Exam Complete
|
BC MCQ Set 110 (1997 Official AP)
0.0%
0/34
Results by Question
1
MCQ
Error Rate 100%
Wrong
\(\displaystyle\int_{0}^{1} \sqrt{x}(x + 1) dx =\)
A
\(0\)
B
\(1\)
\(\dfrac{16}{15}\)
Correct Answer
D
\(\dfrac{7}{5}\)
E
\(2\)
Explanation
\(\displaystyle\int_{0}^{1} \left(x^{\dfrac{3}{2}} + x^{\dfrac{1}{2}}\right) dx = \dfrac{2}{5} + \dfrac{2}{3} = \dfrac{16}{15}\).
2
MCQ
Error Rate 100%
Wrong
\(x = e^{2t}\), \(y = \sin(2t)\), \(\dfrac{dy}{dx} =\)
A
\(4 e^{2t} \cos(2t)\)
B
\(\dfrac{e^{2t}}{\cos(2t)}\)
C
\(\dfrac{\sin(2t)}{2 e^{2t}}\)
D
\(\dfrac{\cos(2t)}{2 e^{2t}}\)
\(\dfrac{\cos(2t)}{e^{2t}}\)
Correct Answer
Explanation
\(\dfrac{dy}{dt} = 2 \cos(2t)\), \(\dfrac{dx}{dt} = 2 e^{2t}\). Ratio: \(\cos(2t)/e^{2t}\).
3
MCQ
Error Rate 100%
Wrong
\(f(x) = 3 x^5 - 4 x^3 - 3 x\) has rel max at
\(-1\)
Correct Answer
B
\(-\sqrt{5}/5\)
C
\(0\)
D
\(\sqrt{5}/5\)
E
\(1\)
Explanation
\(f' = 15 x^4 - 12 x^2 - 3 = 3(5 x^4 - 4 x^2 - 1) = 3(5 x^2 + 1)(x^2 - 1)\). Critical \(x = \pm 1\). At \(x=-1\): max.
4
MCQ
Error Rate 100%
Wrong
\(\dfrac{d}{dx} (x e^{\ln x^2}) =\)
A
\(1 + 2 x\)
B
\(x + x^2\)
\(3 x^2\)
Correct Answer
D
\(x^3\)
E
\(x^2 + x^3\)
Explanation
\(e^{\ln x^2} = x^2\). So \(x cdot x^2 = x^3\). \(\dfrac{d}{dx} = 3 x^2\).
5
MCQ
Error Rate 100%
Wrong
\(f(x) = (x-1)^{\dfrac{3}{2}} + e^{x-2}/2\), \(f'(2) =\)
A
\(1\)
B
\(\dfrac{3}{2}\)
\(2\)
Correct Answer
D
\(\dfrac{7}{2}\)
E
\((3+e)/2\)
Explanation
\(f' = \left(\dfrac{3}{2}\right) \sqrt{x-1} + e^{x-2}/2\). At 2: \(\dfrac{3}{2} + \dfrac{1}{2} = 2\).
6
MCQ
Error Rate 100%
Wrong
Slope of normal to \(y = \sqrt{16 - x}\) at \((0, 4)\)
\(8\)
Correct Answer
B
\(4\)
C
\(\dfrac{1}{8}\)
D
\(-\dfrac{1}{8}\)
E
\(-8\)
Explanation
\(y' = -1/(2 \sqrt{16-x})\). At \(x=0\): \(-\dfrac{1}{8}\). Normal: \(8\).
7
MCQ
Error Rate 100%
Wrong
\(y = x y + x^2 + 1\), find \(y'\) at \(x = -1\)
A
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
Correct Answer
C
\(-1\)
D
\(-2\)
E
nonexistent
Explanation
Implicit: \(y' = y + x y' + 2x \rightarrow y'(1 - x) = y + 2x\). At \(x=-1\): \(y = -y + 1 + 1 \rightarrow 2y = 2 \rightarrow y = 1\). \(y'(2) = 1 - 2 = -1 \rightarrow y' = -\dfrac{1}{2}\).
