BC MCQ Set 80 (1985 Official AP)

Question 1 of 41 | MCQ · Level 1

The area between \(y = 4 x^3 + 2\) and the x-axis from \(x = 1\) to \(x = 2\) is

\(36\)

\(23\)

\(20\)

\(17\)

\(9\)

Question 2 of 41 | MCQ · Level 2

At what values of \(x\) does \(f(x) = 3 x^5 - 5 x^3 + 15\) have a relative maximum?

\(-1\) only

\(0\) only

\(1\) only

\(-1\) and \(1\) only

\(-1, 0\), and \(1\)

Question 3 of 41 | MCQ · Level 3

\(\displaystyle\int_{1}^{2} \dfrac{x + 1}{x^2 + 2 x} d x =\)

\(\ln 8 - \ln 3\)

\(\dfrac{\ln 8 - \ln 3}{2}\)

\(\ln 8\)

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

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

Question 4 of 41 | MCQ · Level 3

A particle moves so that \(x = t^2 - 1\) and \(y = t^4 - 2 t^3\). At \(t = 1\), its acceleration vector is

\((0, -1)\)

\((0, 12)\)

\((2, -2)\)

\((2, 0)\)

\((2, 8)\)

Question 5 of 41 | MCQ · Level 3

If \(f(x) = \dfrac{x}{\tan} x\), then \(f'\left(\dfrac{\pi}{4}\right) =\)

\(2\)

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

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

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

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

Question 6 of 41 | MCQ · Level 1

\(\int \dfrac{d x}{\sqrt{25 - x^2}} =\)

\(\arcsin\left(\dfrac{x}{5}\right) + C\)

\(\arcsin x + C\)

\(\left(\dfrac{1}{5}\right) \arcsin\left(\dfrac{x}{5}\right) + C\)

\(\sqrt{25 - x^2} + C\)

\(2 \sqrt{25 - x^2} + C\)

Question 7 of 41 | MCQ · Level 2

If \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{f(x) - f(2)}{x - 2} = 0\), which must be true?

limit doesn't exist

\(f\) not defined at 2

\(f'(2) = 0\)

\(f\) continuous at 0

\(f(2) = 0\)

Question 8 of 41 | MCQ · Level 3

If \(x y^2 + 2 x y = 8\), then at point \((1, 2)\), \(y'\) is

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

\(-\dfrac{4}{3}\)

\(-1\)

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

\(0\)

Question 9 of 41 | MCQ · Level 3

For \(-1 < x < 1\), if \(f(x) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1} x^{2n-1}}{2n - 1}\), then \(f'(x) =\)

\(\sum (-1)^{n+1} x^{2n-2}\)

\(\sum (-1)^n x^{2n-2}\)

\(\sum (-1)^{2n} x^{2n}\)

\(\sum (-1)^n x^{2n}\)

\(\sum (-1)^{n+1} x^{2n}\)

Question 10 of 41 | MCQ · Level 2

\(\dfrac{d}{d x} \ln\left(\dfrac{1}{1 - x}\right) =\)

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

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

\(1 - x\)

\(x - 1\)

\((1 - x)^2\)

Question 11 of 41 | MCQ · Level 3

\(\int \dfrac{d x}{(x - 1)(x + 2)} =\)

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

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

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

\((\ln|x-1|)(\ln|x+2|) + C\)

\(\ln|(x-1)(x+2)^2| + C\)

Question 12 of 41 | MCQ · Level 3

Let \(f(x) = x^3 - 3 x^2\). MVT \(c\) values on \([0, 3]\)?

\(0\) only

\(2\) only

\(3\) only

\(0\) and \(3\)

\(2\) and \(3\)

Question 13 of 41 | MCQ · Level 3

Which series converge? I. \(\sum \dfrac{1}{n}^2\) II. \(\sum \dfrac{1}{n}\) III. \(\sum (-1)^{n+1}/3^{n-1}\)

I only

III only

I and III only

II and III only

I, II, and III

Question 14 of 41 | MCQ · Level 1

If \(v(t) = 2 t - 4\) and at \(t = 0\) position is 4, then \(x(t) =\)

\(t^2 - 4 t\)

\(t^2 - 4 t - 4\)

\(t^2 - 4 t + 4\)

\(2 t^2 - 4 t\)

\(2 t^2 - 4 t + 4\)

Question 15 of 41 | MCQ · Level 2

Which function shows 'continuous at 0 ⟹ differentiable at 0' is false?

