BC MCQ Set 110 (1997 Official AP)

Question 1 of 34 | MCQ · Level 1

\(\displaystyle\int_{0}^{1} \sqrt{x}(x + 1) dx =\)

\(0\)

\(1\)

\(\dfrac{16}{15}\)

\(\dfrac{7}{5}\)

\(2\)

Question 2 of 34 | MCQ · Level 2

\(x = e^{2t}\), \(y = \sin(2t)\), \(\dfrac{dy}{dx} =\)

\(4 e^{2t} \cos(2t)\)

\(\dfrac{e^{2t}}{\cos(2t)}\)

\(\dfrac{\sin(2t)}{2 e^{2t}}\)

\(\dfrac{\cos(2t)}{2 e^{2t}}\)

\(\dfrac{\cos(2t)}{e^{2t}}\)

Question 3 of 34 | MCQ · Level 2

\(f(x) = 3 x^5 - 4 x^3 - 3 x\) has rel max at

\(-1\)

\(-\sqrt{5}/5\)

\(0\)

\(\sqrt{5}/5\)

\(1\)

Question 4 of 34 | MCQ · Level 2

\(\dfrac{d}{dx} (x e^{\ln x^2}) =\)

\(1 + 2 x\)

\(x + x^2\)

\(3 x^2\)

\(x^3\)

\(x^2 + x^3\)

Question 5 of 34 | MCQ · Level 3

\(f(x) = (x-1)^{\dfrac{3}{2}} + e^{x-2}/2\), \(f'(2) =\)

\(1\)

\(\dfrac{3}{2}\)

\(2\)

\(\dfrac{7}{2}\)

\((3+e)/2\)

Question 6 of 34 | MCQ · Level 2

Slope of normal to \(y = \sqrt{16 - x}\) at \((0, 4)\)

\(8\)

\(4\)

\(\dfrac{1}{8}\)

\(-\dfrac{1}{8}\)

\(-8\)

Question 7 of 34 | MCQ · Level 3

\(y = x y + x^2 + 1\), find \(y'\) at \(x = -1\)

\(\dfrac{1}{2}\)

\(-\dfrac{1}{2}\)

\(-1\)

\(-2\)

nonexistent

Question 8 of 34 | MCQ · Level 2

\(\displaystyle\int_{1}^{\infty} x/(1+x^2)^2 dx\)

\(-\dfrac{1}{2}\)

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

\(\dfrac{1}{4}\)

\(\dfrac{1}{2}\)

divergent

Question 9 of 34 | MCQ · Level 3

\(a(t) = 2t - 7\), \(v(0) = 6\). Particle farthest right at \(t =\)

\(0\)

\(1\)

\(2\)

\(3\)

\(4\)

Question 10 of 34 | MCQ · Level 2

Sum of geometric \(\dfrac{3}{2} + \dfrac{9}{16} + \dfrac{27}{128} + ...\)

\(1.60\)

\(2.35\)

\(2.40\)

\(2.45\)

\(2.50\)

Question 11 of 34 | MCQ · Level 3

Length of \(x = \cos^3 t\), \(y = \sin^3 t\) for \(0 \leq t \leq \dfrac{\pi}{2}\)

\(\int \sqrt{3 \cos^2 t + 3 \sin^2 t} dt\)

\(\int \sqrt{-3 \cos^2 t \sin t + 3 \sin^2 t \cos t} dt\)

\(\int \sqrt{9 \cos^4 t + 9 \sin^4 t} dt\)

\(\int \sqrt{9 \cos^4 t \sin^2 t + 9 \sin^4 t \cos^2 t} dt\)

\(\int \sqrt{\cos^6 t + \sin^6 t} dt\)

Question 12 of 34 | MCQ · Level 2

\(\operatorname*{lim}\limits_{h\rightarrow 0}\dfrac{e^h - 1}{2 h}\)

\(0\)

\(\dfrac{1}{2}\)

\(1\)

\(e\)

nonexistent

Question 13 of 34 | MCQ · Level 3

Third-degree Taylor of \(f(x) = \ln(3 - x)\) about \(x = 2\)

\(-(x-2) + (x-2)^\dfrac{2}{2} - (x-2)^\dfrac{3}{3}\)

