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BC MCQ Set 80
41 Questions
Question 1 of 41
BC MCQ Set 80 0/41
Question 1 of 41   |  Integration and Accumulation of Change  · Level 1
The area between \(y = 4 x^3 + 2\) and the x-axis from \(x = 1\) to \(x = 2\) is
A
\(36\)
✕
B
\(23\)
✕
C
\(20\)
✕
D
\(17\)
✕
E
\(9\)
✕
Question 2 of 41   |  Analytical Applications of Differentiation  · Level 2
At what values of \(x\) does \(f(x) = 3 x^5 - 5 x^3 + 15\) have a relative maximum?
A
\(-1\) only
✕
B
\(0\) only
✕
C
\(1\) only
✕
D
\(-1\) and \(1\) only
✕
E
\(-1, 0\), and \(1\)
✕
Question 3 of 41   |  Integration and Accumulation of Change  · Level 3
\(\displaystyle\int_{1}^{2} \dfrac{x + 1}{x^2 + 2 x} d x =\)
A
\(\ln 8 - \ln 3\)
✕
B
\(\dfrac{\ln 8 - \ln 3}{2}\)
✕
C
\(\ln 8\)
✕
D
\(\dfrac{3 \ln 2}{2}\)
✕
E
\(\dfrac{3 \ln 2 + 2}{2}\)
✕
Question 4 of 41   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
A particle moves so that \(x = t^2 - 1\) and \(y = t^4 - 2 t^3\). At \(t = 1\), its acceleration vector is
A
\((0, -1)\)
✕
B
\((0, 12)\)
✕
C
\((2, -2)\)
✕
D
\((2, 0)\)
✕
E
\((2, 8)\)
✕
Question 5 of 41   |  Differentiation: Definition and Fundamental Properties  · Level 3
If \(f(x) = \dfrac{x}{\tan} x\), then \(f'\left(\dfrac{\pi}{4}\right) =\)
A
\(2\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(1 + \dfrac{\pi}{2}\)
✕
D
\(\dfrac{\pi}{2} - 1\)
✕
E
\(1 - \dfrac{\pi}{2}\)
✕
Question 6 of 41   |  MCQ  · Level 1
\(\int \dfrac{d x}{\sqrt{25 - x^2}} =\)
A
\(\arcsin\left(\dfrac{x}{5}\right) + C\)
✕
B
\(\arcsin x + C\)
✕
C
\(\left(\dfrac{1}{5}\right) \arcsin\left(\dfrac{x}{5}\right) + C\)
✕
D
\(\sqrt{25 - x^2} + C\)
✕
E
\(2 \sqrt{25 - x^2} + C\)
✕
Question 7 of 41   |  Differentiation: Definition and Fundamental Properties  · Level 2
If \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{f(x) - f(2)}{x - 2} = 0\), which must be true?
A
limit doesn't exist
✕
B
\(f\) not defined at 2
✕
C
\(f'(2) = 0\)
✕
D
\(f\) continuous at 0
✕
E
\(f(2) = 0\)
✕
Question 8 of 41   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
If \(x y^2 + 2 x y = 8\), then at point \((1, 2)\), \(y'\) is
A
\(-\dfrac{5}{2}\)
✕
B
\(-\dfrac{4}{3}\)
✕
C
\(-1\)
✕
D
\(-\dfrac{1}{2}\)
✕
E
\(0\)
✕
Question 9 of 41   |  Infinite Sequences and Series  · Level 3
For \(-1 < x < 1\), if \(f(x) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} x^{2n-1}}{2n - 1}\), then \(f'(x) =\)
A
\(\sum (-1)^{n+1} x^{2n-2}\)
✕
B
\(\sum (-1)^n x^{2n-2}\)
✕
C
\(\sum (-1)^{2n} x^{2n}\)
✕
D
\(\sum (-1)^n x^{2n}\)
✕
E
\(\sum (-1)^{n+1} x^{2n}\)
✕
Question 10 of 41   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
\(\dfrac{d}{d x} \ln\left(\dfrac{1}{1 - x}\right) =\)
A
\(\dfrac{1}{1 - x}\)
✕
B
\(\dfrac{1}{x - 1}\)
✕
C
\(1 - x\)
✕
D
\(x - 1\)
✕
E
\((1 - x)^2\)
✕
Question 11 of 41   |  Integration and Accumulation of Change  · Level 3
\(\int \dfrac{d x}{(x - 1)(x + 2)} =\)
A
\(\dfrac{1}{3} \ln|\dfrac{x-1}{x+2}| + C\)
✕
B
\(\dfrac{1}{3} \ln|\dfrac{x+2}{x-1}| + C\)
✕
C
\(\dfrac{1}{3} \ln|(x-1)(x+2)| + C\)
✕
D
\((\ln|x-1|)(\ln|x+2|) + C\)
✕
E
\(\ln|(x-1)(x+2)^2| + C\)
✕
Question 12 of 41   |  Analytical Applications of Differentiation  · Level 3
Let \(f(x) = x^3 - 3 x^2\). MVT \(c\) values on \([0, 3]\)?
