BC MCQ Set 120

Question 1 of 38 | Analytical Applications of Differentiation · Level 2

$f(x) = x^3 + 3 x^2 - 9 x + 7$ increasing for

$-3 < x < 1$

$-1 < x < 1$

$x < -3$ or $x > 1$

$x < -1$ or $x > 3$

All real

Question 2 of 38 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 1

Slope of line $x = 5 t + 2$, $y = 3 t$ for $-3 \leq t \leq 3$

$\dfrac{3}{5}$

$\dfrac{5}{3}$

$3$

$5$

$13$

Question 3 of 38 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

Slope of $y^2 + (xy + 1)^3 = 0$ at $(2, -1)$

$-\dfrac{3}{2}$

$-\dfrac{3}{4}$

$0$

$\dfrac{3}{4}$

$\dfrac{3}{2}$

Question 4 of 38 | Integration and Accumulation of Change · Level 3

$\int 1/(x^2 - 6 x + 8) d x =$

$\left(\dfrac{1}{2}\right) \ln|\dfrac{x-4}{x-2}| + C$

$\left(\dfrac{1}{2}\right) \ln|\dfrac{x-2}{x-4}| + C$

$\left(\dfrac{1}{2}\right) \ln|(x-2)(x-4)| + C$

$\left(\dfrac{1}{2}\right) \ln|(x-4)(x+2)| + C$

$\ln|(x-2)(x-4)| + C$

Question 5 of 38 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

$h(x) = f(g(x))$, $h''(x) =$

$f''(g(x))[g'(x)]^2 + f'(g(x)) g''(x)$

$f''(g(x)) g'(x) + f'(g(x)) g''(x)$

$f''(g(x))[g'(x)]^2$

$f''(g(x)) g''(x)$

$f''(g(x))$

Question 6 of 38 | Integration and Accumulation of Change · Level 2

$\displaystyle\int_{1}^{e} ((x^2-1)/x) d x =$

$e - \dfrac{1}{e}$

$e^2 - e$

$e^2/2 - e + \dfrac{1}{2}$

$e^2 - 2$

$e^2/2 - \dfrac{3}{2}$

Question 7 of 38 | Differential Equations · Level 3

$\dfrac{d y}{d x} = \sin x \cos^2 x$, $y\left(\dfrac{\pi}{2}\right) = 0$. $y(0) =$

$-1$

$-\dfrac{1}{3}$

$0$

$\dfrac{1}{3}$

$1$

Question 8 of 38 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

$x = t^3 - t$, $y = (2 t - 1)^3$. Acceleration vector at $t = 1$

$(0, 1)$

$(2, 3)$

$(2, 6)$

$(6, 12)$

$(6, 24)$

Question 9 of 38 | Integration and Accumulation of Change · Level 1

$f$ linear, $\displaystyle\int_{a}^{b} f''(x) d x =$

$0$

$1$

$\dfrac{ab}{2}$

$b - a$

$(b^2-a^2)/2$

Question 10 of 38 | Limits and Continuity · Level 2

$f(x) = \ln x$ for \(0

$\ln 2$

$\ln 8$

$\ln 16$

$4$

nonexistent

Question 11 of 38 | Infinite Sequences and Series · Level 3

5th degree Taylor of $\sin x$ at 1

$1 - \dfrac{1}{2} + \dfrac{1}{24}$

$1 - \dfrac{1}{2} + \dfrac{1}{4}$

$1 - \dfrac{1}{3} + \dfrac{1}{5}$

$1 - \dfrac{1}{4} + \dfrac{1}{8}$

$1 - \dfrac{1}{6} + \dfrac{1}{120}$

Question 12 of 38 | Integration and Accumulation of Change · Level 2

$\int x \cos x d x =$

$x \sin x - \cos x + C$

$x \sin x + \cos x + C$

$-x \sin x + \cos x + C$

$x \sin x + C$

$\left(\dfrac{1}{2}\right) x^2 \sin x + C$

Question 13 of 38 | Analytical Applications of Differentiation · Level 3

$f(x) = 3 x^5 - 5 x^4$, inflection points of $f$

$-1$

$0$

$1$

$0$ and $1$

$-1, 0,$ and $1$

Question 14 of 38 | Infinite Sequences and Series · Level 3

Series convergent?
I. $\sum n/(n+2)$
II. $\sum \cos(n \pi)/n$
III. $\sum \dfrac{1}{n}$

