AB MCQ Set 60 (SparkNotes full)

Question 1 of 37 | MCQ · Level 2

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

\(-\dfrac{9}{25}\)

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

\(\dfrac{1}{25}\)

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

\(\dfrac{9}{25}\)

Question 2 of 37 | MCQ · Level 2

\(\operatorname*{lim}\limits_{x \rightarrow \dfrac{\pi}{3}} \dfrac{\cos x - \dfrac{1}{2}}{x - \dfrac{\pi}{3}}\) is

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

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

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

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

nonexistent

Question 3 of 37 | MCQ · Level 2

If \(f(x) = \sec(2 x)\), then \(f'(x) =\)

\(\tan^2(2 x)\)

\(2 \tan^2(2 x)\)

\(\tan(2 x) \sec(2 x)\)

\(2 \tan(2 x) \sec(2 x)\)

\(-2 \sin(2 x) \sec^2(2 x)\)

Question 4 of 37 | MCQ · Level 2

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

\(0\)

\(\dfrac{5}{6}\)

\(\dfrac{9}{10}\)

\(1\)

\(2\)

Question 5 of 37 | MCQ · Level 2

What is the slope of the tangent line to the graph of \(\dfrac{x y + 1}{y + 2} = 1\) at point \((2, 1)\)?

\(-1\)

\(-0.5\)

\(0.5\)

\(1\)

nonexistent

Question 6 of 37 | MCQ · Level 2

What is the x-coordinate of the point of inflection for the graph of \(y = x^3 + 3 x^2 - 1\)?

\(-2\)

\(-1\)

\(0\)

\(1\)

\(2\)

Question 7 of 37 | MCQ · Level 3

\(\displaystyle\int_{0}^{x} \sin t \cos^2 t d t =\)

\(-\cos^3 x\)

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

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

\(\cos^3 x\)

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

Question 8 of 37 | MCQ · Level 2

A particle moves along the x-axis so that its position at time \(t > 0\) is given by \(x(t) = 3 \ln t - 6 t + 7\). At what time \(t > 0\) is the velocity of the particle equal to zero?

\(\ln 0.5\)

\(0.5\)

\(\ln 2\)

\(2\)

\(3\)

Question 9 of 37 | MCQ · Level 3

If \(F(x) = \displaystyle\int_{1}^{\ln x} \sqrt{\cos t} d t\), then \(F'(x) =\)

\(\sqrt{\cos x}\)

\(\sqrt{\cos\left(\dfrac{1}{x}\right)}\)

\(\sqrt{\cos(\ln x)}\)

\(\dfrac{\sqrt{\cos(\ln x)}}{x}\)

\(\dfrac{\sin(\ln x)}{2 \sqrt{\cos(\ln x)}}\)

Question 10 of 37 | MCQ · Level 3

The function \(f\) is differentiable on the closed interval \([1, 4]\), and it has values as follows: \(f(1) = 3\), \(f(2) = k\), and \(f(4) = 5 k + 2\). For which of the following values of \(k\) must there exist two points \(a\) and \(b\) on the open interval \((1, 4)\) with \(f'(a) = f'(b)\)?

\(-4\)

\(-\dfrac{5}{3}\)

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

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

\(3\)

Question 11 of 37 | MCQ · Level 3

Which of the following is a solution to the differential equation \(\dfrac{d y}{d x} = \dfrac{\sec^2 x}{y^2}\)?

\(y = \sec x\)

\(y = \sqrt[3]{3 \tan x} + 1\)

\(y = \sqrt[3]{3 \tan x + 1}\)

\(y = 3 \sqrt[3]{\tan x} + 1\)

\(y = 3 \sqrt[3]{\tan x + 1}\)

Question 12 of 37 | MCQ · Level 3

If \(g(x) = f(x^2)\), what is \(g''(x)\)?

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

\(4 x f''(x^2)\)

\((4 x^2 + 2 x) f''(x^2)\)

\(4 x^2 f''(x^2) + 2 f'(x^2)\)

\(4 x^2 f''(x^2) + 2 x f'(x^2)\)

Question 13 of 37 | MCQ · Level 2

What's the instantaneous rate of change of \(y = \ln(\cos x)\) at the point \(x = \dfrac{\pi}{6}\)?

\(-\sqrt{3}\)

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

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

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

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

Question 14 of 37 | MCQ · Level 3

For what values of \(x\) is the function \(f(x) = 1 + x^2 - 2 x^4\) increasing?

