AB MCQ Set 110

Question 1 of 42 | Integration and Accumulation of Change · Level 1

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

\(-\dfrac{7}{8}\)

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

\(\dfrac{15}{64}\)

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

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

Question 2 of 42 | Differentiation: Definition and Fundamental Properties · Level 2

If \(f(x) = (2 x + 1)^4\), then the 4th derivative of \(f(x)\) at \(x = 0\) is

\(0\)

\(24\)

\(48\)

\(240\)

\(384\)

Question 3 of 42 | Integration and Accumulation of Change · Level 1

If \(\dfrac{d y}{d x} = \cos(2 x)\), then \(y =\)

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

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

\(\dfrac{1}{2} \sin(2 x) + C\)

\(\dfrac{1}{2} \sin^2(2 x) + C\)

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

Question 4 of 42 | Limits and Continuity · Level 2

\(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{4 n^2}{n^2 + 10000 n}\) is

\(0\)

\(\dfrac{1}{2500}\)

\(1\)

\(4\)

nonexistent

Question 5 of 42 | Differentiation: Definition and Fundamental Properties · Level 1

If \(f(x) = x\), then \(f'(5) =\)

\(0\)

\(\dfrac{1}{5}\)

\(1\)

\(5\)

\(\dfrac{25}{2}\)

Question 6 of 42 | Integration and Accumulation of Change · Level 2

Which of the following is equal to \(\ln 4\)?

\(\ln 3 + \ln 1\)

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

\(\displaystyle\int_{1}^{4} e^t d t\)

\(\displaystyle\int_{1}^{4} \ln x d x\)

\(\displaystyle\int_{1}^{4} \dfrac{1}{t} d t\)

Question 7 of 42 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

The slope of the line tangent to the graph of \(y = \ln\left(\dfrac{x}{2}\right)\) at \(x = 4\) is

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

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

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

\(1\)

\(4\)

Question 8 of 42 | Integration and Accumulation of Change · Level 2

If \(\displaystyle\int_{-1}^1 e^{-x^2} d x = k\), then \(\displaystyle\int_{-1}^0 e^{-x^2} d x =\)

\(-2 k\)

\(-k\)

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

\(\dfrac{k}{2}\)

\(2 k\)

Question 9 of 42 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If \(y = 10^{x^2 - 1}\), then \(\dfrac{d y}{d x} =\)

\((\ln 10) 10^{x^2 - 1}\)

\((2 x) 10^{x^2 - 1}\)

\((x^2 - 1) 10^{x^2 - 2}\)

\(2 x (\ln 10) 10^{x^2 - 1}\)

\(x^2 (\ln 10) 10^{x^2 - 1}\)

Question 10 of 42 | Contextual Applications of Differentiation · Level 1

The position of a particle moving along a straight line at any time \(t\) is given by \(s(t) = t^2 + 4 t + 4\). What is the acceleration of the particle when \(t = 4\)?

\(0\)

\(2\)

\(4\)

\(8\)

\(12\)

Question 11 of 42 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If \(f(g(x)) = \ln(x^2 + 4)\), \(f(x) = \ln(x^2)\), and \(g(x) > 0\) for all real \(x\), then \(g(x) =\)

\(\dfrac{1}{\sqrt{x^2 + 4}}\)

\(\dfrac{1}{x^2 + 4}\)

\(\sqrt{x^2 + 4}\)

\(x^2 + 4\)

\(x + 2\)

Question 12 of 42 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If \(x^2 + x y + y^3 = 0\), then \(\dfrac{d y}{d x} =\)

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

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

\(\dfrac{-2 x}{1 + 3 y^2}\)

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

\(-\dfrac{2 x + y}{x + 3 y^2 - 1}\)

Question 13 of 42 | Applications of Integration · Level 2

The velocity of a particle moving on a line at time \(t\) is \(v = 3 t^{\dfrac{1}{2}} + 5 t^{\dfrac{3}{2}}\) m/s. How many meters did it travel from \(t = 0\) to \(t = 4\)?

