BC MCQ Set 100 (1993 Official AP)

Question 1 of 43 | MCQ · Level 1

Area enclosed by $y = x^2$ and $y = x$

$\dfrac{1}{6}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{5}{6}$

$1$

Question 2 of 43 | MCQ · Level 2

$f(x) = 2 x^2 + 1$, $\operatorname*{lim}\limits_{x \rightarrow 0} (f(x) - f(0))/x^2 =$

$0$

$1$

$2$

$4$

nonexistent

Question 3 of 43 | MCQ · Level 2

$Q(x) = \displaystyle\int_{0}^{x} p(t) dt$ where $p$ degree $n$. Degree of $Q$?

$0$

$1$

$n - 1$

$n$

$n + 1$

Question 4 of 43 | MCQ · Level 2

Particle on $xy = 10$, $x = 2$, $\dfrac{dy}{dt} = 3$. $\dfrac{dx}{dt} = ?$

$-\dfrac{5}{2}$

$-\dfrac{6}{5}$

$0$

$\dfrac{4}{5}$

$\dfrac{6}{5}$

Question 5 of 43 | MCQ · Level 3

$x = t^2 + 1$, $y = t^3$. $d^\dfrac{2y}{dx}^2 =$

$\dfrac{3}{4 t}$

$\dfrac{3}{2 t}$

$3 t$

$6 t$

$\dfrac{3}{2}$

Question 6 of 43 | MCQ · Level 3

$\displaystyle\int_{0}^{1} x^3 e^{x^4} dx =$

$\dfrac{1}{4}(e - 1)$

$\dfrac{1}{4} e$

$e - 1$

$e$

$4(e - 1)$

Question 7 of 43 | MCQ · Level 1

$f(x) = \ln(e^{2x})$, $f'(x) =$

$1$

$2$

$2 x$

$e^{-2x}$

$2 e^{-2x}$

Question 8 of 43 | MCQ · Level 3

$f(x) = 1 + x^{\dfrac{2}{3}}$. NOT true?

$f$ continuous everywhere

$f$ has min at 0

$f$ increasing for $x > 0$

$f'$ exists everywhere

$f''$ negative for $x > 0$

Question 9 of 43 | MCQ · Level 2

Continuous at $x=1$? I. $\ln x$ II. $e^x$ III. $\ln(e^x - 1)$

I only

II only

I and II only

II and III only

I, II, and III

Question 10 of 43 | MCQ · Level 3

$\displaystyle\int_{4}^{\infty} \dfrac{-2 x}{\sqrt[3]{9 - x^2}} dx =$

$7^{\dfrac{2}{3}}$

$\left(\dfrac{3}{2}\right)\left(7^{\dfrac{2}{3}}\right)$

$9^{\dfrac{2}{3}} + 7^{\dfrac{2}{3}}$

$\left(\dfrac{3}{2}\right)\left(9^{\dfrac{2}{3}} + 7^{\dfrac{2}{3}}\right)$

nonexistent

Question 11 of 43 | MCQ · Level 3

$x(t) = \sin(2t) - \cos(3t)$. $a(\pi) =$

$9$

$\dfrac{1}{9}$

$0$

$-\dfrac{1}{9}$

$-9$

Question 12 of 43 | MCQ · Level 2

$\dfrac{dy}{dx} = x^2 y$, $y$ could be

$3 \ln\left(\dfrac{x}{3}\right)$

$e^{x^\dfrac{3}{3}} + 7$

$2 e^{x^\dfrac{3}{3}}$

$3 e^{2x}$

$x^\dfrac{3}{3} + 1$

Question 13 of 43 | MCQ · Level 2

$f' = x^4(x-2)(x+3)$. How many rel max?

None

One

Two

Three

Four

Question 14 of 43 | MCQ · Level 2

$f(x) = e^{\tan^2 x}$, $f'(x) =$

$e^{\tan^2 x}$

$\sec^2 x e^{\tan^2 x}$

$\tan^2 x e^{\tan^2 x - 1}$

$2 \tan x \sec^2 x e^{\tan^2 x}$

$2 \tan x e^{\tan^2 x}$

Question 15 of 43 | MCQ · Level 2

Series diverge? I. $\sum 2/(k^2+1)$ II. $\sum \left(\dfrac{6}{7}\right)^k$ III. $\sum (-1)^\dfrac{k}{k}$

