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BC MCQ Set 90 (1988 Official AP)
44 Questions
Question 1 of 44
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BC MCQ Set 90 (1988 Official AP) 0/44
Question 1 of 44   |  MCQ  · Level 1
Area in first quadrant enclosed by \(y = x(1 - x)\) and x-axis
A
\(\dfrac{1}{6}\)
✕
B
\(\dfrac{1}{3}\)
✕
C
\(\dfrac{2}{3}\)
✕
D
\(\dfrac{5}{6}\)
✕
E
\(1\)
✕
Question 2 of 44   |  MCQ  · Level 2
\(\displaystyle\int_{0}^{1} x(x^2 + 2)^2 dx =\)
A
\(\dfrac{19}{2}\)
✕
B
\(\dfrac{19}{3}\)
✕
C
\(\dfrac{9}{2}\)
✕
D
\(\dfrac{19}{6}\)
✕
E
\(\dfrac{1}{6}\)
✕
Question 3 of 44   |  MCQ  · Level 3
If \(f(x) = \ln(\sqrt{x})\), then \(f''(x) =\)
A
\(-\dfrac{2}{x^2}\)
✕
B
\(-\dfrac{1}{2 x^2}\)
✕
C
\(-\dfrac{1}{2 x}\)
✕
D
\(-\dfrac{1}{2 x^{\dfrac{3}{2}}}\)
✕
E
\(\dfrac{2}{x^2}\)
✕
Question 4 of 44   |  MCQ  · Level 3
Derivative of \(u \dfrac{v}{w}\)
A
\((u v' + u' v)/w'\)
✕
B
\((u' v' w - u v w')/w^2\)
✕
C
\((u v w' - u v' w - u' v w)/w^2\)
✕
D
\((u' v w + u v' w + u v w')/w^2\)
✕
E
\((u v' w + u' v w - u v w')/w^2\)
✕
Question 5 of 44   |  MCQ  · Level 3
\(f(x) = \sin x\) for \(x<0\), \(x^2\) for \(0\leq x<1\), \(2-x\) for \(1\leq x<2\), \(x-3\) for \(x\geq 2\). NOT continuous at?
A
\(0\) only
✕
B
\(1\) only
✕
C
\(2\) only
✕
D
\(0\) and \(2\) only
✕
E
\(0, 1\), and \(2\)
✕
Question 6 of 44   |  MCQ  · Level 3
\(y^2 - 2 x y = 16\), \(\dfrac{dy}{dx} =\)
A
\(\dfrac{x}{y - x}\)
✕
B
\(\dfrac{y}{x - y}\)
✕
C
\(\dfrac{y}{y - x}\)
✕
D
\(\dfrac{y}{2 y - x}\)
✕
E
\(\dfrac{2 y}{x - y}\)
✕
Question 7 of 44   |  MCQ  · Level 2
\(\displaystyle\int_{2}^{\infty} \dfrac{dx}{x}^2 =\)
A
\(\dfrac{1}{2}\)
✕
B
\(\ln 2\)
✕
C
\(1\)
✕
D
\(2\)
✕
E
nonexistent
✕
Question 8 of 44   |  MCQ  · Level 1
If \(f(x) = e^x\), then \(\ln(f'(2)) =\)
A
\(2\)
✕
B
\(0\)
✕
C
\(\dfrac{1}{e^2}\)
✕
D
\(2 e\)
✕
E
\(e^2\)
✕
Question 9 of 44   |  MCQ  · Level 2
\(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\sin(x+h) - \sin x}{h}\)
A
\(0\)
✕
B
\(1\)
✕
C
\(\sin x\)
✕
D
\(\cos x\)
✕
E
nonexistent
✕
Question 10 of 44   |  MCQ  · Level 2
If \(x + 7 y = 29\) is normal to \(f\) at \((1, 4)\), then \(f'(1) =\)
A
\(7\)
✕
B
\(\dfrac{1}{7}\)
✕
C
\(-\dfrac{1}{7}\)
✕
D
\(-\dfrac{7}{29}\)
✕
E
\(-7\)
✕
Question 11 of 44   |  MCQ  · Level 2
Particle constant accel 3 m/s². Velocity 10 at \(t=2\). Distance during \(v\) from 4 to 10?