None

II only

III only

I and II only

I and III only

Question 15 of 38 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

Area inside $r = 4 \sin \theta$ outside $r = 2$

$\left(\dfrac{1}{2}\right) \displaystyle\int_{0}^{\pi} (4 \sin \theta - 2)^2 d \theta$

$\left(\dfrac{1}{2}\right) \displaystyle\int_{\dfrac{\pi}{4}}^{3 \dfrac{\pi}{4}} (4 \sin \theta - 2)^2 d \theta$

$\left(\dfrac{1}{2}\right) \displaystyle\int_{\dfrac{\pi}{6}}^{5 \dfrac{\pi}{6}} (4 \sin \theta - 2)^2 d \theta$

$\left(\dfrac{1}{2}\right) \displaystyle\int_{\dfrac{\pi}{6}}^{5 \dfrac{\pi}{6}} (16 \sin^2 \theta - 4) d \theta$

$\left(\dfrac{1}{2}\right) \displaystyle\int_{0}^{\pi} (16 \sin^2 \theta - 4) d \theta$

Question 16 of 38 | MCQ · Level 2

When $x=8$, rate of $\sqrt[3]{x}$ is $\dfrac{1}{k}$ times rate of $x$. $k =$

$3$

$4$

$6$

$8$

$12$

Question 17 of 38 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

Length of $x = \left(\dfrac{1}{3}\right) t^3$, $y = \left(\dfrac{1}{2}\right) t^2$ for $0 \leq t \leq 1$

$\int \sqrt{t^2 + 1} d t$

$\int \sqrt{t^2 + t} d t$

$\int \sqrt{t^4 + t^2} d t$

$\left(\dfrac{1}{2}\right) \int \sqrt{4 + t^4} d t$

$\left(\dfrac{1}{6}\right) \int t^2 \sqrt{4 t^2 + 9} d t$

Question 18 of 38 | Infinite Sequences and Series · Level 3

$\operatorname*{lim}\limits_{b \rightarrow \infty} \displaystyle\int_{1}^{b} d x/x^p$ finite. Then which true?

$\sum 1/n^p$ converges

$\sum 1/n^p$ diverges

$\sum 1/n^{p-2}$ converges

$\sum 1/n^{p-1}$ converges

$\sum 1/n^{p+1}$ diverges

Question 19 of 38 | Analytical Applications of Differentiation · Level 2

$f$ continuous on $[a,b]$ has rel max at $c$, $a < c < b$. Which true?
I. $f'(c)$ exists
II. If $f'(c)$ exists, $f'(c)=0$
III. If $f''(c)$ exists, $f''(c) \leq 0$

II only

III only

I and II only

I and III only

II and III only

Question 20 of 38 | Integration and Accumulation of Change · Level 3

$\displaystyle\int_{0}^{\infty} x^2 e^{-x^3} d x$

$-\dfrac{1}{3}$

$0$

$\dfrac{1}{3}$

$1$

divergent

Question 21 of 38 | Differential Equations · Level 4

Logistic: $\dfrac{d P}{d t} = P\left(2 - \dfrac{P}{5000}\right)$, $P(0)=3000$. $\lim P(t) =$

\$2,500$

\$3,000$

\$4,200$

\$5,000$

\$10,000$

Question 22 of 38 | Infinite Sequences and Series · Level 3

$f(x) = \sum a_n x^n$, $f'(1) =$

$0$

$a_1$

$\sum a_n$

$\sum n a_n$

$\sum n a_n^{n-1}$

Question 23 of 38 | Contextual Applications of Differentiation · Level 3

$\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{\displaystyle\int_{1}^{x} e^{t^2} d t}{x^2 - 1}$

$0$

$1$

$\dfrac{e}{2}$

$e$

nonexistent

Question 24 of 38 | Infinite Sequences and Series · Level 3

[Calc] Both $\sum (-1)^{kn}/n$ and $\sum \left(\dfrac{k}{4}\right)^n$ converge for integer $k > 1$. $k =$