None

II only

III only

I and III

II and III

Question 16 of 43 | MCQ · Level 2

Slope of tangent to $\ln(xy) = x$ at $x = 1$

$0$

$1$

$e$

$e^2$

$1 - e$

Question 17 of 43 | MCQ · Level 2

$e^{f(x)} = 1 + x^2$, $f'(x) =$

$\dfrac{1}{1+x^2}$

$\dfrac{2 x}{1+x^2}$

$2 x(1+x^2)$

$2 x e^{1+x^2}$

$2 x \ln(1+x^2)$

Question 18 of 43 | MCQ · Level 4

$a(t) = e^{-2t}$, $v(0) = \dfrac{5}{2}$, $x(0) = \dfrac{17}{4}$. $x(t) =$

$-e^{-2t}/2 + 3$

$e^{-2t}/4 + 4$

$4 e^{-2t} + \left(\dfrac{9}{2}\right) t + \dfrac{1}{4}$

$e^{-2t}/2 + 3 t + \dfrac{15}{4}$

$e^{-2t}/4 + 3 t + 4$

Question 19 of 43 | MCQ · Level 3

$y = \sqrt[3]{x^2+8}/\sqrt[4]{2x+1}$. $y'(0) =$

$-1$

$-\dfrac{1}{2}$

$0$

$\dfrac{1}{2}$

$1$

Question 20 of 43 | MCQ · Level 2

$f(x) = x^2 e^x$ decreasing for

$x < -2$

$-2 < x < 0$

$x > -2$

$x < 0$

$x > 0$

Question 21 of 43 | MCQ · Level 2

Length of $x = t^2$, $y = t$ from $t=0$ to $t=4$

$\displaystyle\int_{0}^{4} \sqrt{4 t + 1} dt$

$2 \displaystyle\int_{0}^{4} \sqrt{t^2 + 1} dt$

$\displaystyle\int_{0}^{4} \sqrt{2 t^2 + 1} dt$

$\displaystyle\int_{0}^{4} \sqrt{4 t^2 + 1} dt$

$2 \pi \displaystyle\int_{0}^{4} \sqrt{4 t^2 + 1} dt$

Question 22 of 43 | MCQ · Level 3

$f, g$ diff, $g(x) \neq 0$ for $x \neq 0$, $\operatorname*{lim}\limits_{x\rightarrow 0} f = \operatorname*{lim}\limits_{x\rightarrow 0} g = 0$, $lim f'/g'$ exists. Then $lim \dfrac{f}{g}$ is

$0$

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

$lim f'(x)/g'(x)$

$(f'g - fg')/f^2$

nonexistent

Question 23 of 43 | MCQ · Level 3

$x = e^t$, $y = t e^{-t}$. Slope at $x = 3$

$20.086$

$0.342$

$-0.005$

$-0.011$

$-0.033$

Question 24 of 43 | MCQ · Level 2

$y = \arctan(e^{2x})$, $\dfrac{dy}{dx} =$

$\dfrac{2 e^{2x}}{\sqrt{1 - e^{4x}}}$

$\dfrac{2 e^{2x}}{1 + e^{4x}}$

$\dfrac{e^{2x}}{1 + e^{4x}}$

$\dfrac{1}{\sqrt{1 - e^{4x}}}$

$\dfrac{1}{1 + e^{4x}}$

Question 25 of 43 | MCQ · Level 3

Interval of convergence of $\displaystyle\sum_{n=0}^\infty (x-1)^\dfrac{n}{3}^n$

$-3 < x \leq 3$

$-3 \leq x \leq 3$

$-2 < x < 4$

$-2 \leq x < 4$

$0 \leq x \leq 2$

Question 26 of 43 | MCQ · Level 3

Position vector $(\ln(t^2+2t), 2t^2)$. $v(2) =$

$\left(\dfrac{3}{4}, 8\right)$

$\left(\dfrac{3}{4}, 4\right)$

$\left(\dfrac{1}{8}, 8\right)$

$\left(\dfrac{1}{8}, 4\right)$

$\left(-\dfrac{5}{16}, 4\right)$

Question 27 of 43 | MCQ · Level 3

$\int x \sec^2 x dx =$

$x \tan x + C$

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

$\sec^2 x + 2 \sec^2 x \tan x + C$

$x \tan x - \ln|\cos x| + C$

$x \tan x + \ln|\cos x| + C$

Question 28 of 43 | MCQ · Level 3

Volume rotating $y = \sec x$ about x-axis from 0 to $\dfrac{\pi}{3}$

$\pi/\sqrt{3}$

$\pi$

$\pi \sqrt{3}$

$8 \dfrac{\pi}{3}$

$\pi \ln\left(\dfrac{1}{2} + \sqrt{3}\right)$

Question 29 of 43 | MCQ · Level 3

$s_n = ((5+n)^\dfrac{100}{5}^{n+1})(5^n/(4+n)^100)$. Limit?