A
\(20\)
✕
B
\(14\)
✕
C
\(7\)
✕
D
\(6\)
✕
E
\(3\)
✕
Question 12 of 44   |  MCQ  · Level 3
Series for \(\sin(2 x)\)
A
\(x - x^\dfrac{3}{3}! + x^\dfrac{5}{5}! - ...\)
✕
B
\(2 x - (2x)^\dfrac{3}{3}! + (2x)^\dfrac{5}{5}! - ...\)
✕
C
\(-(2x)^\dfrac{2}{2}! + (2x)^\dfrac{4}{4}! - ...\)
✕
D
\(x^\dfrac{2}{2}! + x^\dfrac{4}{4}! + ...\)
✕
E
\(2 x + (2x)^\dfrac{3}{3}! + (2x)^\dfrac{5}{5}! + ...\)
✕
Question 13 of 44   |  MCQ  · Level 2
If \(F(x) = \displaystyle\int_{1}^{x^2} \sqrt{1 + t^3} dt\), then \(F'(x) =\)
A
\(2 x \sqrt{1 + x^6}\)
✕
B
\(2 x \sqrt{1 + x^3}\)
✕
C
\(\sqrt{1 + x^6}\)
✕
D
\(\sqrt{1 + x^3}\)
✕
E
\(\displaystyle\int_{1}^{x^2} \dfrac{3 t^2}{2 \sqrt{1 + t^3}} dt\)
✕
Question 14 of 44   |  MCQ  · Level 3
\(x = t^2 + 1\), \(y = \ln(2t + 3)\). Acceleration vector?
A
\(\left(2 t, \dfrac{2}{2 t + 3}\right)\)
✕
B
\((2 t, -\dfrac{4}{(2 t + 3)^2})\)
✕
C
\((2, \dfrac{4}{(2 t + 3)^2})\)
✕
D
\((2, \dfrac{2}{(2 t + 3)^2})\)
✕
E
\((2, -\dfrac{4}{(2 t + 3)^2})\)
✕
Question 15 of 44   |  MCQ  · Level 2
\(\int x e^{2 x} d x =\)
A
\(\dfrac{x e^{2x}}{2} - \dfrac{e^{2x}}{4} + C\)
✕
B
\(\dfrac{x e^{2x}}{2} - \dfrac{e^{2x}}{2} + C\)
✕
C
\(\dfrac{x e^{2x}}{2} + \dfrac{e^{2x}}{4} + C\)
✕
D
\(\dfrac{x e^{2x}}{2} + \dfrac{e^{2x}}{2} + C\)
✕
E
\(\dfrac{x^2 e^{2x}}{4} + C\)
✕
Question 16 of 44   |  MCQ  · Level 3
\(\displaystyle\int_{2}^{3} \dfrac{3}{(x-1)(x+2)} dx =\)
A
\(-\dfrac{33}{20}\)
✕
B
\(-\dfrac{9}{20}\)
✕
C
\(\ln\left(\dfrac{5}{2}\right)\)
✕
D
\(\ln\left(\dfrac{8}{5}\right)\)
✕
E
\(\ln\left(\dfrac{2}{5}\right)\)
✕
Question 17 of 44   |  MCQ  · Level 3
Trapezoidal approx of \(\displaystyle\int_{-4}^2 e^{-x}/2 dx\) with 3 subdivisions
A
\(e^2 + e^0 + e^{-2}\)
✕
B
\(e^4 + e^2 + e^0\)
✕
C
\(e^4 + 2 e^2 + 2 e^0 + e^{-2}\)
✕
D
\(\dfrac{1}{2}(e^4 + e^2 + e^0 + e^{-2})\)
✕
E
\(\dfrac{1}{2}(e^4 + 2 e^2 + 2 e^0 + e^{-2})\)
✕
Question 18 of 44   |  MCQ  · Level 2
Polynomial with rel max \((-2,4)\), rel min \((1,1)\), rel max \((5,7)\) no other crit. How many zeros?