$6$

$5$

$4$

$3$

$2$

Question 25 of 38 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2

[Calc] $f(t) = (e^{-t}, \cos t)$, $f''(t) =$

$(-e^{-t} + \sin t, ?)$

$(e^{-t}, -\cos t)$

$(-e^{-t}, -\sin t)$

$(e^{-t}, \cos t)$

$(e^{-t}, -\cos t)$

Question 26 of 38 | Contextual Applications of Differentiation · Level 3

[Calc] Radius decreasing 0.1 cm/s. $\dfrac{d A}{d t}$ in terms of $C$

$-(0.2) \pi C$

$-(0.1) C$

$-(0.1) C/(2 \pi)$

$(0.1)^2 C$

$(0.1)^2 \pi C$

Question 27 of 38 | Limits and Continuity · Level 2

$f(x) = (x-1)\dfrac{x^2-4}{x^2-a}$ continuous everywhere, positive $a$

None

$1$ only

$2$ only

$4$ only

$1$ and $4$ only

Question 28 of 38 | Applications of Integration · Level 3

[Calc] $R$ enclosed by $y = 1 + \ln(\cos^4 x)$, x-axis, $x = \pm \dfrac{2}{3}$. Area approx

$0$

$1$

$2$

$3$

$4$

Question 29 of 38 | Differential Equations · Level 2

$\dfrac{d y}{d x} = \sqrt{1 - y^2}$, $\dfrac{d^2 y}{d x^2} =$

$-2 y$

$-y$

$-\dfrac{y}{\sqrt{1-y^2}}$

$y$

$\dfrac{1}{2}$

Question 30 of 38 | Integration and Accumulation of Change · Level 2

$f(x) = g(x) + 7$ for $3 \leq x \leq 5$. $\displaystyle\int_{3}^{5} [f(x) + g(x)] d x =$

$2 \int g + 7$

$2 \int g + 14$

$2 \int g + 28$

$\int g + 7$

$\int g + 14$

Question 31 of 38 | Infinite Sequences and Series · Level 4

[Calc] Taylor for $\ln x$ at 1: 3 nonzero terms. Max $|\ln x - f(x)|$ on $[0.3, 1.7]$

$0.030$

$0.039$

$0.145$

$0.153$

$0.529$

Question 32 of 38 | Infinite Sequences and Series · Level 4

$\sum (x+2)^n/\sqrt{n}$ converges for

$-3 < x < -1$

$-3 \leq x < -1$

$-3 \leq x \leq -1$

$-1 \leq x < 1$

$-1 \leq x \leq 1$

Question 33 of 38 | Integration and Accumulation of Change · Level 2

[Calc] $f(2)=10, f(5)=30, f(7)=40, f(8)=20$. Trapezoidal $\displaystyle\int_{2}^{8} f$

$110$

$130$

$160$

$190$

$210$

Question 34 of 38 | Applications of Integration · Level 3

[Calc] Base $x+2y=8$, semicircular cross-sections. Volume

$12.566$

$14.661$

$16.755$

$67.021$

$134.041$

Question 35 of 38 | Differentiation: Definition and Fundamental Properties · Level 3

[Calc] Tangent to $f(x) = x^4 + 2 x^2$ where $f'(x) = 1$

$y = 8 x - 5$

$y = x + 7$

$y = x + 0.763$

$y = x - 0.122$

$y = x - 2.146$

Question 36 of 38 | Infinite Sequences and Series · Level 3

[Calc] Maclaurin series $1 - x + x^2/2! - x^3/3! + ... = e^{-x}$ intersects $y = x^3$ at $x =$

$0.773$

$0.865$

$0.929$

$1.000$

$1.857$

Question 37 of 38 | Integration and Accumulation of Change · Level 3

[Calc] $a(t) = 5, 2, 8, 3$ at $t=0,2,4,6$. $v(0)=11$. Left Riemann sum estimate $v(6)$

$26$

$30$

$37$

$39$

$41$

Question 38 of 38 | Contextual Applications of Differentiation · Level 3

[Calc] $f(x) = x^2 - 2 x + 3$. Tangent at $x=2$ approximates $f$ within 0.5. Greatest $x$

$2.4$

$2.5$

$2.6$

$2.7$

$2.8$

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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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