\(x^{-\dfrac{4}{3}}\)

\(x^{-\dfrac{1}{3}}\)

\(x^{\dfrac{1}{3}}\)

\(x^{\dfrac{4}{3}}\)

\(x^3\)

Question 16 of 41 | MCQ · Level 2

If \(f(x) = x \ln(x^2)\), then \(f'(x) =\)

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

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

\(\ln(x^2) + \dfrac{1}{x}\)

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

\(\dfrac{1}{x}\)

Question 17 of 41 | MCQ · Level 1

\(\int \sin(2 x + 3) d x =\)

\(-2 \cos(2 x + 3) + C\)

\(-\cos(2 x + 3) + C\)

\(-\dfrac{1}{2} \cos(2 x + 3) + C\)

\(\dfrac{1}{2} \cos(2 x + 3) + C\)

\(\cos(2 x + 3) + C\)

Question 18 of 41 | MCQ · Level 4

If \(g(x) = e^{f(x)}\) and \(g''(x) = h(x) e^{f(x)}\), then \(h(x) =\)

\(f'(x) + f''(x)\)

\(f'(x) + (f''(x))^2\)

\((f'(x) + f''(x))^2\)

\((f'(x))^2 + f''(x)\)

\(2 f'(x) + f''(x)\)

Question 19 of 41 | MCQ · Level 3

If \(\int f(x) \sin x d x = -f(x) \cos x + \int 3 x^2 \cos x d x\), then \(f(x)\) could be

\(3 x^2\)

\(x^3\)

\(-x^3\)

\(\sin x\)

\(\cos x\)

Question 20 of 41 | MCQ · Level 2

Area of circle increases at \(96 \pi\) m²/s. When area is \(64 \pi\), find \(\dfrac{dr}{dt}\).

\(6\)

\(8\)

\(16\)

\(4 \sqrt{3}\)

\(12 \sqrt{3}\)

Question 21 of 41 | MCQ · Level 3

\(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\displaystyle\int_{1}^{1+h} \sqrt{x^5 + 8} dx}{h}\) is

\(0\)

\(1\)

\(3\)

\(2 \sqrt{2}\)

nonexistent

Question 22 of 41 | MCQ · Level 3

Area enclosed by polar curve \(r = \sin(2 \theta)\) for \(0 \leq \theta \leq \dfrac{\pi}{2}\)

\(0\)

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

\(1\)

\(\dfrac{\pi}{8}\)

\(\dfrac{\pi}{4}\)

Question 23 of 41 | MCQ · Level 3

\(x(t) = t e^{-2 t}\). Particle at rest when?

No values

\(0\) only

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

\(1\) only

\(0\) and \(\dfrac{1}{2}\)

Question 24 of 41 | MCQ · Level 4

For \(0 < x < \dfrac{\pi}{2}\), if \(y = (\sin x)^x\), then \(\dfrac{dy}{dx} =\)

\(x \ln(\sin x)\)

\((\sin x)^x \cot x\)

\(x (\sin x)^{x-1} \cos x\)

\((\sin x)^x (x \cos x + \sin x)\)

\((\sin x)^x (x \cot x + \ln(\sin x))\)

Question 25 of 41 | MCQ · Level 3

An antiderivative of \(f(x) = e^{x + e^x}\) is

\(\dfrac{e^{x+e^x}}{1+e^x}\)

\((1+e^x) e^{x+e^x}\)

\(e^{1+e^x}\)

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

\(e^{e^x}\)

Question 26 of 41 | MCQ · Level 3

\(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{4}} \dfrac{\sin\left(x - \dfrac{\pi}{4}\right)}{x - \dfrac{\pi}{4}}\) is

\(0\)

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

\(\dfrac{\pi}{4}\)

\(1\)

nonexistent

Question 27 of 41 | MCQ · Level 3

If \(x = t^3 - t\) and \(y = \sqrt{3 t + 1}\), \(\dfrac{dy}{dx}\) at \(t=1\) is

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

\(\dfrac{3}{8}\)

\(\dfrac{3}{4}\)

\(\dfrac{8}{3}\)

\(8\)

Question 28 of 41 | MCQ · Level 4

What are all \(x\) for which \(\displaystyle\sum_{n=1}^\infty (x-1)^\dfrac{n}{n}\) converges?

\(-1 \leq x < 1\)

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

\(0 < x < 2\)

\(0 \leq x < 2\)

\(0 \leq x \leq 2\)

Question 29 of 41 | MCQ · Level 3

Equation of normal to \(y = x^3 + 3 x^2 + 7 x - 1\) at \(x = -1\)

\(4 x + y = -10\)

\(x - 4 y = 23\)

\(4 x - y = 2\)

\(x + 4 y = 25\)

\(x + 4 y = -25\)

Question 30 of 41 | MCQ · Level 2

If \(\dfrac{dy}{dt} = -2 y\) and \(y = 1\) at \(t = 0\), find \(t\) for \(y = \dfrac{1}{2}\).