A
One
✕
B
Two
✕
C
Three
✕
D
Four
✕
E
Five
✕
Question 19 of 44   |  MCQ  · Level 2
Definition of \(\operatorname*{lim}\limits_{x\rightarrow a} f(x) = L\)
A
\(0 < |x-a| < \epsilon\), then \(|f - L| < \delta\)
✕
B
\(0 < |f - L| < \epsilon\), then \(|x-a| < \delta\)
✕
C
\(|f-L| < \delta\), then \(0 < |x-a| < \epsilon\)
✕
D
\(0 < |x-a| < \delta\) and \(|f-L| < \epsilon\)
✕
E
\(0 < |x-a| < \delta\), then \(|f - L| < \epsilon\)
✕
Question 20 of 44   |  MCQ  · Level 2
Average value of \(\dfrac{1}{x}\) on \([1, 3]\)
A
\(\dfrac{1}{2}\)
✕
B
\(\dfrac{2}{3}\)
✕
C
\(\dfrac{\ln 2}{2}\)
✕
D
\(\dfrac{\ln 3}{2}\)
✕
E
\(\ln 3\)
✕
Question 21 of 44   |  MCQ  · Level 4
\(f(x) = (x^2 + 1)^x\), \(f'(x) =\)
A
\(x(x^2+1)^{x-1}\)
✕
B
\(2 x^2 (x^2+1)^{x-1}\)
✕
C
\(x \ln(x^2+1)\)
✕
D
\(\ln(x^2+1) + \dfrac{2 x^2}{x^2 + 1}\)
✕
E
\((x^2+1)^x [\ln(x^2+1) + \dfrac{2 x^2}{x^2+1}]\)
✕
Question 22 of 44   |  MCQ  · Level 4
Area of loop of \(r = 4 \cos(3 \theta)\)
A
\(16 \displaystyle\int_{-\dfrac{\pi}{3}}^{\dfrac{\pi}{3}} \cos(3 \theta) d \theta\)
✕
B
\(8 \displaystyle\int_{-\dfrac{\pi}{6}}^{\dfrac{\pi}{6}} \cos(3 \theta) d \theta\)
✕
C
\(8 \displaystyle\int_{-\dfrac{\pi}{3}}^{\dfrac{\pi}{3}} \cos^2(3 \theta) d \theta\)
✕
D
\(16 \displaystyle\int_{-\dfrac{\pi}{6}}^{\dfrac{\pi}{6}} \cos^2(3 \theta) d \theta\)
✕
E
\(8 \displaystyle\int_{-\dfrac{\pi}{6}}^{\dfrac{\pi}{6}} \cos^2(3 \theta) d \theta\)
✕
Question 23 of 44   |  MCQ  · Level 3
MVT for \(f(x) = x^3 - 2 x^2\) on \([0, 2]\). \(c =\)
A
\(0\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(1\)
✕
D
\(\dfrac{4}{3}\)
✕
E
\(2\)
✕
Question 24 of 44   |  MCQ  · Level 3
Base of solid: region in first quadrant under \(y = 4 x^2\), line \(x = 1\). Sections perpendicular to x-axis are squares. Volume?
A
\(\dfrac{4 \pi}{3}\)
✕
B
\(\dfrac{16 \pi}{5}\)
✕
C
\(\dfrac{4}{3}\)
✕
D
\(\dfrac{16}{5}\)
✕
E
\(\dfrac{64}{5}\)
✕
Question 25 of 44   |  MCQ  · Level 3
\(f''\) exists and \(f(x) > 0\). Which is NOT necessarily true?
A
\(\displaystyle\int_{-1}^1 f dx > 0\)
✕
B
\(\displaystyle\int_{-1}^1 2 f dx = 2 \displaystyle\int_{-1}^1 f dx\)
✕
C
\(\displaystyle\int_{-1}^1 f dx = 2 \displaystyle\int_{0}^{1} f dx\)
✕
D
\(\displaystyle\int_{-1}^1 f dx = -\displaystyle\int_{1}^{-1} f dx\)
✕
E
\(\displaystyle\int_{-1}^1 f dx = \displaystyle\int_{-1}^0 f dx + \displaystyle\int_{0}^{1} f dx\)
✕
Question 26 of 44   |  MCQ  · Level 3
If \(y = x^3 + a x^2 + b x - 4\) has inflection at \((1, -6)\), what is \(b\)?
