BC MCQ Set 60

Question 1 of 44 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

The asymptotes of the graph of the parametric equations \(x = \dfrac{1}{t}\), \(y = \dfrac{t}{t + 1}\) are

\(x = 0\), \(y = 0\)

\(x = 0\) only

\(x = -1\), \(y = 0\)

\(x = -1\) only

\(x = 0\), \(y = 1\)

Question 2 of 44 | Analytical Applications of Differentiation · Level 3

What are the coordinates of the inflection point on the graph of \(y = (x + 1) \arctan x\)?

\((-1, 0)\)

\((0, 0)\)

\((0, 1)\)

\(\left(1, \dfrac{\pi}{4}\right)\)

\(\left(1, \dfrac{\pi}{2}\right)\)

Question 3 of 44 | Analytical Applications of Differentiation · Level 3

The Mean Value Theorem guarantees the existence of a special point on the graph of \(y = \sqrt{x}\) between \((0, 0)\) and \((4, 2)\). What are the coordinates of this point?

\((2, 1)\)

\((1, 1)\)

\((2, \sqrt{2})\)

\(\left(\dfrac{1}{2}, \dfrac{1}{\sqrt{2}}\right)\)

None of the above

Question 4 of 44 | Integration and Accumulation of Change · Level 2

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

\(1\)

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

\(2\)

\(4\)

\(6\)

Question 5 of 44 | Differentiation: Composite, Implicit, and Inverse Functions · 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 | Differentiation: Definition and Fundamental Properties · 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 | Analytical Applications of Differentiation · 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 | Differentiation: Composite, Implicit, and Inverse Functions · Level 3

If \(h(x) = f^2(x) - g^2(x)\), \(f'(x) = -g(x)\), and \(g'(x) = f(x)\), then \(h'(x) =\)

\(0\)

\(1\)

\(-4 f(x) g(x)\)

\((-g(x))^2 - (f(x))^2\)

\(-2(-g(x) + f(x))\)

Question 9 of 44 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3

The area of the closed region bounded by the polar graph of \(r = \sqrt{3 + \cos \theta}\) is given by the integral

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

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

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

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

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

Question 10 of 44 | Integration and Accumulation of Change · Level 3

\(\displaystyle\int_{0}^{1} \dfrac{x^2}{x^2 + 1} d x =\)

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

\(\ln 2\)

\(0\)

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

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

Question 11 of 44 | Analytical Applications of Differentiation · 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 | Integration and Accumulation of Change · Level 3

If \(F(x) = \displaystyle\int_{0}^{x} e^{-t^2} d t\), then \(F'(x) =\)

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

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

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

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

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

Question 13 of 44 | Applications of Integration · 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 | Differentiation: Composite, Implicit, and Inverse Functions · Level 2

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

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

\(6 x^2 - 2 x + 4\)

\(x^2\)

\(x\)

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

Question 15 of 44 | Analytical Applications of Differentiation · 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 | Analytical Applications of Differentiation · 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 | Differentiation: Definition and Fundamental Properties · 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 | Contextual Applications of Differentiation · 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 | Differentiation: Composite, Implicit, and Inverse Functions · 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 | Analytical Applications of Differentiation · 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 | Integration and Accumulation of Change · Level 3

If \(f(x) = \displaystyle\int_{0}^{x} \dfrac{1}{\sqrt{t^3 + 2}} d t\), which of the following is FALSE?

\(f(0) = 0\)

\(f\) is continuous at \(x\) for all \(x \geq 0\).

\(f(1) > 0\)

\(f'(1) = \dfrac{1}{\sqrt{3}}\)

\(f(-1) > 0\)

Question 22 of 44 | Differential Equations · Level 4

If the graph of \(y = f(x)\) contains the point \((0, 2)\), \(\dfrac{d y}{d x} = \dfrac{-x}{y e^{x^2}}\), and \(f(x) > 0\) for all \(x\), then \(f(x) =\)

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

\(\sqrt{3} + e^{-x}\)

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

\(\sqrt{3 + e^{-x^2}}\)

\(\sqrt{3 + e^{x^2}}\)

Question 23 of 44 | Differentiation: Composite, Implicit, and Inverse Functions · 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 | Applications of Integration · 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 | Integration and Accumulation of Change · 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 | Integration and Accumulation of Change · 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 | Limits and Continuity · Level 3

What is \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{e^{2 x} - 1}{\tan x}\)?