8
MCQ
Error Rate 100%
Wrong
\(\displaystyle\int_{1}^{\infty} x/(1+x^2)^2 dx\)
A
\(-\dfrac{1}{2}\)
B
\(-\dfrac{1}{4}\)
\(\dfrac{1}{4}\)
Correct Answer
D
\(\dfrac{1}{2}\)
E
divergent
Explanation
\(u = 1 + x^2\), \(du = 2x dx\). \(\left(\dfrac{1}{2}\right) \displaystyle\int_{2}^{\infty} \dfrac{du}{u}^2 = -1/(2u)|_2^\infty = \dfrac{1}{4}\).
9
MCQ
Error Rate 100%
Wrong
\(a(t) = 2t - 7\), \(v(0) = 6\). Particle farthest right at \(t =\)
A
\(0\)
\(1\)
Correct Answer
C
\(2\)
D
\(3\)
E
\(4\)
Explanation
\(v = t^2 - 7 t + 6 = (t-1)(t-6)\). \(v = 0\) at \(t = 1\) (in interval). Farthest right when v changes from + to -.
10
MCQ
Error Rate 100%
Wrong
Sum of geometric \(\dfrac{3}{2} + \dfrac{9}{16} + \dfrac{27}{128} + ...\)
A
\(1.60\)
B
\(2.35\)
\(2.40\)
Correct Answer
D
\(2.45\)
E
\(2.50\)
Explanation
\(a = \dfrac{3}{2}\), \(r = \dfrac{3}{8}\). Sum \(= \dfrac{\dfrac{3}{2}}{1 - \dfrac{3}{8}} = \dfrac{\dfrac{3}{2}}{\dfrac{5}{8}} = \dfrac{12}{5} = 2.40\).
11
MCQ
Error Rate 100%
Wrong
Length of \(x = \cos^3 t\), \(y = \sin^3 t\) for \(0 \leq t \leq \dfrac{\pi}{2}\)
A
\(\int \sqrt{3 \cos^2 t + 3 \sin^2 t} dt\)
B
\(\int \sqrt{-3 \cos^2 t \sin t + 3 \sin^2 t \cos t} dt\)
C
\(\int \sqrt{9 \cos^4 t + 9 \sin^4 t} dt\)
\(\int \sqrt{9 \cos^4 t \sin^2 t + 9 \sin^4 t \cos^2 t} dt\)
Correct Answer
E
\(\int \sqrt{\cos^6 t + \sin^6 t} dt\)
Explanation
\(\dfrac{dx}{dt} = -3 \cos^2 t \sin t\), \(\dfrac{dy}{dt} = 3 \sin^2 t \cos t\). Length integral with sum of squares.
12
MCQ
Error Rate 100%
Wrong
\(\operatorname*{lim}\limits_{h\rightarrow 0}\dfrac{e^h - 1}{2 h}\)
A
\(0\)
\(\dfrac{1}{2}\)
Correct Answer
C
\(1\)
D
\(e\)
E
nonexistent
Explanation
Derivative of \(e^x\) at 0 = 1. So \(\dfrac{1}{2}\).
13
MCQ
Error Rate 100%
Wrong
Third-degree Taylor of \(f(x) = \ln(3 - x)\) about \(x = 2\)
A
\(-(x-2) + (x-2)^\dfrac{2}{2} - (x-2)^\dfrac{3}{3}\)
\(-(x-2) - (x-2)^\dfrac{2}{2} - (x-2)^\dfrac{3}{3}\)
Correct Answer
C
\((x-2) + (x-2)^2 + (x-2)^3\)
D
\((x-2) + (x-2)^\dfrac{2}{2} + (x-2)^\dfrac{3}{3}\)
E
\((x-2) - (x-2)^\dfrac{2}{2} + (x-2)^\dfrac{3}{3}\)
Explanation
\(f(2) = 0\), \(f'(x) = -1/(3-x)\), \(f'(2) = -1\). \(f''(x) = -1/(3-x)^2\), \(f''(2) = -1\). \(f'''(x) = -2/(3-x)^3\), \(f'''(2) = -2\). \(T_3 = -(x-2) - (x-2)^\dfrac{2}{2} - (x-2)^\dfrac{3}{3}\).