A
\(-3\)
✕
B
\(0\)
✕
C
\(1\)
✕
D
\(3\)
✕
E
Cannot be determined
✕
Question 27 of 44   |  MCQ  · Level 3
\(\dfrac{d}{dx} \ln|\cos\left(\dfrac{\pi}{x}\right)| =\)
A
\(\dfrac{-\pi}{x^2 \cos\left(\dfrac{\pi}{x}\right)}\)
✕
B
\(-\tan\left(\dfrac{\pi}{x}\right)\)
✕
C
\(\dfrac{1}{\cos\left(\dfrac{\pi}{x}\right)}\)
✕
D
\(\dfrac{\pi}{x} \tan\left(\dfrac{\pi}{x}\right)\)
✕
E
\(\dfrac{\pi}{x^2} \tan\left(\dfrac{\pi}{x}\right)\)
✕
Question 28 of 44   |  MCQ  · Level 3
Region in Q1 enclosed by \(x = 0\), \(y = 5\), \(y = x^2 + 1\). Revolved about y-axis.
A
\(6 \pi\)
✕
B
\(8 \pi\)
✕
C
\(\dfrac{34 \pi}{3}\)
✕
D
\(16 \pi\)
✕
E
\(\dfrac{544 \pi}{15}\)
✕
Question 29 of 44   |  MCQ  · Level 3
\(\displaystyle\sum_{i=n}^{\infty} \left(\dfrac{1}{3}\right)^i =\)
A
\(\dfrac{3}{2} - \left(\dfrac{1}{3}\right)^n\)
✕
B
\(\dfrac{3}{2}[1 - \left(\dfrac{1}{3}\right)^n]\)
✕
C
\(\dfrac{3}{2}\left(\dfrac{1}{3}\right)^n\)
✕
D
\(\dfrac{2}{3}\left(\dfrac{1}{3}\right)^n\)
✕
E
\(\dfrac{2}{3}\left(\dfrac{1}{3}\right)^{n+1}\)
✕
Question 30 of 44   |  MCQ  · Level 3
\(\displaystyle\int_{0}^{2} \sqrt{4 - x^2} dx =\)
A
\(\dfrac{8}{3}\)
✕
B
\(\dfrac{16}{3}\)
✕
C
\(\pi\)
✕
D
\(2 \pi\)
✕
E
\(4 \pi\)
✕
Question 31 of 44   |  MCQ  · Level 4
General solution of \(y' = y + x^2\) is \(y =\)
A
\(C e^x\)
✕
B
\(C e^x + x^2\)
✕
C
\(-x^2 - 2 x - 2 + C\)
✕
D
\(e^x - x^2 - 2 x - 2 + C\)
✕
E
\(C e^x - x^2 - 2 x - 2\)
✕
Question 32 of 44   |  MCQ  · Level 3
Length of \(y = x^3\) from 0 to 2
A
\(\displaystyle\int_{0}^{2} \sqrt{1 + x^6} dx\)
✕
B
\(\displaystyle\int_{0}^{2} \sqrt{1 + 3 x^2} dx\)
✕
C
\(\pi \displaystyle\int_{0}^{2} \sqrt{1 + 9 x^4} dx\)
✕
D
\(2 \pi \displaystyle\int_{0}^{2} \sqrt{1 + 9 x^4} dx\)
✕
E
\(\displaystyle\int_{0}^{2} \sqrt{1 + 9 x^4} dx\)
✕
Question 33 of 44   |  MCQ  · Level 3
Curve \(x = t^3 + t\), \(y = t^4 + 2 t^2\). Tangent at \(t=1\)
A
\(y = 2 x\)
✕
B
\(y = 8 x\)
✕
C
\(y = 2 x - 1\)
✕
D
\(y = 4 x - 5\)
✕
E
\(y = 8 x + 13\)
✕
Question 34 of 44   |  MCQ  · Level 2
\(\operatorname*{lim}\limits_{x \rightarrow \infty} x^\dfrac{k}{e}^x\) where \(k\) positive integer
A
\(0\)
✕
B
\(1\)
✕
C
\(e\)
✕
D
\(k!\)
✕
E
nonexistent
✕
Question 35 of 44   |  MCQ  · Level 3
Region between \(y=1\) and \(y = \sin x\) from 0 to \(\dfrac{\pi}{2}\) revolved about x-axis
A
\(2 \pi \int x \sin x dx\)
✕
B
\(2 \pi \int x \cos x dx\)
✕
C
\(\pi \int (1 - \sin x)^2 dx\)
✕
D
\(\pi \int \sin^2 x dx\)
✕
E
\(\pi \int (1 - \sin^2 x) dx\)
✕
Question 36 of 44   |  MCQ  · Level 3
Person 2m, lamppost 8m. Shadow lengthens at 4/9 m/s. Person walks at?