\(-(x-2) - (x-2)^\dfrac{2}{2} - (x-2)^\dfrac{3}{3}\)

\((x-2) + (x-2)^2 + (x-2)^3\)

\((x-2) + (x-2)^\dfrac{2}{2} + (x-2)^\dfrac{3}{3}\)

\((x-2) - (x-2)^\dfrac{2}{2} + (x-2)^\dfrac{3}{3}\)

Question 14 of 34 | MCQ · Level 3

Vertical tangent of \(x = t^3 - t^2 - 1\), \(y = t^4 + 2 t^2 - 8 t\) at

\(0\) only

\(1\) only

\(0\) and \(\dfrac{2}{3}\) only

\(0, \dfrac{2}{3}, 1\)

No value

Question 15 of 34 | MCQ · Level 3

\(\sum (x-2)^n/(n cdot 3^n)\) converges for

\(-3 \leq x \leq 3\)

\(-3 < x < 3\)

\(-1 < x \leq 5\)

\(-1 \leq x \leq 5\)

\(-1 \leq x < 5\)

Question 16 of 34 | MCQ · Level 3

Area inside \(r = 2 \cos \theta\) outside \(r = \cos \theta\)

\(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\)

\(3 \displaystyle\int_{0}^{\pi} \cos^2 \theta d \theta\)

\(\left(\dfrac{3}{2}\right) \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\)

\(3 \displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos \theta d \theta\)

\(3 \displaystyle\int_{0}^{\pi} \cos \theta d \theta\)

Question 17 of 34 | MCQ · Level 3

Triangle with hypotenuse 5, opposite x. \(\theta\) increases at 3 rad/min. Find \(\dfrac{dx}{dt}\) when \(x=3\).

\(3\)

\(\dfrac{15}{4}\)

\(4\)

\(9\)

\(12\)

Question 18 of 34 | MCQ · Level 4

Coefficient of \(x^7\) in Taylor of \(f\) where \(f'(x) = \sin(x^2)\)

\(\dfrac{1}{7}!\)

\(\dfrac{1}{7}\)

\(0\)

\(-\dfrac{1}{42}\)

\(-\dfrac{1}{7}!\)

Question 19 of 34 | MCQ · Level 3

\(\operatorname*{lim}\limits_{n \rightarrow \infty} \displaystyle\sum_{i=1}^n \sqrt{x_i} \Delta x\) for partition of \([a,b]\)

\(\left(\dfrac{2}{3}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\)

\(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\)

\(\left(\dfrac{3}{2}\right)\left(b^{\dfrac{3}{2}} - a^{\dfrac{3}{2}}\right)\)

\(b^{\dfrac{1}{2}} - a^{\dfrac{1}{2}}\)

\(2\left(b^{\dfrac{1}{2}} - a^{\dfrac{1}{2}}\right)\)

Question 20 of 34 | MCQ · Level 3

[Calc] Which sequences converge? I. \(\{5n/(2n-1)\}\) II. \(\{e^\dfrac{n}{n}\}\) III. \(\{e^n/(1+e^n)\}\)

I only

II only

I and II only

I and III only

I, II, and III

Question 21 of 34 | MCQ · Level 3

[Calc] Region enclosed by \(y = x\) and \(y = 4 x - x^2\) revolved about y-axis

\(\pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) dx\)

\(\pi \displaystyle\int_{0}^{3} (x^2 - (4x - x^2)^2) dx\)

\(\pi \displaystyle\int_{0}^{3} (3 x - x^2)^2 dx\)

\(2 \pi \displaystyle\int_{0}^{3} (x^3 - 3 x^2) dx\)

\(2 \pi \displaystyle\int_{0}^{3} (3 x^2 - x^3) dx\)

Question 22 of 34 | MCQ · Level 3

[Calc] \(\operatorname*{lim}\limits_{h\rightarrow 0}(\ln(e+h) - 1)/h\)

\(f'(e)\) where \(f(x) = \ln x\)

\(f'(e)\) where \(f(x) = \ln \dfrac{x}{x}\)

\(f'(1)\) where \(f(x) = \ln x\)

\(f'(1)\) where \(f(x) = \ln(x+e)\)

\(f'(0)\) where \(f(x) = \ln x\)

Question 23 of 34 | MCQ · Level 3

[Calc] \(y(t) = \left(\dfrac{1}{6}\right) \cos(5t) - \left(\dfrac{1}{4}\right) \sin(5t)\). Velocity = 0 how many times in \([0, 4]\)?

