AB MCQ Set 160 (Theorems + Related Rates)

Question 1 of 9 | Analytical Applications of Differentiation · 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. $f(x) = |x^2 - 2 x|$

I only

II only

III only

I and II

I and III

Question 2 of 9 | Differentiation: Definition and Fundamental Properties · 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 $x = 3$

I only

II only

I and II

I and III

I, II, and III

Question 3 of 9 | Analytical Applications of Differentiation · Level 3

How many values of $c$ satisfy the conclusion of the Mean Value Theorem for $f(x) = x^2 + 1$ on $[-1, 1]$?

$0$

$1$

$2$

$3$

$4$

Question 4 of 9 | Contextual Applications of Differentiation · 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?

$0.5$ ft/sec

$\dfrac{5}{8}$ ft/sec

$\dfrac{2}{3}$ ft/sec

$\dfrac{4}{3}$ ft/sec

$\dfrac{8}{3}$ ft/sec

Question 5 of 9 | Contextual Applications of Differentiation · Level 3

Spherical balloon: $\dfrac{d V}{d t} = 8$ in³/s. How fast is diameter increasing when $V = 36 \pi$ in³? $(V = \left(\dfrac{4}{3}\right) \pi r^3)$

$4/(9 \pi)$

$2/(3 \pi)$

$2/(9 \pi)$

$8/(27 \pi)$

$2/(27 \pi)$

Question 6 of 9 | Contextual Applications of Differentiation · Level 3

Sand falling at $10$ m³/s, height = (1/2) diameter. Find $\dfrac{d h}{d t}$ when $h = 5$. $(V = \left(\dfrac{1}{3}\right) \pi r^2 h)$

$1/(25 \pi)$

$2/(5 \pi)$

$4/(5 \pi)$

$8/(5 \pi)$

$250 \pi$

Question 7 of 9 | Contextual Applications of Differentiation · Level 2

Sphere: $\dfrac{d V}{d t} = 3 \pi$ cm³/s. $\dfrac{d r}{d t}$ when $r = \dfrac{1}{2}$? $(V = \left(\dfrac{4}{3}\right) \pi r^3)$

$\pi$

$3$

$2$

$1$

$\dfrac{1}{2}$

Question 8 of 9 | Contextual Applications of Differentiation · Level 3

Balloon rises at $10$ ft/s. Observer 40 ft away. Rate of change of angle of elevation when balloon at 30 ft.

$\dfrac{3}{20}$

$\dfrac{4}{25}$

$\dfrac{1}{5}$

$\dfrac{1}{3}$

$\dfrac{25}{64}$

Question 9 of 9 | Contextual Applications of Differentiation · Level 3

Point on $y = x^2 + 1$, x-coord increases at $1.5$ units/s. Rate of distance from origin when at $(1, 2)$.

$7 \dfrac{\sqrt{5}}{10}$

$\sqrt{5}$

$3 \dfrac{\sqrt{5}}{2}$

$3 \sqrt{5}$

$\dfrac{15}{2}$

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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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