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AB MCQ Set 30
22 Questions
Question 1 of 22
AB MCQ Set 30 0/22
Question 1 of 22   |  Limits and Continuity  · Level 2
\(\operatorname*{lim}\limits_{x \rightarrow 3^-} \dfrac{|x - 3|}{3 - x} =\)
A
\(-\infty\)
✕
B
\(-1\)
✕
C
\(0\)
✕
D
\(1\)
✕
E
\(\infty\)
✕
Question 2 of 22   |  Differentiation: Definition and Fundamental Properties  · Level 2
If \(f(2) = 7\) and \(f'(2) = -3\), then the equation of the tangent to the curve \(y = f(x)\) at \(x = 2\) is
A
\(y = -3 x + 13\)
✕
B
\(y = -3 x + 23\)
✕
C
\(y = x\)
✕
D
\(y = 2 x - 17\)
✕
E
\(y = 7 x - 17\)
✕
Question 3 of 22   |  Analytical Applications of Differentiation  · Level 2
On the interval \(1 < x < 2\), the curve \(y = x^3 - 6 x^2 + 9 x + 1\) is
A
increasing and concave up
✕
B
increasing and concave down
✕
C
decreasing and concave up
✕
D
decreasing and concave down
✕
E
horizontal
✕
Question 4 of 22   |  Analytical Applications of Differentiation  · Level 3
The minimum value of the function \(f(x) = \sqrt[3]{x^2 + 4 a x + 12 a^2}\), \(a > 0\), is
A
\(-2 a\)
✕
B
\(\sqrt[3]{6 a^2}\)
✕
C
\(2 \sqrt[3]{a^2}\)
✕
D
\(2 a\)
✕
E
none of the above
✕
Question 5 of 22   |  Limits and Continuity  · Level 2
\(\operatorname*{lim}\limits_{x \rightarrow -\infty} \dfrac{10 - 2^x}{10 + 2^{-x}} =\)
A
\(-1\)
✕
B
\(0\)
✕
C
\(1\)
✕
D
\(10\)
✕
E
\(\infty\)
✕
Question 6 of 22   |  Differentiation: Definition and Fundamental Properties  · Level 3
The x-coordinate of the point where the tangent to the parabola \(y = a x^2\) at \(x = p\) (not a vertex) intersects the x-axis is
A
\(\dfrac{p}{2}\)
✕
B
\(\dfrac{p^2}{2}\)
✕
C
\(\dfrac{a p}{2}\)
✕
D
\(\dfrac{a p^2}{2}\)
✕
E
\(\dfrac{a}{p^2}\)
✕
Question 7 of 22   |  Differentiation: Definition and Fundamental Properties  · Level 2
The table below shows some of the values of two differentiable functions \(f\) and \(g\) and their derivatives. If \(h(x) = f(x) g(x)\), then \(h'(5) = \)
\(x\) \(f(x)\) \(g(x)\) \(f'(x)\) \(g'(x)\)
3 -3 6 -5 1
4 0 3 -3 9
5 3 -2 4 5
A
\(2\)
✕
B
\(7\)
✕
C
\(14\)
✕
D
\(20\)
✕
E
\(26\)
✕
Question 8 of 22   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
Using the values in the table from the previous problem (\(f(3)=-3\), \(f(4)=0\), \(f(5)=3\), \(g(3)=6\), \(g(4)=3\), \(g(5)=-2\), \(f'(3)=-5\), \(f'(4)=-3\), \(f'(5)=4\), \(g'(3)=1\), \(g'(4)=9\), \(g'(5)=5\)), if \(h(x) = f(g(x))\), then \(h'(4) =\)
A
\(-45\)
✕
B
\(-27\)
✕
C
\(-15\)
✕
D
\(0\)
✕
E
\(25\)
✕
Question 9 of 22   |  Contextual Applications of Differentiation  · Level 2
If \(f(x)\) is a continuous function and \(f(2) = 7\) and \(f'(2) = -3\), then \(f(2.01)\) is approximately
A
\(-6.03\)
✕
B
\(6.92\)
✕
C
\(6.97\)
✕
D
\(7.01\)
✕
E
\(7.03\)
✕
Question 10 of 22   |  Analytical Applications of Differentiation  · Level 4
Consider the curve \(y = 2 x^3 - 3(k + 1) x^2 + 6 k x\), \(k > 1\). On the interval \(1 < x < k\),
A
\(y'\) is positive, and \(y''\) is first positive, then negative
✕
B
\(y'\) is positive, and \(y''\) is first negative, then positive
✕
C
\(y'\) is negative, and \(y''\) is first positive, then negative
✕
D
\(y'\) is negative, and \(y''\) is first negative, then positive
✕
E
Neither the sign of \(y'\) nor the sign of \(y''\) can be determined without knowing the value of \(k\).
✕
Question 11 of 22   |  Differentiation: Definition and Fundamental Properties  · Level 2
If \(f(x) = 2^x\) and \(2^{3.03} \approx 8.168\), which of the following is closest to \(f'(3)\)?
A
\(.168\)
✕
B
\(.97\)
✕
C
\(1\)
✕
D
\(3\)
✕
E
\(5.6\)
✕
Question 12 of 22   |  Analytical Applications of Differentiation  · Level 3
Pictured above (in source) is the graph of \(f'(x)\), which is positive on \((-4, 4)\) and reaches its maximum at \(x = 0\). For what values of \(x\) is the graph of \(f(x)\) concave down?