\(32\)

\(40\)

\(64\)

\(80\)

\(184\)

Question 14 of 42 | Limits and Continuity · Level 2

The domain of \(f(x) = \ln(x^2 - 4)\) is the set of all real \(x\) such that

\(|x| < 2\)

\(|x| \leq 2\)

\(|x| > 2\)

\(|x| \geq 2\)

\(x\) is a real number

Question 15 of 42 | Analytical Applications of Differentiation · Level 2

\(f(x) = x^3 - 3 x^2\) has a relative maximum at \(x =\)

\(-2\)

\(0\)

\(1\)

\(2\)

\(4\)

Question 16 of 42 | Integration and Accumulation of Change · Level 3

\(\displaystyle\int_{0}^{1} x e^{-x} d x =\)

\(1 - 2 e\)

\(-1\)

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

\(1\)

\(2 e - 1\)

Question 17 of 42 | MCQ · Level 2

If \(y = \cos^2 x - \sin^2 x\), then \(y' =\)

\(-1\)

\(0\)

\(-2 \sin(2 x)\)

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

\(2(\cos x - \sin x)\)

Question 18 of 42 | Limits and Continuity · Level 2

If \(f(x_1) + f(x_2) = f(x_1 + x_2)\) for all real \(x_1, x_2\), which could define \(f\)?

\(f(x) = x + 1\)

\(f(x) = 2 x\)

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

\(f(x) = e^x\)

\(f(x) = x^2\)

Question 19 of 42 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If \(y = \arctan(\cos x)\), then \(\dfrac{d y}{d x} =\)

\(\dfrac{-\sin x}{1 + \cos^2 x}\)

\(-(arcsec(\cos x))^2 \sin x\)

\((arcsec(\cos x))^2\)

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

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

Question 20 of 42 | Limits and Continuity · Level 3

If domain of \(f(x) = 1/(1-x^2)\) is \(\{x: |x| > 1\}\), what is the range of \(f\)?

\(\{x: -\infty < x < -1\}\)

\(\{x: -\infty < x < 0\}\)

\(\{x: -\infty < x < 1\}\)

\(\{x: -1 < x < \infty\}\)

\(\{x: 0 < x < \infty\}\)

Question 21 of 42 | Integration and Accumulation of Change · Level 2

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

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

\(1\)

\(2\)

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

\(\ln 3\)

Question 22 of 42 | Differentiation: Definition and Fundamental Properties · Level 2

\(\dfrac{d}{d x}\left(\dfrac{1}{x^3} - \dfrac{1}{x} + x^2\right)\) at \(x = -1\) is

\(-6\)

\(-4\)

\(0\)

\(2\)

\(6\)

Question 23 of 42 | Integration and Accumulation of Change · Level 2

If \(\displaystyle\int_{-2}^2 (x^7 + k) d x = 16\), then \(k =\)

\(-12\)

\(-4\)

\(0\)

\(4\)

\(12\)

Question 24 of 42 | Differentiation: Definition and Fundamental Properties · Level 2

If \(f(x) = e^x\), which is equal to \(f'(e)\)?

\(\operatorname*{lim}\limits_{h\rightarrow 0} \dfrac{e^{x+h}}{h}\)

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

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

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

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

Question 25 of 42 | Limits and Continuity · Level 2

The graph of \(y^2 = x^2 + 9\) is symmetric to which of the following?
I. x-axis
II. y-axis
III. origin

I only

II only

III only

I and II only

I, II, and III

Question 26 of 42 | Integration and Accumulation of Change · Level 2

\(\displaystyle\int_{0}^{3} |x - 1| d x =\)

\(0\)

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

\(2\)

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

\(6\)

Question 27 of 42 | Contextual Applications of Differentiation · Level 2

If position is \(-5 t^2\) on \([0, 3]\), average velocity is

\(-45\)

\(-30\)

\(-15\)

\(-10\)

\(-5\)

Question 28 of 42 | Limits and Continuity · Level 2

Which of the following are continuous for all real \(x\)?
I. \(y = x^{\dfrac{2}{3}}\)
II. \(y = e^x\)
III. \(y = \tan x\)

None

I only

II only

I and II

I and III

Question 29 of 42 | Integration and Accumulation of Change · Level 2

\(\int \tan(2 x) d x =\)

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

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

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

\(2 \ln|\cos(2 x)| + C\)

\(\dfrac{1}{2} \sec(2 x) \tan(2 x) + C\)

Question 30 of 42 | Contextual Applications of Differentiation · Level 3

Volume of cone \(V = \left(\dfrac{1}{3}\right) \pi r^2 h\). Both \(r\) and \(h\) increase at \(\dfrac{1}{2}\) cm/sec. When \(h = 9\), \(r = 6\), find \(\dfrac{d V}{d t}\).