$\dfrac{1}{5}$

$1$

$\dfrac{5}{4}$

$\left(\dfrac{5}{4}\right)^100$

Doesn't converge

Question 30 of 43 | MCQ · Level 2

$\displaystyle\int_{a}^{b} f = 5$, $\displaystyle\int_{a}^{b} g = -1$. Which must true? I. $f > g$ II. $\int (f+g) = 4$ III. $\int fg = -5$

I only

II only

III only

II and III only

I, II, and III

Question 31 of 43 | MCQ · Level 2

$\displaystyle\int_{0}^{\pi} \sin x dx$ equals which?

$\displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \cos x dx$

$\displaystyle\int_{0}^{\pi} \cos x dx$

$\displaystyle\int_{-\pi}^0 \sin x dx$

$\displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \sin x dx$

$\displaystyle\int_{\pi}^{2 \pi} \sin x dx$

Question 32 of 43 | MCQ · Level 3

[Calc] 40-foot ladder, $Q$ moves at $\dfrac{3}{4}$ as fast as $P$. Find $RQ$.

$\left(\dfrac{6}{5}\right) \sqrt{10}$

$\left(\dfrac{8}{5}\right) \sqrt{10}$

$80/\sqrt{7}$

$24$

$32$

Question 33 of 43 | MCQ · Level 2

$F(x) = \displaystyle\int_{0}^{x} f(t) dt$, $F(a) = -2$, $F(b) = -2$, $a < b$. Which true?

$f(x) = 0$ for some $x \in (a,b)$

$f(x) > 0$ for all

$f(x) < 0$ for all

$F(x) \leq 0$ for all

$F(x) = 0$ for some

Question 34 of 43 | MCQ · Level 3

Cylinder: height + circumference = 30. Max volume radius?

$3$

$10$

$20$

$\dfrac{30}{\pi}^2$

$\dfrac{10}{\pi}$

Question 35 of 43 | MCQ · Level 2

$f(x) = x$ for $x \leq 1$, $\dfrac{1}{x}$ for $x > 1$. $\displaystyle\int_{0}^{e} f dx =$

$0$

$\dfrac{3}{2}$

$2$

$e$

$e + \dfrac{1}{2}$

Question 36 of 43 | MCQ · Level 3

[Calc] Epidemic exponential. 1000 at $t=0$, 1200 at $t=7$. At $t=12$?

$343$

\$1,343$

\$1,367$

\$1,400$

\$2,057$

Question 37 of 43 | MCQ · Level 2

$\dfrac{dy}{dx} = \dfrac{1}{x}$. Average rate of change on $[1, 4]$

$-\dfrac{1}{4}$

$\left(\dfrac{1}{2}\right) \ln 2$

$\left(\dfrac{2}{3}\right) \ln 2$

$\dfrac{2}{5}$

$2$

Question 38 of 43 | MCQ · Level 3

[Calc] $y = \ln(1 + 2 x - x^2)$, Simpson's with 2 subintervals on $[0, 2]$. Wait region in Q1, so x from 0 to where $1+2x-x^2 = 1$, i.e., 0 and 2.

$0.462$

$0.693$

$0.924$

$0.986$

$1.850$

Question 39 of 43 | MCQ · Level 3

$f(x) = \displaystyle\int_{-2}^{x^2 - 3 x} e^{t^2} dt$. Min at $x =$

No value

$\dfrac{1}{2}$

$\dfrac{3}{2}$

$2$

$3$

Question 40 of 43 | MCQ · Level 4

$\operatorname*{lim}\limits_{x \rightarrow 0} (1 + 2 x)^{\csc x}$

$0$

$1$

$2$

$e$

$e^2$

Question 41 of 43 | MCQ · Level 3

Coefficient of $x^6$ in Taylor of $\sin(x^2)$

$-\dfrac{1}{6}$

$0$

$\dfrac{1}{120}$

$\dfrac{1}{6}$

$1$

Question 42 of 43 | MCQ · Level 2

MVT for integrals: $\displaystyle\int_{a}^{b} f =$

$f\dfrac{c}{b-a}$

$\dfrac{f(b)-f(a)}{b-a}$

$f(b) - f(a)$

$f'(c)(b-a)$

$f(c)(b-a)$

Question 43 of 43 | MCQ · Level 3

[Calc] $f(x) = \displaystyle\sum_{k=1}^\infty (\sin^2 x)^k$. $f(1) =$

$0.369$

$0.585$

$2.400$

$2.426$

$3.426$

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Area & Circumference

Volume

Triangles

Other Facts

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Volume

Triangles

Other Facts

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