Zero

Three

Five

Six

Seven

Question 24 of 34 | MCQ · Level 3

[Calc] \(f(x) = \cos(2x) + \ln(3x)\), least value of \(x\) for inflection?

\(0.56\)

\(0.93\)

\(1.18\)

\(2.38\)

\(2.44\)

Question 25 of 34 | MCQ · Level 2

[Calc] \(f\) continuous on \([-3, 6]\), \(f(-3) = -1\), \(f(6) = 3\). IVT guarantees:

\(f(0) = 0\)

\(f'(c) = \dfrac{4}{9}\)

\(-1 \leq f(x) \leq 3\)

\(f(c) = 1\) for some \(c\)

\(f(c) = 0\) for some \(c\) between \(-1\) and \(3\)

Question 26 of 34 | MCQ · Level 3

[Calc] \(\displaystyle\int_{0}^{x} (t^2 - 2t) dt \geq \displaystyle\int_{2}^{x} t dt\) for \(0 \leq x \leq 4\). Greatest x?

\(1.35\)

\(1.38\)

\(1.41\)

\(1.48\)

\(1.59\)

Question 27 of 34 | MCQ · Level 4

[Calc] \(\dfrac{dy}{dx} = (1 + \ln x) y\), \(y(1) = 1\). \(y =\)

\(e^{(x^2-1)/x^2}\)

\(1 + \ln x\)

\(\ln x\)

\(e^{2x + x \ln x - 2}\)

\(e^{x \ln x}\)

Question 28 of 34 | MCQ · Level 3

\(\int x^2 \sin x dx =\)

\(-x^2 \cos x - 2 x \sin x - 2 \cos x + C\)

\(-x^2 \cos x + 2 x \sin x - 2 \cos x + C\)

\(-x^2 \cos x + 2 x \sin x + 2 \cos x + C\)

\(-x^\dfrac{3}{3} \cos x + C\)

\(2 x \cos x + C\)

Question 29 of 34 | MCQ · Level 3

[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 \(\dfrac{5}{2}\)

None

I only

III only

I and III only

II and III only

Question 30 of 34 | MCQ · Level 2

\(\int dx/((x-1)(x+3)) =\)

\(\left(\dfrac{1}{4}\right) \ln|\dfrac{x-1}{x+3}| + C\)

\(\left(\dfrac{1}{4}\right) \ln|\dfrac{x+3}{x-1}| + C\)

\(\left(\dfrac{1}{2}\right) \ln|(x-1)(x+3)| + C\)

\(\left(\dfrac{1}{2}\right) \ln|\dfrac{2x+2}{(x-1)(x+3)}| + C\)

\(\ln|(x-1)(x+3)| + C\)

Question 31 of 34 | MCQ · Level 3

Base of solid: \(y = 2 - x^2\) in Q1 with axes. Cross-sections perp to y-axis are squares. Volume?

\(\pi \displaystyle\int_{0}^{2} (2 - y)^2 dy\)

\(\displaystyle\int_{0}^{2} (2 - y) dy\)

\(\pi \displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 dx\)

\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2)^2 dx\)

\(\displaystyle\int_{0}^{\sqrt{2}} (2 - x^2) dx\)

Question 32 of 34 | MCQ · Level 3

[Calc] \(f(x) = \displaystyle\int_{0}^{x^2} \sin t dt\). How many points in \([0, \sqrt{\pi}]\) where instantaneous rate equals avg rate?

Zero

One

Two

Three

Four

Question 33 of 34 | MCQ · Level 3

[Calc] \(f\) antiderivative of \(x^2/(1+x^5)\), \(f(1) = 0\). \(f(4) =\)

\(-0.012\)

\(0\)

\(0.016\)

\(0.376\)

\(0.629\)

Question 34 of 34 | MCQ · Level 2

[Calc] Spring: 10 lb stretches 4 inches. Work to stretch 6 inches?

\(60\)

\(45\)

\(40\)

\(15\)

\(7.2\)

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Area & Circumference

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