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

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

\(\dfrac{\ln 2}{2}\)

\(\dfrac{\sqrt{2}}{2}\)

\(\ln 2\)

Question 31 of 41 | MCQ · Level 4

Surface area generated by revolving \(x = y^3\) from \(y=0\) to \(y=1\) about y-axis

\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + 9 y^4} d y\)

\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + y^6} d y\)

\(2 \pi \displaystyle\int_{0}^{1} y^3 \sqrt{1 + 3 y^2} d y\)

\(2 \pi \displaystyle\int_{0}^{1} y \sqrt{1 + 9 y^4} d y\)

\(2 \pi \displaystyle\int_{0}^{1} y \sqrt{1 + y^6} d y\)

Question 32 of 41 | MCQ · Level 3

Region in first quadrant between x-axis and \(y = 6 x - x^2\) rotated around y-axis. Volume?

\(\displaystyle\int_{0}^{6} \pi (6x - x^2)^2 dx\)

\(\displaystyle\int_{0}^{6} 2 \pi x (6x - x^2) dx\)

\(\displaystyle\int_{0}^{6} \pi x (6x - x^2)^2 dx\)

\(\displaystyle\int_{0}^{6} \pi (3 + \sqrt{9-y})^2 dy\)

\(\displaystyle\int_{0}^{9} \pi (3 + \sqrt{9-y})^2 dy\)

Question 33 of 41 | MCQ · Level 3

\(\displaystyle\int_{-1}^1 \dfrac{3}{x^2} d x\) is

\(-6\)

\(-3\)

\(0\)

\(6\)

nonexistent

Question 34 of 41 | MCQ · Level 4

General solution of \(\dfrac{dy}{dx} + y = x e^{-x}\)

\(y = \left(x^\dfrac{2}{2}\right) e^{-x} + C e^{-x}\)

\(y = \left(x^\dfrac{2}{2}\right) e^{-x} + e^{-x} + C\)

\(y = -e^{-x} + C/(1+x)\)

\(y = x e^{-x} + C e^{-x}\)

\(y = C_1 e^x + C_2 x e^{-x}\)

Question 35 of 41 | MCQ · Level 4

\(\operatorname*{lim}\limits_{x \rightarrow \infty} (1 + 5 e^x)^{\dfrac{1}{x}}\)

\(0\)

\(1\)

\(e\)

\(e^5\)

nonexistent

Question 36 of 41 | MCQ · Level 4

Base of solid: region under \(y = e^{-x}\), axes, \(x = 3\). Cross-sections perpendicular to x-axis are squares. Volume?

\(\dfrac{1 - e^{-6}}{2}\)

\(\dfrac{1}{2} e^{-6}\)

\(e^{-6}\)

\(e^{-3}\)

\(1 - e^{-3}\)

Question 37 of 41 | MCQ · Level 3

If \(u = \dfrac{x}{2}\), then \(\displaystyle\int_{2}^{4} \dfrac{1 - \left(\dfrac{x}{2}\right)^2}{x} dx =\)

\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{u} du\)

\(\displaystyle\int_{2}^{4} \dfrac{1 - u^2}{u} du\)

\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{2u} du\)

\(\displaystyle\int_{1}^{2} \dfrac{1 - u^2}{4u} du\)

\(\displaystyle\int_{2}^{4} \dfrac{1 - u^2}{2u} du\)

Question 38 of 41 | MCQ · Level 2

Length of \(y = \left(\dfrac{2}{3}\right) x^{\dfrac{3}{2}}\) from \(x = 0\) to \(x = 3\)?

\(\dfrac{8}{3}\)

\(4\)

\(\dfrac{14}{3}\)

\(\dfrac{16}{3}\)

\(7\)

Question 39 of 41 | MCQ · Level 3

Coefficient of \(x^3\) in Taylor series for \(e^{3x}\) about \(x = 0\)

\(\dfrac{1}{6}\)

\(\dfrac{1}{3}\)

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

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

\(\dfrac{9}{2}\)

Question 40 of 41 | MCQ · Level 3

Slope \(3 x^2 y\), curve through \((0, 8)\). Equation:

\(y = 8 e^{x^3}\)

\(y = x^3 + 8\)

\(y = e^{x^3} + 7\)

\(y = \ln(x+1) + 8\)

\(y^2 = x^3 + 8\)

Question 41 of 41 | MCQ · Level 4

\(\operatorname*{lim}\limits_{n \rightarrow \infty} \left(\dfrac{1}{n}\right) [\left(\dfrac{1}{n}\right)^2 + \left(\dfrac{2}{n}\right)^2 + ... + \left(\dfrac{3n}{n}\right)^2]\)

\(\displaystyle\int_{0}^{1} \dfrac{1}{x}^2 dx\)

\(3 \displaystyle\int_{0}^{1} \left(\dfrac{1}{x}\right)^2 dx\)

\(\displaystyle\int_{0}^{3} \left(\dfrac{1}{x}\right)^2 dx\)

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

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

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Area & Circumference

Volume

Triangles

Other Facts

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Area & Circumference

Volume

Triangles

Other Facts

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