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

\(10 \pi\)

\(24 \pi\)

\(54 \pi\)

\(108 \pi\)

Question 31 of 42 | Integration and Accumulation of Change · Level 2

\(\displaystyle\int_{0}^{\dfrac{\pi}{3}} \sin(3 x) d x =\)

\(-2\)

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

\(0\)

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

\(2\)

Question 32 of 42 | Applications of Integration · Level 3

The area in the first quadrant enclosed by \(y = x^3 + 8\) and \(y = x + 8\) is

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

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

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

\(1\)

\(\dfrac{65}{4}\)

Question 33 of 42 | Analytical Applications of Differentiation · Level 3

If max of \(f\) is 5, min is \(-7\), which must be true?
I. max of \(f(|x|)\) is 5
II. max of \(|f(x)|\) is 7
III. min of \(f(|x|)\) is 0

I only

II only

I and II only

II and III only

I, II, and III

Question 34 of 42 | Limits and Continuity · Level 2

\(\operatorname*{lim}\limits_{x \rightarrow 0} (x \csc x)\) is

\(-\infty\)

\(-1\)

\(0\)

\(1\)

\(\infty\)

Question 35 of 42 | Integration and Accumulation of Change · Level 3

If \(f, g\) have continuous first/second derivatives and \(f(x) \leq g(x)\) for all real \(x\), which must be true?
I. \(f' \leq g'\)
II. \(f'' \leq g''\)
III. \(\displaystyle\int_{0}^{1} f \leq \displaystyle\int_{0}^{1} g\)

None

I only

III only

I and II only

I, II, and III

Question 36 of 42 | Analytical Applications of Differentiation · Level 3

If \(f(x) = \ln \dfrac{x}{x}\) for \(x > 0\), which is true?

\(f\) increasing for all \(x > 0\).

\(f\) increasing for \(x > 1\).

\(f\) decreasing for \(0 < x < 1\).

\(f\) decreasing for \(1 < x < e\).

\(f\) decreasing for \(x > e\).

Question 37 of 42 | Integration and Accumulation of Change · Level 2

If \(2 \leq f(x) \leq 4\) on \([0, 2]\), max possible \(\displaystyle\int_{0}^{2} f(x) d x\) is

\(0\)

\(2\)

\(4\)

\(8\)

\(16\)

Question 38 of 42 | Limits and Continuity · Level 2

If \(\operatorname*{lim}\limits_{x\rightarrow a} f(x) = L\), which must be true?

\(f'(a)\) exists.

\(f\) is continuous at \(a\).

\(f(a)\) is defined.

\(f(a) = L\)

None of the above

Question 39 of 42 | Integration and Accumulation of Change · Level 2

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

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

\(\sqrt{1 + x^2} - \sqrt{5}\)

\(\sqrt{1 + x^2}\)

\(\dfrac{x}{\sqrt{1+x^2}} - \dfrac{1}{\sqrt{5}}\)

\(\dfrac{1}{2 \sqrt{1+x^2}} - \dfrac{1}{2 \sqrt{5}}\)

Question 40 of 42 | Analytical Applications of Differentiation · Level 3

Equation of tangent to \(y = x^3 + 3 x^2 + 2\) at its point of inflection

\(y = -6 x - 6\)

\(y = -3 x + 1\)

\(y = 2 x + 10\)

\(y = 3 x - 1\)

\(y = 4 x + 1\)

Question 41 of 42 | Applications of Integration · Level 3

Average value of \(f(x) = x^2 \sqrt{x^3 + 1}\) on \([0, 2]\) is

\(\dfrac{26}{9}\)

\(\dfrac{13}{3}\)

\(\dfrac{26}{3}\)

\(13\)

\(26\)

Question 42 of 42 | Applications of Integration · Level 3

Region enclosed by \(y = x^2\), line \(x = 2\), x-axis revolved about y-axis. Volume?

\(8 \pi\)

\(\dfrac{32 \pi}{5}\)

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

\(4 \pi\)

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

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