AB MCQ Set 90 (1969 Official AP)

Question 1 of 44 | MCQ · Level 1

Which of the following defines a function \(f\) for which \(f(-x) = -f(x)\)?

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

\(f(x) = \sin x\)

\(f(x) = \cos x\)

\(f(x) = \log x\)

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

Question 2 of 44 | MCQ · Level 2

\(\ln(x - 2) < 0\) if and only if

\(x < 3\)

\(0 < x < 3\)

\(2 < x < 3\)

\(x > 2\)

\(x > 3\)

Question 3 of 44 | MCQ · Level 3

If \(\begin{cases} f(x) = \dfrac{\sqrt{2 x + 5} - \sqrt{x + 7}}{x - 2} \text{for} x \neq 2 \\ f(2) = k \end{cases}\) and if \(f\) is continuous at \(x = 2\), then \(k = \)

\(0\)

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

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

\(1\)

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

Question 4 of 44 | MCQ · Level 2

\(\displaystyle\int_{0}^{8} \dfrac{d x}{\sqrt{1 + x}} =\)

\(1\)

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

\(2\)

\(4\)

\(6\)

Question 5 of 44 | MCQ · Level 2

If \(3 x^2 + 2 x y + y^2 = 2\), then the value of \(\dfrac{d y}{d x}\) at \(x = 1\) is

\(-2\)

\(0\)

\(2\)

\(4\)

not defined

Question 6 of 44 | MCQ · Level 2

What is \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{8 \left(\dfrac{1}{2} + h\right)^8 - 8 \left(\dfrac{1}{2}\right)^8}{h}\)?

\(0\)

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

\(1\)

The limit does not exist.

It cannot be determined from the information given.

Question 7 of 44 | MCQ · Level 3

For what value of \(k\) will \(x + \dfrac{k}{x}\) have a relative maximum at \(x = -2\)?

\(-4\)

\(-2\)

\(2\)

\(4\)

None of these

Question 8 of 44 | MCQ · Level 2

If \(p(x) = (x + 2)(x + k)\) and if the remainder is \(12\) when \(p(x)\) is divided by \(x - 1\), then \(k = \)

\(2\)

\(3\)

\(6\)

\(11\)

\(13\)

Question 9 of 44 | MCQ · Level 3

When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is

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

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

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

\(1\)

\(\pi\)

Question 10 of 44 | MCQ · Level 2

The set of all points \((e^t, t)\), where \(t\) is a real number, is the graph of \(y = \)

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

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

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

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

\(\ln x\)

Question 11 of 44 | MCQ · Level 3

The point on the curve \(x^2 + 2 y = 0\) that is nearest the point \(\left(0, -\dfrac{1}{2}\right)\) occurs where \(y\) is

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

\(0\)

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

\(-1\)

none of the above

Question 12 of 44 | MCQ · Level 3

If \(f(x) = \dfrac{4}{x - 1}\) and \(g(x) = 2 x\), then the solution set of \(f(g(x)) = g(f(x))\) is

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

\(\{2\}\)

\(\{3\}\)

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

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

Question 13 of 44 | MCQ · Level 4

The region bounded by the x-axis and the part of the graph of \(y = \cos x\) between \(x = -\dfrac{\pi}{2}\) and \(x = \dfrac{\pi}{2}\) is separated into two regions by the line \(x = k\). If the area of the region for \(-\dfrac{\pi}{2} \leq x \leq k\) is three times the area of the region for \(k \leq x \leq \dfrac{\pi}{2}\), then \(k = \)

\(\arcsin\left(\dfrac{1}{4}\right)\)

\(\arcsin\left(\dfrac{1}{3}\right)\)

\(\dfrac{\pi}{6}\)

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

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

Question 14 of 44 | MCQ · Level 2

If the function \(f\) is defined by \(f(x) = x^5 - 1\), then \(f^{-1}\), the inverse function of \(f\), is defined by \(f^{-1}(x) = \)

\(\dfrac{1}{\sqrt[5]{x} + 1}\)

\(\dfrac{1}{\sqrt[5]{x + 1}}\)

\(\sqrt[5]{x - 1}\)

\(\sqrt[5]{x} - 1\)

\(\sqrt[5]{x + 1}\)

Question 15 of 44 | MCQ · Level 3

If \(f'(x)\) and \(g'(x)\) exist and \(f'(x) > g'(x)\) for all real \(x\), then the graph of \(y = f(x)\) and the graph of \(y = g(x)\)

intersect exactly once.

intersect no more than once.

do not intersect.

could intersect more than once.

have a common tangent at each point of intersection.