A
\(0\) only
✕
B
\(2\) only
✕
C
\(3\) only
✕
D
\(0\) and \(3\)
✕
E
\(2\) and \(3\)
✕
Question 13 of 41   |  Infinite Sequences and Series  · Level 3
Which series converge?
I. \(\sum 1/n^2\)
II. \(\sum \dfrac{1}{n}\)
III. \(\sum (-1)^{n+1}/3^{n-1}\)
A
I only
✕
B
III only
✕
C
I and III only
✕
D
II and III only
✕
E
I, II, and III
✕
Question 14 of 41   |  MCQ  · Level 1
If \(v(t) = 2 t - 4\) and at \(t = 0\) position is 4, then \(x(t) =\)
A
\(t^2 - 4 t\)
✕
B
\(t^2 - 4 t - 4\)
✕
C
\(t^2 - 4 t + 4\)
✕
D
\(2 t^2 - 4 t\)
✕
E
\(2 t^2 - 4 t + 4\)
✕
Question 15 of 41   |  MCQ  · Level 2
Which function shows 'continuous at 0 ⟹ differentiable at 0' is false?
A
\(x^{-\dfrac{4}{3}}\)
✕
B
\(x^{-\dfrac{1}{3}}\)
✕
C
\(x^{\dfrac{1}{3}}\)
✕
D
\(x^{\dfrac{4}{3}}\)
✕
E
\(x^3\)
✕
Question 16 of 41   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
If \(f(x) = x \ln(x^2)\), then \(f'(x) =\)
A
\(\ln(x^2) + 1\)
✕
B
\(\ln(x^2) + 2\)
✕
C
\(\ln(x^2) + \dfrac{1}{x}\)
✕
D
\(1/x^2\)
✕
E
\(\dfrac{1}{x}\)
✕
Question 17 of 41   |  Integration and Accumulation of Change  · Level 1
\(\int \sin(2 x + 3) d x =\)
A
\(-2 \cos(2 x + 3) + C\)
✕
B
\(-\cos(2 x + 3) + C\)
✕
C
\(-\dfrac{1}{2} \cos(2 x + 3) + C\)
✕
D
\(\dfrac{1}{2} \cos(2 x + 3) + C\)
✕
E
\(\cos(2 x + 3) + C\)
✕
Question 18 of 41   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 4
If \(g(x) = e^{f(x)}\) and \(g''(x) = h(x) e^{f(x)}\), then \(h(x) =\)
A
\(f'(x) + f''(x)\)
✕
B
\(f'(x) + (f''(x))^2\)
✕
C
\((f'(x) + f''(x))^2\)
✕
D
\((f'(x))^2 + f''(x)\)
✕
E
\(2 f'(x) + f''(x)\)
✕
Question 19 of 41   |  Integration and Accumulation of Change  · Level 3
If \(\int f(x) \sin x d x = -f(x) \cos x + \int 3 x^2 \cos x d x\), then \(f(x)\) could be
A
\(3 x^2\)
✕
B
\(x^3\)
✕
C
\(-x^3\)
✕
D
\(\sin x\)
✕
E
\(\cos x\)
✕
Question 20 of 41   |  Contextual Applications of Differentiation  · Level 2
Area of circle increases at \(96 \pi\) m²/s. When area is \(64 \pi\), find \(\dfrac{d r}{d t}\).
A
\(6\)
✕
B
\(8\)
✕
C
\(16\)
✕
D
\(4 \sqrt{3}\)
✕
E
\(12 \sqrt{3}\)
✕
Question 21 of 41   |  Integration and Accumulation of Change  · Level 3
\(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\displaystyle\int_{1}^{1+h} \sqrt{x^5 + 8} d x}{h}\) is
A
\(0\)
✕
B
\(1\)
✕
C
\(3\)
✕
D
\(2 \sqrt{2}\)
✕
E
nonexistent
✕
Question 22 of 41   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
Area enclosed by polar curve \(r = \sin(2 \theta)\) for \(0 \leq \theta \leq \dfrac{\pi}{2}\)
A
\(0\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(1\)
✕
D
\(\dfrac{\pi}{8}\)
✕
E
\(\dfrac{\pi}{4}\)
✕
Question 23 of 41   |  Contextual Applications of Differentiation  · Level 3
\(x(t) = t e^{-2 t}\). Particle at rest when?
