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AB MCQ Set 160 (Theorems + Related Rates)
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Question 1 of 9
AB MCQ Set 160 (Theorems + Related Rates) 0/9
Question 1 of 9   |  MCQ  · Level 3
Which of the following functions satisfy the hypothesis of the Mean Value Theorem on the interval \([0, 2]\)?
I. \(f(x) = \sin(\pi x) + \cos(2 x)\)
II. \(f(x) = \sqrt[3]{x - 1}\)
III. \(f(x) = |x^2 - 2 x|\)
A
I only
✕
B
II only
✕
C
III only
✕
D
I and II
✕
E
I and III
✕
Question 2 of 9   |  MCQ  · Level 3
If \(\operatorname*{lim}\limits_{h \rightarrow 0} (f(3 + h) - f(3))/h = 0\), then which of the following must be true?
I. \(f\) has derivative at \(x = 3\)
II. \(f\) is continuous at \(x = 3\)
III. \(f\) has a critical value at \(x = 3\)
A
I only
✕
B
II only
✕
C
I and II
✕
D
I and III
✕
E
I, II, and III
✕
Question 3 of 9   |  MCQ  · Level 3
How many values of \(c\) satisfy the conclusion of the Mean Value Theorem for \(f(x) = x^2 + 1\) on \([-1, 1]\)?
A
\(0\)
✕
B
\(1\)
✕
C
\(2\)
✕
D
\(3\)
✕
E
\(4\)
✕
Question 4 of 9   |  MCQ  · Level 3
A 20-foot ladder leans against a wall. Top moves down at \(0.5\) ft/sec. How fast is foot moving when foot is 12 ft from wall?
A
\(0.5\) ft/sec
✕
B
\(\dfrac{5}{8}\) ft/sec
✕
C
\(\dfrac{2}{3}\) ft/sec
✕
D
\(\dfrac{4}{3}\) ft/sec
✕
E
\(\dfrac{8}{3}\) ft/sec
✕
Question 5 of 9   |  MCQ  · Level 3
Spherical balloon: \(\dfrac{d V}{d t} = 8\) in³/s. How fast is diameter increasing when \(V = 36 \pi\) in³? \((V = \left(\dfrac{4}{3}\right) \pi r^3)\)
A
\(4/(9 \pi)\)
✕
B
\(2/(3 \pi)\)
✕
C
\(2/(9 \pi)\)
✕
D
\(8/(27 \pi)\)
✕
E
\(2/(27 \pi)\)
✕
Question 6 of 9   |  MCQ  · Level 3
Sand falling at \(10\) m³/s, height = (1/2) diameter. Find \(\dfrac{d h}{d t}\) when \(h = 5\). \((V = \left(\dfrac{1}{3}\right) \pi r^2 h)\)
A
\(1/(25 \pi)\)
✕
B
\(2/(5 \pi)\)
✕
C
\(4/(5 \pi)\)
✕
D
\(8/(5 \pi)\)
✕
E
\(250 \pi\)
✕
Question 7 of 9   |  MCQ  · Level 2
Sphere: \(\dfrac{d V}{d t} = 3 \pi\) cm³/s. \(\dfrac{d r}{d t}\) when \(r = \dfrac{1}{2}\)? \((V = \left(\dfrac{4}{3}\right) \pi r^3)\)
A
\(\pi\)
✕
B
\(3\)
✕
C
\(2\)
✕
D
\(1\)
✕
E
\(\dfrac{1}{2}\)
✕
Question 8 of 9   |  MCQ  · Level 3
Balloon rises at \(10\) ft/s. Observer 40 ft away. Rate of change of angle of elevation when balloon at 30 ft.
A
\(\dfrac{3}{20}\)
✕
B
\(\dfrac{4}{25}\)
✕
C
\(\dfrac{1}{5}\)
✕
D
\(\dfrac{1}{3}\)
✕
E
\(\dfrac{25}{64}\)
✕
Question 9 of 9   |  MCQ  · Level 3
Point on \(y = x^2 + 1\), x-coord increases at \(1.5\) units/s. Rate of distance from origin when at \((1, 2)\).
A
\(7 \dfrac{\sqrt{5}}{10}\)
✕
B
\(\sqrt{5}\)
✕
C
\(3 \dfrac{\sqrt{5}}{2}\)
✕
D
\(3 \sqrt{5}\)
✕
E
\(\dfrac{15}{2}\)
✕

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Graphing Calculator
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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