14
MCQ
Error Rate 100%
Wrong
Vertical tangent of \(x = t^3 - t^2 - 1\), \(y = t^4 + 2 t^2 - 8 t\) at
A
\(0\) only
B
\(1\) only
\(0\) and \(\dfrac{2}{3}\) only
Correct Answer
D
\(0, \dfrac{2}{3}, 1\)
E
No value
Explanation
Vertical: \(\dfrac{dx}{dt} = 3 t^2 - 2 t = t(3t - 2) = 0\) at \(t = 0, \dfrac{2}{3}\). Both with \(\dfrac{dy}{dt} \neq 0\).
15
MCQ
Error Rate 100%
Wrong
\(\sum (x-2)^n/(n cdot 3^n)\) converges for
A
\(-3 \leq x \leq 3\)
B
\(-3 < x < 3\)
C
\(-1 < x \leq 5\)
D
\(-1 \leq x \leq 5\)
\(-1 \leq x < 5\)
Correct Answer
Explanation
Center 2, radius 3. At \(x = -1\): alternating harmonic, converges. At \(x = 5\): harmonic, diverges. So \([-1, 5)\).
16
MCQ
Error Rate 100%
Wrong
Area inside \(r = 2 \cos \theta\) outside \(r = \cos \theta\)
\(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\)
Correct Answer
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
\(3 \displaystyle\int_{0}^{\pi} \cos \theta d \theta\)
Explanation
\(\left(\dfrac{1}{2}\right) \int [(2 \cos \theta)^2 - (\cos \theta)^2] d \theta = \left(\dfrac{1}{2}\right) \int 3 \cos^2 \theta d \theta\). Symmetric, double over \([0, \dfrac{\pi}{2}]\): \(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\).
17
MCQ
Error Rate 100%
Wrong
Triangle with hypotenuse 5, opposite x. \(\theta\) increases at 3 rad/min. Find \(\dfrac{dx}{dt}\) when \(x=3\).
A
\(3\)
B
\(\dfrac{15}{4}\)
C
\(4\)
D
\(9\)
\(12\)
Correct Answer
Explanation
\(x = 5 \sin \theta\). \(\dfrac{dx}{dt} = 5 \cos \theta cdot 3\). When \(x=3\): \(\sin \theta = \dfrac{3}{5}\), \(\cos \theta = \dfrac{4}{5}\). \(\dfrac{dx}{dt} = 5\left(\dfrac{4}{5}\right)(3) = 12\).
18
MCQ
Error Rate 100%
Wrong
Coefficient of \(x^7\) in Taylor of \(f\) where \(f'(x) = \sin(x^2)\)
A
\(\dfrac{1}{7}!\)
B
\(\dfrac{1}{7}\)
C
\(0\)
\(-\dfrac{1}{42}\)
Correct Answer
E
\(-\dfrac{1}{7}!\)
Explanation
\(\sin(x^2) = x^2 - x^\dfrac{6}{6} + ...\). Integrate: \(f = x^\dfrac{3}{3} - x^\dfrac{7}{42} + ...\). Coeff of \(x^7\): \(-\dfrac{1}{42}\).
19
MCQ
Error Rate 100%
Wrong
\(\operatorname*{lim}\limits_{n \rightarrow \infty} \displaystyle\sum_{i=1}^n \sqrt{x_i} \Delta x\) for partition of \([a,b]\)
\(\left(\dfrac{2}{3}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\)
Correct Answer
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
\(2\left(b^{\dfrac{1}{2}} - a^{\dfrac{1}{2}}\right)\)
Explanation
\(\displaystyle\int_{a}^{b} \sqrt{x} dx = \left(\dfrac{2}{3}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\).