Question 16 of 44 | MCQ · Level 3

The graph of \(y = 5 x^4 - x^5\) has a point of inflection at

\((0, 0)\) only

\((3, 162)\) only

\((4, 256)\) only

\((0, 0)\) and \((3, 162)\)

\((0, 0)\) and \((4, 256)\)

Question 17 of 44 | MCQ · Level 2

If \(f(x) = 2 + |x - 3|\) for all \(x\), then the value of the derivative \(f'(x)\) at \(x = 3\) is

\(-1\)

\(0\)

\(1\)

\(2\)

nonexistent

Question 18 of 44 | MCQ · Level 3

A point moves on the x-axis in such a way that its velocity at time \(t\) \((t > 0)\) is given by \(v = \dfrac{\ln t}{t}\). At what value of \(t\) does \(v\) attain its maximum?

\(1\)

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

\(e\)

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

There is no maximum value for \(v\).

Question 19 of 44 | MCQ · Level 2

An equation for a tangent to the graph of \(y = \arcsin \dfrac{x}{2}\) at the origin is

\(x - 2 y = 0\)

\(x - y = 0\)

\(x = 0\)

\(y = 0\)

\(\pi x - 2 y = 0\)

Question 20 of 44 | MCQ · Level 2

At \(x = 0\), which of the following is true of the function \(f\) defined by \(f(x) = x^2 + e^{-2 x}\)?

\(f\) is increasing.

\(f\) is decreasing.

\(f\) is discontinuous.

\(f\) has a relative minimum.

\(f\) has a relative maximum.

Question 21 of 44 | MCQ · Level 1

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

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

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

\(2 x\)

\(1\)

\(2\)

Question 22 of 44 | MCQ · Level 2

The area of the region bounded by the curve \(y = e^{2 x}\), the x-axis, the y-axis, and the line \(x = 2\) is equal to

\(\dfrac{e^4}{2} - e\)

\(\dfrac{e^4}{2} - 1\)

\(\dfrac{e^4}{2} - \dfrac{1}{2}\)

\(2 e^4 - e\)

\(2 e^4 - 2\)

Question 23 of 44 | MCQ · Level 2

If \(\sin x = e^y\), \(0 < x < \pi\), what is \(\dfrac{d y}{d x}\) in terms of \(x\)?

\(-\tan x\)

\(-\cot x\)

\(\cot x\)

\(\tan x\)

\(\csc x\)

Question 24 of 44 | MCQ · Level 3

A region in the plane is bounded by the graph of \(y = \dfrac{1}{x}\), the x-axis, the line \(x = m\), and the line \(x = 2 m\), \(m > 0\). The area of this region

is independent of \(m\).

increases as \(m\) increases.

decreases as \(m\) increases.

decreases as \(m\) increases when \(m < \dfrac{1}{2}\); increases as \(m\) increases when \(m > \dfrac{1}{2}\).

increases as \(m\) increases when \(m < \dfrac{1}{2}\); decreases as \(m\) increases when \(m > \dfrac{1}{2}\).

Question 25 of 44 | MCQ · Level 2

\(\displaystyle\int_{0}^{1} \sqrt{x^2 - 2 x + 1} d x\) is

\(-1\)

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

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

\(1\)

none of the above

Question 26 of 44 | MCQ · Level 2

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

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

\(\sec^2 x + C\)

\(\ln|\sec x| + C\)

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

\(\sec x \tan x + C\)

Question 27 of 44 | MCQ · Level 2

The function defined by \(f(x) = \sqrt{3} \cos x + 3 \sin x\) has an amplitude of

\(3 - \sqrt{3}\)

\(\sqrt{3}\)

\(2 \sqrt{3}\)

\(3 + \sqrt{3}\)

\(3 \sqrt{3}\)

Question 28 of 44 | MCQ · Level 3

\(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \dfrac{\cos x}{\sin x} d x =\)

\(\ln \sqrt{2}\)

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

\(\ln \sqrt{3}\)

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

\(\ln e\)

Question 29 of 44 | MCQ · Level 3

If a function \(f\) is continuous for all \(x\) and if \(f\) has a relative maximum at \((-1, 4)\) and a relative minimum at \((3, -2)\), which of the following statements must be true?

The graph of \(f\) has a point of inflection somewhere between \(x = -1\) and \(x = 3\).

\(f'(-1) = 0\)

The graph of \(f\) has a horizontal asymptote.

The graph of \(f\) has a horizontal tangent line at \(x = 3\).

The graph of \(f\) intersects both axes.