\(-1 < x < 1\)

\(x < -\dfrac{1}{2}\) and \(0 < x < \dfrac{1}{2}\)

\(-\dfrac{1}{2} < x < 0\) and \(x > \dfrac{1}{2}\)

\(x < -1\) and \(x > 1\)

\(x > 0\)

Question 15 of 37 | MCQ · Level 2

Which of the following is an equation of the tangent line to the graph of \(y = x^4 - x^3 - x^2 + x + 1\) at the point \((1, 1)\)?

\(y = 1\)

\(y = x\)

\(y = -2 x + 3\)

\(y = 2 x - 1\)

\(y = -x + 2\)

Question 16 of 37 | MCQ · Level 2

The function \(f\) is continuous on the closed interval \([0, 10]\) with values \(f(0)=1\), \(f(1)=-1\), \(f(3)=4\), \(f(7)=2\), \(f(10)=3\). Using the subintervals \([0,1]\), \([1,3]\), \([3,7]\), and \([7,10]\), what is the left Riemann sum estimate for \(\displaystyle\int_{0}^{10} f(x) d x\)?

\(15\)

\(17.5\)

\(20\)

\(21\)

\(22.5\)

Question 17 of 37 | MCQ · Level 3

The function \(f\) is given by \(f(x) = \begin{cases} \ln 2 x & \text{if } 0 < x < 2 \\ 2 \ln x & \text{if } x \geq 2 \end{cases}\). The limit \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x)\) is

\(0\)

\(0.5\)

\(1\)

\(2 \ln 2\)

nonexistent

Question 18 of 37 | MCQ · Level 4

The differentiable function \(f(x)\) achieves its maximum when \(x = 0\). Which of the following statements must be true? I. The function \(g(x) = x f(x)\) has a critical point when \(x = 0\). II. The function \(h(x) = (f(x))^2\) achieves its maximum at \(x = 0\). III. The function \(k(x) = f(x^2)\) achieves its maximum at \(x = 0\).

None

III only

I and II only

I and III only

II and III only

Question 19 of 37 | MCQ · Level 2

\(\displaystyle\int_{1}^{8} \dfrac{d x}{\sqrt[3]{x}} =\)

\(-\dfrac{63}{128}\)

\(\dfrac{63}{128}\)

\(1\)

\(3\)

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

Question 20 of 37 | MCQ · Level 2

If \(x + y = 2\), what is the minimum value of \(x^2 + y^2\)?

\(0\)

\(1\)

\(2\)

\(4\)

\(8\)

Question 21 of 37 | MCQ · Level 3

A particle moves along the graph of \(y = x + \sin x\). As it passes the point \((2 \pi, 2 \pi)\), the particle's y-coordinate is increasing at a rate of \(2\) units per second. How fast is the x-coordinate of the particle changing at this point (in units per second)?

\(0\)

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

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

\(1\)

\(2\)

Question 22 of 37 | MCQ · Level 3

A particle moves along the x-axis. Its position at time \(t\) is given by \(x(t) = \dfrac{t^4}{24} - \dfrac{t^3}{2} + 2 t^2 - 1\). What is the maximum acceleration of the particle on the interval \(0 \leq t \leq 4\)?

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

\(3\)

\(\dfrac{10}{3}\)

\(\dfrac{11}{3}\)

\(4\)

Question 23 of 37 | MCQ · Level 4

The function \(f\) is given by \(f(x) = \displaystyle\int_{0}^{x} (t^2 - 3) e^t d t\). For which value of \(x\) does \(f\) have a relative minimum?

\(-3\)

\(-\sqrt{3}\)

\(0\)

\(1\)

\(\sqrt{3}\)

Question 24 of 37 | MCQ · Level 2

[Calculator] What is the average value of the function \(f(x) = \sin \sqrt{x}\) on the interval \([1, 3]\)?

\(0.146\)

\(0.914\)

\(0.964\)

\(0.987\)

\(1.928\)

Question 25 of 37 | MCQ · Level 3

[Calculator] Right triangle \(A B C\) has its right angle at \(A\). Leg \(A B\) is decreasing at a constant rate of \(3\) cm per minute. Leg \(A C\) is increasing at a constant rate of \(4\) cm per minute. How is the hypotenuse \(B C\) changing at the moment when \(A B = 12\) and \(A C = 5\)?

decreasing at \(16\) cm/min

decreasing at \(\dfrac{16}{13}\) cm/min

not changing

increasing at \(\dfrac{33}{13}\) cm/min

increasing at \(16\) cm/min

Question 26 of 37 | MCQ · Level 3

[Calculator] If \(g'(x) = x(\ln x)^2\) and \(g(2) = 3\), what is \(g(3)\)?