20
MCQ
Error Rate 100%
Wrong
[Calc] Which sequences converge? I. \(\{5n/(2n-1)\}\) II. \(\{e^\dfrac{n}{n}\}\) III. \(\{e^n/(1+e^n)\}\)
A
I only
B
II only
C
I and II only
I and III only
Correct Answer
E
I, II, and III
Explanation
I: \(\rightarrow \dfrac{5}{2}\). II: \(\rightarrow \infty\). III: \(\rightarrow 1\).
21
MCQ
Error Rate 100%
Wrong
[Calc] Region enclosed by \(y = x\) and \(y = 4 x - x^2\) revolved about y-axis
A
\(\pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) dx\)
B
\(\pi \displaystyle\int_{0}^{3} (x^2 - (4x - x^2)^2) dx\)
C
\(\pi \displaystyle\int_{0}^{3} (3 x - x^2)^2 dx\)
D
\(2 \pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) dx\)
\(2 \pi \displaystyle\int_{0}^{3} (3 x^2 - x^3) dx\)
Correct Answer
Explanation
Shell: \(V = 2 \pi \int x cdot (top - bottom) dx = 2 \pi \displaystyle\int_{0}^{3} x(4x - x^2 - x) dx = 2 \pi \displaystyle\int_{0}^{3} (3 x^2 - x^3) dx\).
22
MCQ
Error Rate 100%
Wrong
[Calc] \(\operatorname*{lim}\limits_{h\rightarrow 0}(\ln(e+h) - 1)/h\)
\(f'(e)\) where \(f(x) = \ln x\)
Correct Answer
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 \(f(x) = \ln x\)
Explanation
Definition of derivative of \(\ln x\) at \(x = e\): \(\left(\dfrac{d}{dx} \ln x\right)|_{x=e} = \dfrac{1}{e}\).
23
MCQ
Error Rate 100%
Wrong
[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
Six
Correct Answer
E
Seven
Explanation
\(y' = -\left(\dfrac{5}{6}\right) \sin(5t) - \left(\dfrac{5}{4}\right) \cos(5t)\). Set 0: \(\tan(5t) = -\dfrac{3}{2}\). Period of tan is \(\pi\), so \(5t\) has period \(\pi\). \(5(4) = 20\). \(\dfrac{20}{\pi} \approx 6.37\). So 6 zeros.
24
MCQ
Error Rate 100%
Wrong
[Calc] \(f(x) = \cos(2x) + \ln(3x)\), least value of \(x\) for inflection?
A
\(0.56\)
\(0.93\)
Correct Answer
C
\(1.18\)
D
\(2.38\)
E
\(2.44\)
Explanation
\(f'' = -4 \cos(2x) - \dfrac{1}{x}^2 = 0\). Numerical.
25
MCQ
Error Rate 100%
Wrong
[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
\(-1 \leq f(x) \leq 3\)
\(f(c) = 1\) for some \(c\)
Correct Answer
E
\(f(c) = 0\) for some \(c\) between \(-1\) and \(3\)
Explanation
IVT: \(f\) takes every value between \(-1\) and \(3\), including \(1\).
26
MCQ
Error Rate 100%
Wrong
[Calc] \(\displaystyle\int_{0}^{x} (t^2 - 2t) dt \geq \displaystyle\int_{2}^{x} t dt\) for \(0 \leq x \leq 4\). Greatest x?
A
\(1.35\)
\(1.38\)
Correct Answer
C
\(1.41\)
D
\(1.48\)
E
\(1.59\)
Explanation
Numerical.