\(-1\)

\(0\)

\(1\)

\(2\)

The limit does not exist.

Question 28 of 44 | Integration and Accumulation of Change · Level 4

\(\displaystyle\int_{0}^{1} (4 - x^2)^{-\dfrac{3}{2}} d x =\)

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

\(\dfrac{2 \sqrt{3} - 3}{4}\)

\(\dfrac{\sqrt{3}}{12}\)

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

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

Question 29 of 44 | Infinite Sequences and Series · Level 3

\(\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n x^n}{n!}\) is the Taylor series about zero for which of the following functions?

\(\sin x\)

\(\cos x\)

\(e^x\)

\(e^{-x}\)

\(\ln(1 + x)\)

Question 30 of 44 | Differential Equations · 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 | Infinite Sequences and Series · Level 3

For what values of \(x\) does the series \(1 + 2^x + 3^x + 4^x + ... + n^x + ...\) converge?

No values of \(x\)

\(x < -1\)

\(x \geq -1\)

\(x > -1\)

All values of \(x\)

Question 32 of 44 | Applications of Integration · 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 | Differential Equations · 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 | Applications of Integration · 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 | Contextual Applications of Differentiation · 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 | Limits and Continuity · Level 4

Of the following choices of \(\delta\), which is the largest that could be used successfully with an arbitrary \(\epsilon\) in an epsilon-delta proof of \(\operatorname*{lim}\limits_{x \rightarrow 2}(1 - 3 x) = -5\)?

\(\delta = 3 \epsilon\)

\(\delta = \epsilon\)

\(\delta = \dfrac{\epsilon}{2}\)

\(\delta = \dfrac{\epsilon}{4}\)

\(\delta = \dfrac{\epsilon}{5}\)

Question 37 of 44 | Differentiation: Composite, Implicit, and Inverse Functions · Level 4

If \(f(x) = (x^2 + 1)^{2 - 3 x}\), then \(f'(1) =\)

\(-\dfrac{1}{2} \ln(8 e)\)

\(-\ln(8 e)\)

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

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

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

Question 38 of 44 | Differentiation: Composite, Implicit, and Inverse Functions · 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 | Integration and Accumulation of Change · 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 | Integration and Accumulation of Change · 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 | Integration and Accumulation of Change · Level 3

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

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

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

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

\(4 \cos x - 2 x \sin x + C\)

\((2 - x^2) \cos x - 4 \sin x + C\)

Question 42 of 44 | Applications of Integration · Level 3

Which of the following integrals gives the length of the graph of \(y = \tan x\) between \(x = a\) and \(x = b\), where \(0 < a < b < \dfrac{\pi}{2}\)?

\(\displaystyle\int_{a}^{b} \sqrt{x^2 + \tan^2 x} d x\)

\(\displaystyle\int_{a}^{b} \sqrt{x + \tan x} d x\)

\(\displaystyle\int_{a}^{b} \sqrt{1 + \sec^2 x} d x\)

\(\displaystyle\int_{a}^{b} \sqrt{1 + \tan^2 x} d x\)

\(\displaystyle\int_{a}^{b} \sqrt{1 + \sec^4 x} d x\)

Question 43 of 44 | Differential Equations · Level 4

If \(f''(x) - f'(x) - 2 f(x) = 0\), \(f'(0) = -2\), and \(f(0) = 2\), then \(f(1) =\)

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

\(1\)

\(0\)

\(e^2\)

\(2 e^{-1}\)

Question 44 of 44 | Infinite Sequences and Series · Level 4

The complete interval of convergence of the series \(\displaystyle\sum_{k=1}^{\infty} \dfrac{(x + 1)^k}{k^2}\) is

\(0 < x < 2\)

\(0 \leq x \leq 2\)

\(-2 < x \leq 0\)

\(-2 \leq x < 0\)

\(-2 \leq x \leq 0\)

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