\(2.163\)

\(2.660\)

\(2.780\)

\(5.163\)

\(5.660\)

Question 27 of 37 | MCQ · Level 2

\(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{x^3 - 111 x^2 + 3 x - 2}{1 - 2 x + 22 x^2 + 3 x^3} =\)

\(-2\)

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

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

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

\(1\)

Question 28 of 37 | MCQ · Level 3

[Calculator] The derivative of a function \(f\) is given by \(f'(x) = \sin(\cos x) - 0.1 x\). How many critical points does \(f\) have on the open interval \((0, 8)\)?

None

One

Two

Three

Four

Question 29 of 37 | MCQ · Level 4

The function \(f\) is given by \(f(x) = \begin{cases} \dfrac{x^3}{|x|} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}\). Which of the following statements are true? I. \(f\) is continuous at the point \(x = 0\). II. \(f\) is differentiable at the point \(x = 0\). III. \(x = 0\) is a point of inflection for a graph of \(f\).

I only

III only

I and II only

I and III only

I, II, and III

Question 30 of 37 | MCQ · Level 3

[Calculator] What is the area of the region bounded by the graphs of \(y = e^{x^2}\) and \(y = \tan x\) and the vertical lines \(x = -\dfrac{1}{2}\) and \(x = 1\)?

\(1.523\)

\(2.358\)

\(2.493\)

\(4.783\)

\(7.409\)

Question 31 of 37 | MCQ · Level 3

[Calculator] What is the x-value of the point at which the tangent line to the graph of \(y = e^{x^2} + x\) is perpendicular to the line \(3 y = 2 x + 1\)?

\(-0.732\)

\(-0.589\)

\(-0.162\)

\(0.236\)

\(0.361\)

Question 32 of 37 | MCQ · Level 3

The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the graph of \(y = 1 - x^3\). If cross-sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

\(0.252\)

\(0.505\)

\(1.010\)

\(2.020\)

\(2.356\)

Question 33 of 37 | MCQ · Level 4

The region enclosed by the graphs of \(y = e^{x - 1}\) and \(y = -x\) and the vertical lines \(x = 0\) and \(x = 2\) is rotated about the line \(y = -3\). Which of the following gives the volume of the generated solid?

\(\pi \displaystyle\int_{0}^{2} [(e^{x-1} - 3)^2 - (-x - 3)^2] d x\)

\(\pi \displaystyle\int_{0}^{2} [(e^{x-1} + 3)^2 - (-x + 3)^2] d x\)

\(\pi \displaystyle\int_{0}^{2} [(e^{x-1})^2 - (-x)^2 - 3^2] d x\)

\(\pi \displaystyle\int_{-2}^e [(\ln y - 2)^2 - (-y - 3)^2] d y\)

\(\pi \displaystyle\int_{-2}^e [(\ln y + 4)^2 - (-y + 3)^2] d y\)

Question 34 of 37 | MCQ · Level 3

[Calculator] Which of the following is an equation of the tangent line to the graph of the function \(f(x) = e^x + x^2\) at the point where \(f'(x) = 2\)?

\(y = 2 x - 0.630\)

\(y = 2 x + 0.537\)

\(y = 2 x + 0.839\)

\(y = 2 x + 0.926\)

\(y = 2 x + 1.469\)

Question 35 of 37 | MCQ · Level 3

The function \(f\) is twice differentiable on the closed interval \([0, 2]\) with \(f(0) = 1\), \(f(1) = 3\), \(f(2) = 1\). Which of the following statements is FALSE?

The equation \(f(x) = 2\) has at least two solutions on \((0, 2)\).

The equation \(f'(x) = 0\) has a solution on \((0, 2)\).

There exists a point \(c\) on \((0, 2)\) such that \(f'(c) = 2\).

\(f''(x) > 0\) for all \(x\) on the open interval \((0, 2)\).

\(f\) has a relative maximum on the open interval \((0, 2)\).

Question 36 of 37 | MCQ · Level 2

The function \(f\) is continuous on \([0, 3]\) with \(f(0) = 2\), \(f(1) = 5\), \(f(2) = 4\), \(f(3) = 3\). Using the subintervals \([0, 1]\), \([1, 2]\), \([2, 3]\), what is the trapezoidal approximation to \(\displaystyle\int_{0}^{3} f(x) d x\)?

\(11\)

\(11.5\)

\(12\)

\(12.5\)

\(13\)

Question 37 of 37 | MCQ · Level 3

[Calculator] A population of bacteria given by \(y(t)\) grows according to the equation \(\dfrac{d y}{d t} = k y\), where \(k\) is a constant and \(t\) is measured in minutes. If \(y(10) = 10\) and \(y(30) = 25\), what is the value of \(k\)?

\(-2.079\)

\(0.046\)

\(0.107\)

\(0.125\)

\(0.230\)

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