A
\(-2 < x < 2\)
✕
B
\(x < -4\) or \(0 < x < 4\)
✕
C
\(-4 < x < 4\)
✕
D
all values of \(x\)
✕
E
the graph of \(f(x)\) is always concave up
✕
Question 13 of 22   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 2
If \(g(1) = 3\), \(g'(1) = 4\), \(g(2) = 8\), and \(g'(2) = 3\), and \(f(x) = g^2(x)\), then \(f'(2) =\)
A
\(12\)
✕
B
\(16\)
✕
C
\(23\)
✕
D
\(24\)
✕
E
\(48\)
✕
Question 14 of 22   |  Integration and Accumulation of Change  · Level 2
If \(\displaystyle\int_{0}^{4} f(x) d x = 10\), \(\displaystyle\int_{0}^{5} f(x) d x = 9\), and \(\displaystyle\int_{4}^{7} f(x) d x = 1\), then \(\displaystyle\int_{5}^{7} f(x) d x =\)
A
\(-1\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(3\)
✕
E
\(4\)
✕
Question 15 of 22   |  Integration and Accumulation of Change  · Level 3
If \(u = x^2 + 1\), then \(\displaystyle\int_{1}^{2} \dfrac{x^2}{x^2 + 1} d x =\)
A
\(\displaystyle\int_{1}^{2} \dfrac{u - 1}{u} d u\)
✕
B
\(\displaystyle\int_{1}^{2} \dfrac{\sqrt{u - 1}}{u} d u\)
✕
C
\(\displaystyle\int_{2}^{5} \dfrac{u - 1}{u} d u\)
✕
D
\(\displaystyle\int_{2}^{5} \dfrac{\sqrt{u - 1}}{u} d u\)
✕
E
\(\displaystyle\int_{2}^{5} \dfrac{\sqrt{u - 1}}{2 u} d u\)
✕
Question 16 of 22   |  MCQ  · Level 3
The average area of all circles with radii between 3 and 6 is
A
\(\dfrac{25 \pi}{2}\)
✕
B
\(\dfrac{27 \pi}{2}\)
✕
C
\(18 \pi\)
✕
D
\(21 \pi\)
✕
E
\(\dfrac{45 \pi}{2}\)
✕
Question 17 of 22   |  Integration and Accumulation of Change  · Level 3
A rumor spreads continuously at the rate of \(3 t^2 + 6 t\) (where \(t\) is measured in days). How many people hear the rumor on the third day?
A
\(21\)
✕
B
\(34\)
✕
C
\(44\)
✕
D
\(45\)
✕
E
\(54\)
✕
Question 18 of 22   |  Limits and Continuity  · Level 3
If \(\operatorname*{lim}\limits_{x \rightarrow 2}[\ln f(x)] = 1\), then \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) =\)
A
\(0\)
✕
B
\(\ln 2\)
✕
C
\(1\)
✕
D
\(2\)
✕
E
\(e\)
✕
Question 19 of 22   |  Analytical Applications of Differentiation  · Level 4
The graph of \(y = f''(x)\) consists of two straight line segments. It passes through \((0, 3)\), \((3, 0)\), \((6, -3)\), \((9, 0)\). If \(f'(0) = 0\), then in the vicinity of which of the following values of \(x\) is the curve \(y = f(x)\) falling and concave down?
A
\(2\)
✕
B
\(4\)
✕
C
\(6\)
✕
D
\(8\)
✕
E
\(10\)
✕
Question 20 of 22   |  Differentiation: Composite, Implicit, and Inverse Functions  · Level 3
If \(f(x) = \sin 2 x \cos 3 x\) and \(k\) is an odd integer, then \(f'(k \pi) =\)
A
\(-5\)
✕
B
\(-2\)
✕
C
\(-1\)
✕
D
\(1\)
✕
E
\(5\)
✕
Question 21 of 22   |  Integration and Accumulation of Change  · Level 3
If \(F(x) = \displaystyle\int_{1}^{x} \dfrac{4}{1 + \ln t} d t\), then \(F'(e) =\)
A
\(\dfrac{1}{e^2}\)
✕
B
\(\ln 2\)
✕
C
\(2\)
✕
D
\(2 e\)
✕
E
\(e^2\)
✕
Question 22 of 22   |  Differential Equations  · Level 3
If the slope of the tangent to the curve at any point \((x, y)\) on the curve equals \(\dfrac{x}{y}\), what kind of curve can it be?
A
a circle
✕
B
a parabola
✕
C
an ellipse
✕
D
a hyperbola
✕
E
none of the above
✕

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Reference Sheet

Area & Circumference

Circle$A = \pi r^2$,  $C = 2\pi r$
Rectangle$A = lw$
Triangle$A = \tfrac{1}{2}bh$
Trapezoid$A = \tfrac{1}{2}(b_1+b_2)h$

Volume

Box$V = lwh$
Cylinder$V = \pi r^2 h$
Sphere$V = \tfrac{4}{3}\pi r^3$
Cone$V = \tfrac{1}{3}\pi r^2 h$
Pyramid$V = \tfrac{1}{3}lwh$

Triangles

Pythagorean Thm$a^2 + b^2 = c^2$
30-60-90sides: $1,\, \sqrt{3},\, 2$
45-45-90sides: $1,\, 1,\, \sqrt{2}$
Triangle Anglessum $= 180°$

Other Facts

Circle Degrees$360° = 2\pi \text{ rad}$
Exterior Angle= sum of non-adjacent interior angles

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is $2\pi$.

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