Unit 4 check

Unit 4 check: Contextual Applications of Differentiation

10 questions from 7 lessons in Unit 4. Answer each one, and if you miss, follow the link back to where it is taught, then try again. Your score counts first tries.

Question 1 · from 4.1 · Match

No calculator. Tap a function, then the units of its derivative.

Question 2 · from 4.2 · Multiple choice

No calculator. A particle moves along a line with velocity $v (t) = (t - 1) (t - 3)^{2}$ for $0 < t < 5$ . At which times does the particle change direction?

Question 3 · from 4.3 · Decide and justify

No calculator. Let $W (t)$ be the amount of water in a reservoir, in millions of gallons, $t$ days after June 1. On day 6, $W^{'} (6) = - 2.5$ and $W^{''} (6) = 0.4$ . A ranger says the amount of water in the reservoir is increasing on day 6, because $W^{''} (6)$ is positive. Is the ranger right?

Which reason supports your answer?

Question 4 · from 4.4 · Calculate

No calculator. A city's population density is $D = \frac{P}{A}$ people per square kilometer, where $P$ is its population and $A$ its area in square kilometers. Both change with time $t$ , in years. At one instant $P = 120,000$ people and $A = 40$ square kilometers, the population is growing at 3,000 people per year, and the city is annexing land at 1.5 square kilometers per year. Find $\frac{d D}{d t}$ at that instant, in people per square kilometer per year.

Your answer

Question 5 · from 4.5 · Decide and justify

No calculator. A cylindrical candle with a radius of 2 centimeters burns down, losing wax at 3 cubic centimeters per minute. To find how fast its height is changing when the candle is 10 centimeters tall, a classmate writes $V = 4 π h$ , so $\frac{d V}{d t} = 4 π \frac{d h}{d t}$ , puts in $\frac{d V}{d t} = 3$ , gets $\frac{d h}{d t} = \frac{3}{4 π}$ , and says the height is increasing at $\frac{3}{4 π}$ centimeter per minute at that instant. Is the classmate right?

Which reason supports your answer?

Question 6 · from 4.6 · Multiple choice

No calculator. You want to approximate $\sin 0.05$ by hand with a line tangent to the graph of $f (x) = \sin x$ . Which tangent line should you use?

Setup for question 7

The functions $f$ and $g$ are twice differentiable for all $x$ , and $f^{''}$ and $g^{''}$ are continuous. The table gives their values and the values of their first two derivatives at three inputs.

Values of f, g and their derivatives
$x$	$f^{'} (x)$	$f^{''} (x)$	$g (x)$	$g^{'} (x)$	$g^{''} (x)$
1	2	$- 3$	4	$- 1$	2
2	$- 9$	1	0	$- 2$	3
3	0	$- 6$	0	0	$- 4$

Question 7 · from 4.7 · Calculate

No calculator. What is $\lim_{x \to 2} \frac{f (x)}{g (x)}$ ? Give an exact decimal.

Your answer

Question 8 · from 4.1 · Sort it

No calculator. $S (t)$ is the depth of snow on the ground, in inches, $t$ hours after midnight. Tap a sentence, then tap what it gives a value of.

A value of $S$

A value of $S^{'}$

Question 9 · from 4.2 · Sort it

No calculator. Each card gives a particle's velocity $v$ , in meters per second, and acceleration $a$ , in meters per second per second, at one instant. Sort each card by what the particle's speed is doing at that instant.

Speeding up

Slowing down

Question 10 · from 4.3 · Calculate

Graphing calculator required. The depth of the water at a dock is $D (t) = 3 + 1.2 \sin (0.5 t)$ meters, $t$ hours after midnight. With your calculator in radian mode, find $D^{'} (5)$ , the rate of change of the depth at 5 a.m., in meters per hour. Give your answer to three decimal places.

Your answer

0 of 10 answered.