A
\(\dfrac{4}{27}\)
✕
B
\(\dfrac{4}{9}\)
✕
C
\(\dfrac{3}{4}\)
✕
D
\(\dfrac{4}{3}\)
✕
E
\(\dfrac{16}{9}\)
✕
Question 37 of 44   |  MCQ  · Level 3
\(\sum x^\dfrac{n}{n}\) converges for
A
\(-1 \leq x \leq 1\)
✕
B
\(-1 < x \leq 1\)
✕
C
\(-1 \leq x < 1\)
✕
D
\(-1 < x < 1\)
✕
E
All real \(x\)
✕
Question 38 of 44   |  MCQ  · Level 4
\(\dfrac{dy}{dx} = y \sec^2 x\), \(y = 5\) at \(x = 0\), then \(y =\)
A
\(e^{\tan x} + 4\)
✕
B
\(e^{\tan x} + 5\)
✕
C
\(5 e^{\tan x}\)
✕
D
\(\tan x + 5\)
✕
E
\(\tan x + 5 e^x\)
✕
Question 39 of 44   |  MCQ  · Level 2
\(g\) inverse of \(f\), \(g(-2) = 5\), \(f'(5) = -\dfrac{1}{2}\). \(g'(-2) =\)
A
\(2\)
✕
B
\(\dfrac{1}{2}\)
✕
C
\(\dfrac{1}{5}\)
✕
D
\(-\dfrac{1}{5}\)
✕
E
\(-2\)
✕
Question 40 of 44   |  MCQ  · Level 3
\(\operatorname*{lim}\limits_{n \rightarrow \infty} \left(\dfrac{1}{n}\right)[\sqrt{\dfrac{1}{n}} + \sqrt{\dfrac{2}{n}} + ... + \sqrt{\dfrac{n}{n}}]\)
A
\(\left(\dfrac{1}{2}\right) \displaystyle\int_{0}^{1} 1/\sqrt{x} dx\)
✕
B
\(\displaystyle\int_{0}^{1} \sqrt{x} dx\)
✕
C
\(\displaystyle\int_{0}^{1} x dx\)
✕
D
\(\displaystyle\int_{1}^{2} x dx\)
✕
E
\(2 \displaystyle\int_{1}^{2} x \sqrt{x} dx\)
✕
Question 41 of 44   |  MCQ  · Level 2
\(\displaystyle\int_{1}^{4} f(x) dx = 6\), find \(\displaystyle\int_{1}^{4} f(5 - x) dx\)
A
\(6\)
✕
B
\(3\)
✕
C
\(0\)
✕
D
\(-1\)
✕
E
\(-6\)
✕
Question 42 of 44   |  MCQ  · Level 3
Bacteria double in 3 hours. How many hours to triple?
A
\(\dfrac{3 \ln 3}{\ln 2}\)
✕
B
\(\dfrac{2 \ln 3}{\ln 2}\)
✕
C
\(\dfrac{\ln 3}{\ln 2}\)
✕
D
\(\ln\left(\dfrac{27}{2}\right)\)
✕
E
\(\ln\left(\dfrac{9}{2}\right)\)
✕
Question 43 of 44   |  MCQ  · Level 3
Which series converge? I. \(\sum (-1)^{n+1}/(2n+1)\) II. \(\sum \left(\dfrac{1}{n}\right)\left(\dfrac{3}{2}\right)^n\) III. \(\sum 1/(n \ln n)\)
A
I only
✕
B
II only
✕
C
III only
✕
D
I and III only
✕
E
I, II, and III
✕
Question 44 of 44   |  MCQ  · Level 3
Largest rectangle inscribed in \(4 x^2 + 9 y^2 = 36\)
A
\(6 \sqrt{2}\)
✕
B
\(12\)
✕
C
\(24\)
✕
D
\(24 \sqrt{2}\)
✕
E
\(36\)
✕

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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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