Question 30 of 44 | MCQ · Level 3

If \(f'(x) = -f(x)\) and \(f(1) = 1\), then \(f(x) = \)

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

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

\(e^{1 - x}\)

\(e^{-x}\)

\(-e^x\)

Question 31 of 44 | MCQ · Level 2

If \(a, b, c, d\), and \(e\) are real numbers and \(a \neq 0\), then the polynomial equation \(a x^7 + b x^5 + c x^3 + d x + e = 0\) has

only one real root.

at least one real root.

an odd number of nonreal roots.

no real roots.

no positive real roots.

Question 32 of 44 | MCQ · Level 2

What is the average (mean) value of \(3 t^3 - t^2\) over the interval \(-1 \leq t \leq 2\)?

\(\dfrac{11}{4}\)

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

\(8\)

\(\dfrac{33}{4}\)

\(16\)

Question 33 of 44 | MCQ · Level 3

Which of the following is an equation of a curve that intersects at right angles every curve of the family \(y = \dfrac{1}{x} + k\) (where \(k\) takes all real values)?

\(y = -x\)

\(y = -x^2\)

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

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

\(y = \ln x\)

Question 34 of 44 | MCQ · Level 2

At \(t = 0\) a particle starts at rest and moves along a line in such a way that at time \(t\) its acceleration is \(24 t^2\) feet per second per second. Through how many feet does the particle move during the first 2 seconds?

\(32\)

\(48\)

\(64\)

\(96\)

\(192\)

Question 35 of 44 | MCQ · Level 3

The approximate value of \(y = \sqrt{4 + \sin x}\) at \(x = 0.12\), obtained from the tangent to the graph at \(x = 0\), is

\(2.00\)

\(2.03\)

\(2.06\)

\(2.12\)

\(2.24\)

Question 36 of 44 | MCQ · Level 2

Which is the best of the following polynomial approximations to \(\cos 2 x\) near \(x = 0\)?

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

\(1 + x\)

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

\(1 - 2 x^2\)

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

Question 37 of 44 | MCQ · Level 3

\(\int \dfrac{x^2}{e^{x^3}} d x =\)

\(-\dfrac{1}{3} \ln e^{x^3} + C\)

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

\(-\dfrac{1}{3 e^{x^3}} + C\)

\(\dfrac{1}{3} \ln e^{x^3} + C\)

\(\dfrac{x^3}{3 e^{x^3}} + C\)

Question 38 of 44 | MCQ · Level 3

If \(y = \tan u\), \(u = v - \dfrac{1}{v}\), and \(v = \ln x\), what is the value of \(\dfrac{d y}{d x}\) at \(x = e\)?

\(0\)

\(\dfrac{1}{e}\)

\(1\)

\(\dfrac{2}{e}\)

\(\sec^2 e\)

Question 39 of 44 | MCQ · Level 2

If \(n\) is a non-negative integer, then \(\displaystyle\int_{0}^{1} x^n d x = \displaystyle\int_{0}^{1} (1 - x)^n d x\) for

no \(n\)

\(n\) even, only

\(n\) odd, only

nonzero \(n\), only

all \(n\)

Question 40 of 44 | MCQ · Level 3

If \(\begin{cases} f(x) = 8 - x^2 \text{for} -2 \leq x \leq 2 \\ f(x) = x^2 \text{elsewhere} \end{cases}\), then \(\displaystyle\int_{-1}^3 f(x) d x\) is a number between

\(0\) and \(8\)

\(8\) and \(16\)

\(16\) and \(24\)

\(24\) and \(32\)

\(32\) and \(40\)

Question 41 of 44 | MCQ · Level 3

What are all values of \(k\) for which the graph of \(y = x^3 - 3 x^2 + k\) will have three distinct x-intercepts?

All \(k > 0\)

All \(k < 4\)

\(k = 0, 4\)

\(0 < k < 4\)

All \(k\)

Question 42 of 44 | MCQ · Level 1

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

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

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

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

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

\(-\dfrac{1}{5} \cos(2 x + 3) + C\)

Question 43 of 44 | MCQ · Level 2

The fundamental period of the function defined by \(f(x) = 3 - 2 \cos^2 \dfrac{\pi x}{3}\) is

\(1\)

\(2\)

\(3\)

\(5\)

\(6\)

Question 44 of 44 | MCQ · Level 4

If \(\dfrac{d}{d x}(f(x)) = g(x)\) and \(\dfrac{d}{d x}(g(x)) = f(x^2)\), then \(\dfrac{d^2}{d x^2}(f(x^3)) =\)

\(f(x^6)\)

\(g(x^3)\)

\(3 x^2 g(x^3)\)

\(9 x^4 f(x^6) + 6 x g(x^3)\)

\(f(x^6) + g(x^3)\)

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