✕
D
\(2.45\)
✕
E
\(2.50\)
✕
Question 11 of 34   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
Length of \(x = \cos^3 t\), \(y = \sin^3 t\) for \(0 \leq t \leq \dfrac{\pi}{2}\)
A
\(\int \sqrt{3 \cos^2 t + 3 \sin^2 t} d t\)
✕
B
\(\int \sqrt{-3 \cos^2 t \sin t + 3 \sin^2 t \cos t} d t\)
✕
C
\(\int \sqrt{9 \cos^4 t + 9 \sin^4 t} d t\)
✕
D
\(\int \sqrt{9 \cos^4 t \sin^2 t + 9 \sin^4 t \cos^2 t} d t\)
✕
E
\(\int \sqrt{\cos^6 t + \sin^6 t} d t\)
✕
Question 12 of 34   |  Differentiation: Definition and Fundamental Properties  · Level 2
\(\operatorname*{lim}\limits_{h\rightarrow 0}\dfrac{e^h - 1}{2 h}\)
A
\(0\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(1\)
✕
D
\(e\)
✕
E
nonexistent
✕
Question 13 of 34   |  Infinite Sequences and Series  · Level 3
Third-degree Taylor of \(f(x) = \ln(3 - x)\) about \(x = 2\)
A
\(-(x-2) + (x-2)^2/2 - (x-2)^3/3\)
✕
B
\(-(x-2) - (x-2)^2/2 - (x-2)^3/3\)
✕
C
\((x-2) + (x-2)^2 + (x-2)^3\)
✕
D
\((x-2) + (x-2)^2/2 + (x-2)^3/3\)
✕
E
\((x-2) - (x-2)^2/2 + (x-2)^3/3\)
✕
Question 14 of 34   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
Vertical tangent of \(x = t^3 - t^2 - 1\), \(y = t^4 + 2 t^2 - 8 t\) at
A
\(0\) only
✕
B
\(1\) only
✕
C
\(0\) and \(\dfrac{2}{3}\) only
✕
D
\(0, \dfrac{2}{3}, 1\)
✕
E
No value
✕
Question 15 of 34   |  Infinite Sequences and Series  · Level 3
\(\sum (x-2)^n/(n \cdot 3^n)\) converges for
A
\(-3 \leq x \leq 3\)
✕
B
\(-3 < x < 3\)
✕
C
\(-1 < x \leq 5\)
✕
D
\(-1 \leq x \leq 5\)
✕
E
\(-1 \leq x < 5\)
✕
Question 16 of 34   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
Area inside \(r = 2 \cos \theta\) outside \(r = \cos \theta\)
A
\(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\)
✕
B
\(3 \displaystyle\int_{0}^{\pi} \cos^2 \theta d \theta\)
✕
C
\(\left(\dfrac{3}{2}\right) \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\)
✕
D
\(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos \theta d \theta\)
✕
E
\(3 \displaystyle\int_{0}^{\pi} \cos \theta d \theta\)
✕
Question 17 of 34   |  Contextual Applications of Differentiation  · Level 3
Triangle with hypotenuse 5, opposite
x. \(\theta\) increases at 3 rad/min. Find \(\dfrac{d x}{d t}\) when \(x=3\).
A
\(3\)
✕
B
\(\dfrac{15}{4}\)
✕
C
\(4\)
✕
D
\(9\)
✕
E
\(12\)
✕
Question 18 of 34   |  Infinite Sequences and Series  · Level 4
Coefficient of \(x^7\) in Taylor of \(f\) where \(f'(x) = \sin(x^2)\)
A
\(\dfrac{1}{7}!\)
✕
B
\(\dfrac{1}{7}\)
✕
C
\(0\)
✕
D
\(-\dfrac{1}{42}\)
✕
E
\(-\dfrac{1}{7}!\)
✕
Question 19 of 34   |  Integration and Accumulation of Change  · Level 3
\(\operatorname*{lim}\limits_{n \rightarrow \infty} \displaystyle\sum_{i=1}^n \sqrt{x_i} \Delta x\) for partition of \([a,b]\)
A
\(\left(\dfrac{2}{3}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\)
✕
B
\(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\)
✕
C
\(\left(\dfrac{3}{2}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\)
✕
D
\(b^{\dfrac{1}{2}} - a^{\dfrac{1}{2}}\)
✕
E
\(2\left(b^{\dfrac{1}{2}} - a^{\dfrac{1}{2}}\right)\)
✕
Question 20 of 34   |  Infinite Sequences and Series  · Level 3
[Calc] Which sequences converge?
I. \(\{5n/(2n-1)\}\)
II. \(\{e^n/n\}\)
III. \(\{e^n/(1+e^n)\}\)
A
I only
✕
B
II only
✕
C
I and II only
✕
D
I and III only
✕
E
I, II, and III
✕
Question 21 of 34   |  Applications of Integration  · Level 3
[Calc] Region enclosed by \(y = x\) and \(y = 4 x - x^2\) revolved about y-axis
A
\(\pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) d x\)
✕
B
\(\pi \displaystyle\int_{0}^{3} (x^2 - (4x - x^2)^2) d x\)
✕
C
\(\pi \displaystyle\int_{0}^{3} (3 x - x^2)^2 d x\)
✕
D
\(2 \pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) d x\)
✕
E
\(2 \pi \displaystyle\int_{0}^{3} (3 x^2 - x^3) d x\)
✕
Question 22 of 34   |  Differentiation: Definition and Fundamental Properties  · Level 3
[Calc] \(\operatorname*{lim}\limits_{h\rightarrow 0}(\ln(e+h) - 1)/h\)
A
\(f'(e)\) where \(f(x) = \ln x\)
✕
B
\(f'(e)\) where \(f(x) = \ln \dfrac{x}{x}\)
✕
C
\(f'(1)\) where \(f(x) = \ln x\)
✕
D
\(f'(1)\) where \(f(x) = \ln(x+e)\)
✕
E
\(f'(0)\) where \(f(x) = \ln x\)
✕
Question 23 of 34   |  Contextual Applications of Differentiation  · Level 3
[Calc] \(y(t) = \left(\dfrac{1}{6}\right) \cos(5t) - \left(\dfrac{1}{4}\right) \sin(5t)\). Velocity = 0 how many times in \([0, 4]\)?