A
No values
✕
B
\(0\) only
✕
C
\(\dfrac{1}{2}\) only
✕
D
\(1\) only
✕
E
\(0\) and \(\dfrac{1}{2}\)
✕
Question 24 of 41   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 4
For \(0 < x < \dfrac{\pi}{2}\), if \(y = (\sin x)^x\), then \(\dfrac{d y}{d x} =\)
A
\(x \ln(\sin x)\)
✕
B
\((\sin x)^x \cot x\)
✕
C
\(x (\sin x)^{x-1} \cos x\)
✕
D
\((\sin x)^x (x \cos x + \sin x)\)
✕
E
\((\sin x)^x (x \cot x + \ln(\sin x))\)
✕
Question 25 of 41   |  Integration and Accumulation of Change  · Level 3
An antiderivative of \(f(x) = e^{x + e^x}\) is
A
\(\dfrac{e^{x+e^x}}{1+e^x}\)
✕
B
\((1+e^x) e^{x+e^x}\)
✕
C
\(e^{1+e^x}\)
✕
D
\(e^{x+e^x}\)
✕
E
\(e^{e^x}\)
✕
Question 26 of 41   |  Limits and Continuity  · Level 3
\(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{4}} \dfrac{\sin\left(x - \dfrac{\pi}{4}\right)}{x - \dfrac{\pi}{4}}\) is
A
\(0\)
✕
B
\(\dfrac{1}{\sqrt{2}}\)
✕
C
\(\dfrac{\pi}{4}\)
✕
D
\(1\)
✕
E
nonexistent
✕
Question 27 of 41   |  Parametric Equations, Polar Coordinates, and Vector-Valued Functions  · Level 3
If \(x = t^3 - t\) and \(y = \sqrt{3 t + 1}\), \(\dfrac{d y}{d x}\) at \(t=1\) is
A
\(\dfrac{1}{8}\)
✕
B
\(\dfrac{3}{8}\)
✕
C
\(\dfrac{3}{4}\)
✕
D
\(\dfrac{8}{3}\)
✕
E
\(8\)
✕
Question 28 of 41   |  Infinite Sequences and Series  · Level 4
What are all \(x\) for which \(\displaystyle\sum_{n=1}^\infty (x-1)^n / n\) converges?
A
\(-1 \leq x < 1\)
✕
B
\(-1 \leq x \leq 1\)
✕
C
\(0 < x < 2\)
✕
D
\(0 \leq x < 2\)
✕
E
\(0 \leq x \leq 2\)
✕
Question 29 of 41   |  MCQ  · Level 3
Equation of normal to \(y = x^3 + 3 x^2 + 7 x - 1\) at \(x = -1\)
A
\(4 x + y = -10\)
✕
B
\(x - 4 y = 23\)
✕
C
\(4 x - y = 2\)
✕
D
\(x + 4 y = 25\)
✕
E
\(x + 4 y = -25\)
✕
Question 30 of 41   |  Differential Equations  · Level 2
If \(\dfrac{d y}{d t} = -2 y\) and \(y = 1\) at \(t = 0\), find \(t\) for \(y = \dfrac{1}{2}\).
A
\(-\dfrac{\ln 2}{2}\)
✕
B
\(-\dfrac{1}{4}\)
✕
C
\(\dfrac{\ln 2}{2}\)
✕
D
\(\dfrac{\sqrt{2}}{2}\)
✕
E
\(\ln 2\)
✕
Question 31 of 41   |  Applications of Integration  · Level 4
Surface area generated by revolving \(x = y^3\) from \(y=0\) to \(y=1\) about y-axis
A
\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + 9 y^4} d y\)
✕
B
\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + y^6} d y\)
✕
C
\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + 3 y^2} d y\)
✕
D
\(2 \pi \displaystyle\int_{0}^{1} y \sqrt{1 + 9 y^4} d y\)
✕
E
\(2 \pi \displaystyle\int_{0}^{1} y \sqrt{1 + y^6} d y\)
✕
Question 32 of 41   |  Applications of Integration  · Level 3
Region in first quadrant between x-axis and \(y = 6 x - x^2\) rotated around y-axis. Volume?