27
MCQ
Error Rate 100%
Wrong
[Calc] \(\dfrac{dy}{dx} = (1 + \ln x) y\), \(y(1) = 1\). \(y =\)
A
\(e^{(x^2-1)/x^2}\)
B
\(1 + \ln x\)
C
\(\ln x\)
D
\(e^{2x + x \ln x - 2}\)
\(e^{x \ln x}\)
Correct Answer
Explanation
Separable: \(\dfrac{dy}{y} = (1 + \ln x) dx\). \(\ln y = x + x \ln x - x + C = x \ln x + C\). \(y(1) = 0 + C = 0 \rightarrow C = 0\). \(y = e^{x \ln x}\).
28
MCQ
Error Rate 100%
Wrong
\(\int x^2 \sin x dx =\)
A
\(-x^2 \cos x - 2 x \sin x - 2 \cos x + C\)
B
\(-x^2 \cos x + 2 x \sin x - 2 \cos x + C\)
\(-x^2 \cos x + 2 x \sin x + 2 \cos x + C\)
Correct Answer
D
\(-x^\dfrac{3}{3} \cos x + C\)
E
\(2 x \cos x + C\)
Explanation
By parts twice.
29
MCQ
Error Rate 100%
Wrong
[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
I and III only
Correct Answer
E
II and III only
Explanation
I: \((7-2)/2 = \dfrac{5}{2}\) ✓. II: not enough info. III: avg \(f' = \dfrac{f(3)-f(1)}{3-1} = \dfrac{5}{2}\) ✓.
30
MCQ
Error Rate 100%
Wrong
\(\int dx/((x-1)(x+3)) =\)
\(\left(\dfrac{1}{4}\right) \ln|\dfrac{x-1}{x+3}| + C\)
Correct Answer
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
\(\ln|(x-1)(x+3)| + C\)
Explanation
Partial fractions: \(1/((x-1)(x+3)) = \left(\dfrac{1}{4}\right)[1/(x-1) - 1/(x+3)]\). Integral: \(\left(\dfrac{1}{4}\right) \ln|\dfrac{x-1}{x+3}|\).
31
MCQ
Error Rate 100%
Wrong
Base of solid: \(y = 2 - x^2\) in Q1 with axes. Cross-sections perp to y-axis are squares. Volume?
A
\(\pi \displaystyle\int_{0}^{2} (2 - y)^2 dy\)
\(\displaystyle\int_{0}^{2} (2 - y) dy\)
Correct Answer
C
\(\pi \displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 dx\)
D
\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 dx\)
E
\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2) dx\)
Explanation
Side \(= \sqrt{2 - y}\). Area \(= 2 - y\). \(V = \displaystyle\int_{0}^{2} (2 - y) dy\).
32
MCQ
Error Rate 100%
Wrong
[Calc] \(f(x) = \displaystyle\int_{0}^{x^2} \sin t dt\). How many points in \([0, \sqrt{\pi}]\) where instantaneous rate equals avg rate?
A
Zero
B
One
Two
Correct Answer
D
Three
E
Four
Explanation
Numerical: solve \(f'(x) = (f(\sqrt{\pi}) - f(0))/\sqrt{\pi}\). Two solutions.
33
MCQ
Error Rate 100%
Wrong
[Calc] \(f\) antiderivative of \(x^2/(1+x^5)\), \(f(1) = 0\). \(f(4) =\)
A
\(-0.012\)
B
\(0\)
C
\(0.016\)
\(0.376\)
Correct Answer
E
\(0.629\)
Explanation
\(f(4) = \displaystyle\int_{1}^{4} x^2/(1+x^5) dx \approx 0.376\).
34
MCQ
Error Rate 100%
Wrong
[Calc] Spring: 10 lb stretches 4 inches. Work to stretch 6 inches?
A
\(60\)
\(45\)
Correct Answer
C
\(40\)
D
\(15\)
E
\(7.2\)
Explanation
\(F = kx\). \(10 = 4k \rightarrow k = 2.5\). \(W = \displaystyle\int_{0}^{6} 2.5 x dx = 2.5(18) = 45\).