A
Zero
✕
B
Three
✕
C
Five
✕
D
Six
✕
E
Seven
✕
Question 24 of 34   |  Analytical Applications of Differentiation  · Level 3
[Calc] \(f(x) = \cos(2x) + \ln(3x)\), least value of \(x\) for inflection?
A
\(0.56\)
✕
B
\(0.93\)
✕
C
\(1.18\)
✕
D
\(2.38\)
✕
E
\(2.44\)
✕
Question 25 of 34   |  Limits and Continuity  · Level 2
[Calc] \(f\) continuous on \([-3, 6]\), \(f(-3) = -1\), \(f(6) = 3\). IVT guarantees:
A
\(f(0) = 0\)
✕
B
\(f'(c) = \dfrac{4}{9}\)
✕
C
\(-1 \leq f(x) \leq 3\)
✕
D
\(f(c) = 1\) for some \(c\)
✕
E
\(f(c) = 0\) for some \(c\) between \(-1\) and \(3\)
✕
Question 26 of 34   |  Integration and Accumulation of Change  · Level 3
[Calc] \(\displaystyle\int_{0}^{x} (t^2 - 2t) d t \geq \displaystyle\int_{2}^{x} t d t\) for \(0 \leq x \leq 4\). Greatest x?
A
\(1.35\)
✕
B
\(1.38\)
✕
C
\(1.41\)
✕
D
\(1.48\)
✕
E
\(1.59\)
✕
Question 27 of 34   |  Differential Equations  · Level 4
[Calc] \(\dfrac{d y}{d x} = (1 + \ln x) y\), \(y(1) = 1\). \(y =\)
A
\(e^{(x^2-1)/x^2}\)
✕
B
\(1 + \ln x\)
✕
C
\(\ln x\)
✕
D
\(e^{2x + x \ln x - 2}\)
✕
E
\(e^{x \ln x}\)
✕
Question 28 of 34   |  Integration and Accumulation of Change  · Level 3
\(\int x^2 \sin x d x =\)
A
\(-x^2 \cos x - 2 x \sin x - 2 \cos x + C\)
✕
B
\(-x^2 \cos x + 2 x \sin x - 2 \cos x + C\)
✕
C
\(-x^2 \cos x + 2 x \sin x + 2 \cos x + C\)
✕
D
\(-x^3/3 \cos x + C\)
✕
E
\(2 x \cos x + C\)
✕
Question 29 of 34   |  Integration and Accumulation of Change  · Level 3
[Calc] \(f\) twice diff, \(f(1) = 2\), \(f(3) = 7\). Which true on \([1,3]\)?
I. avg rate \(= \dfrac{5}{2}\)
II. avg value of \(f\) is \(\dfrac{9}{2}\)
III. avg of \(f'\) is \(\dfrac{5}{2}\)
A
None
✕
B
I only
✕
C
III only
✕
D
I and III only
✕
E
II and III only
✕
Question 30 of 34   |  Integration and Accumulation of Change  · Level 2
\(\int d x/((x-1)(x+3)) =\)
A
\(\left(\dfrac{1}{4}\right) \ln|\dfrac{x-1}{x+3}| + C\)
✕
B
\(\left(\dfrac{1}{4}\right) \ln|\dfrac{x+3}{x-1}| + C\)
✕
C
\(\left(\dfrac{1}{2}\right) \ln|(x-1)(x+3)| + C\)
✕
D
\(\left(\dfrac{1}{2}\right) \ln|\dfrac{2x+2}{(x-1)(x+3)}| + C\)
✕
E
\(\ln|(x-1)(x+3)| + C\)
✕
Question 31 of 34   |  Applications of Integration  · Level 3
Base of solid: \(y = 2 - x^2\) in Q1 with axes. Cross-sections perp to y-axis are squares. Volume?
A
\(\pi \displaystyle\int_{0}^{2} (2 - y)^2 d y\)
✕
B
\(\displaystyle\int_{0}^{2} (2 - y) d y\)
✕
C
\(\pi \displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 d x\)
✕
D
\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 d x\)
✕
E
\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2) d x\)
✕
Question 32 of 34   |  Integration and Accumulation of Change  · Level 3
[Calc] \(f(x) = \displaystyle\int_{0}^{x^2} \sin t d t\). How many points in \([0, \sqrt{\pi}]\) where instantaneous rate equals avg rate?
A
Zero
✕
B
One
✕
C
Two
✕
D
Three
✕
E
Four
✕
Question 33 of 34   |  Integration and Accumulation of Change  · Level 3
[Calc] \(f\) antiderivative of \(x^2/(1+x^5)\), \(f(1) = 0\). \(f(4) =\)
A
\(-0.012\)
✕
B
\(0\)
✕
C
\(0.016\)
✕
D
\(0.376\)
✕
E
\(0.629\)
✕
Question 34 of 34   |  Applications of Integration  · Level 2
[Calc] Spring: 10 lb stretches 4 inches. Work to stretch 6 inches?
A
\(60\)
✕
B
\(45\)
✕
C
\(40\)
✕
D
\(15\)
✕
E
\(7.2\)
✕

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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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