A
\(\displaystyle\int_{0}^{6} \pi (6x - x^2)^2 d x\)
✕
B
\(\displaystyle\int_{0}^{6} 2 \pi x (6x - x^2) d x\)
✕
C
\(\displaystyle\int_{0}^{6} \pi x (6x - x^2)^2 d x\)
✕
D
\(\displaystyle\int_{0}^{6} \pi (3 + \sqrt{9-y})^2 d y\)
✕
E
\(\displaystyle\int_{0}^{9} \pi (3 + \sqrt{9-y})^2 d y\)
✕
Question 33 of 41   |  Integration and Accumulation of Change  · Level 3
\(\displaystyle\int_{-1}^1 \dfrac{3}{x^2} d x\) is
A
\(-6\)
✕
B
\(-3\)
✕
C
\(0\)
✕
D
\(6\)
✕
E
nonexistent
✕
Question 34 of 41   |  Differential Equations  · Level 4
General solution of \(\dfrac{d y}{d x} + y = x e^{-x}\)
A
\(y = (x^2/2) e^{-x} + C e^{-x}\)
✕
B
\(y = (x^2/2) e^{-x} + e^{-x} + C\)
✕
C
\(y = -e^{-x} + C/(1+x)\)
✕
D
\(y = x e^{-x} + C e^{-x}\)
✕
E
\(y = C_1 e^x + C_2 x e^{-x}\)
✕
Question 35 of 41   |  Limits and Continuity  · Level 4
\(\operatorname*{lim}\limits_{x \rightarrow \infty} (1 + 5 e^x)^{\dfrac{1}{x}}\)
A
\(0\)
✕
B
\(1\)
✕
C
\(e\)
✕
D
\(e^5\)
✕
E
nonexistent
✕
Question 36 of 41   |  Applications of Integration  · Level 4
Base of solid: region under \(y = e^{-x}\), axes, \(x = 3\). Cross-sections perpendicular to x-axis are squares. Volume?
A
\(\dfrac{1 - e^{-6}}{2}\)
✕
B
\(\dfrac{1}{2} e^{-6}\)
✕
C
\(e^{-6}\)
✕
D
\(e^{-3}\)
✕
E
\(1 - e^{-3}\)
✕
Question 37 of 41   |  Integration and Accumulation of Change  · Level 3
If \(u = \dfrac{x}{2}\), then \(\displaystyle\int_{2}^{4} \dfrac{1 - \left(\dfrac{x}{2}\right)^2}{x} d x =\)
A
\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{u} d u\)
✕
B
\(\displaystyle\int_{2}^{4} \dfrac{1 - u^2}{u} d u\)
✕
C
\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{2u} d u\)
✕
D
\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{4u} d u\)
✕
E
\(\displaystyle\int_{2}^{4} \dfrac{1 - u^2}{2u} d u\)
✕
Question 38 of 41   |  Applications of Integration  · Level 2
Length of \(y = \left(\dfrac{2}{3}\right) x^{\dfrac{3}{2}}\) from \(x = 0\) to \(x = 3\)?
A
\(\dfrac{8}{3}\)
✕
B
\(4\)
✕
C
\(\dfrac{14}{3}\)
✕
D
\(\dfrac{16}{3}\)
✕
E
\(7\)
✕
Question 39 of 41   |  Infinite Sequences and Series  · Level 3
Coefficient of \(x^3\) in Taylor series for \(e^{3x}\) about \(x = 0\)
A
\(\dfrac{1}{6}\)
✕
B
\(\dfrac{1}{3}\)
✕
C
\(\dfrac{1}{2}\)
✕
D
\(\dfrac{3}{2}\)
✕
E
\(\dfrac{9}{2}\)
✕
Question 40 of 41   |  Differential Equations  · Level 3
Slope \(3 x^2 y\), curve through \((0, 8)\). Equation:
A
\(y = 8 e^{x^3}\)
✕
B
\(y = x^3 + 8\)
✕
C
\(y = e^{x^3} + 7\)
✕
D
\(y = \ln(x+1) + 8\)
✕
E
\(y^2 = x^3 + 8\)
✕
Question 41 of 41   |  Integration and Accumulation of Change  · Level 4
\(\operatorname*{lim}\limits_{n \rightarrow \infty} \left(\dfrac{1}{n}\right) [\left(\dfrac{1}{n}\right)^2 + \left(\dfrac{2}{n}\right)^2 + ... + \left(\dfrac{3n}{n}\right)^2]\)
A
\(\displaystyle\int_{0}^{1} 1/x^2 d x\)
✕
B
\(3 \displaystyle\int_{0}^{1} \left(\dfrac{1}{x}\right)^2 d x\)
✕
C
\(\displaystyle\int_{0}^{3} \left(\dfrac{1}{x}\right)^2 d x\)
✕
D
\(\displaystyle\int_{0}^{3} x^2 d x\)
✕
E
\(3 \displaystyle\int_{0}^{3} x^2 d x\)
✕

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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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