Unit 5 check

Unit 5 check: Analytical Applications of Differentiation

12 questions from 12 lessons in Unit 5. 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 5.1 · Multiple choice

No calculator. A function $f$ is continuous on $[2, 6]$ and differentiable on $(2, 6)$ , with $f (2) = 7$ and $f (6) = 19$ . Which statement does the Mean Value Theorem guarantee?

Question 2 · from 5.2 · Fill the statement

No calculator. Tap a blank, then tap an answer.

A function $f$ is continuous on $[0, 6]$ . $f (4) \geq f (x)$ for every $x$ near 4, but $f (1) > f (4)$ . So $f (4)$ is a maximum value of $f$ , and it is not the maximum value of $f$ on $[0, 6]$ .

Question 3 · from 5.3 · Decide and justify

No calculator. On the interval $0 < x < 3$ , the graph of $g^{'}$ , the derivative of $g$ , falls steadily from $g^{'} (0) = 5$ to $g^{'} (3) = 1$ . Is $g$ increasing or decreasing on $0 < x < 3$ ?

Which reason supports your answer?

Question 4 · from 5.4 · Calculate

Graphing calculator required. Let $f$ be a function with $f^{'} (x) = \cos x - \frac{x}{2}$ for $0 < x < 4$ . In radian mode, find the value of $x$ at which $f$ has a relative maximum. Give your answer to three decimal places.

Your answer

Question 5 · from 5.5 · Multiple choice

No calculator. Let $f (x) = | x^{2} - 4 |$ on the closed interval $[- 1, 3]$ . Which list gives every candidate for an absolute extremum of $f$ on $[- 1, 3]$ ?

Setup for question 6

A function $f$ is twice differentiable for all real numbers. The table gives the sign of $f^{'} (x)$ and of $f^{''} (x)$ on each interval, and the inputs where either one is 0.

Signs of the first and second derivatives of f
$x$	$f^{'} (x)$	$f^{''} (x)$
$x < - 1$	Positive	Negative
$x = - 1$	0	Negative
$- 1 < x < 1$	Negative	Negative
$x = 1$	Negative	0
$1 < x < 3$	Negative	Positive
$x = 3$	0	Positive
$x > 3$	Positive	Positive

Question 6 · from 5.6 · Multiple choice

No calculator. Which gives all the values of $x$ for which the graph of $f$ is concave up?

Setup for question 7

The function $g$ is twice differentiable for all real numbers, and its only critical points are $x = - 3$ , $x = 1$ and $x = 5$ . The table gives the values of $g$ and its first two derivatives there.

The function g and its derivatives at its critical points
$x$	$g (x)$	$g^{''} (x)$
$- 3$	4	$- 6$
1	$- 2.4$	0
5	$- 8.8$	6

Question 7 · from 5.7 · Calculate

No calculator. What is the absolute minimum value of $g$ on the interval $2 < x < 9$ ?

Your answer

Question 8 · from 5.8 · Fill the statement

No calculator. Tap a blank, then tap an answer.

The graph of $f^{'}$ crosses the $x$ -axis from above to below at $x = 7$ , so $f$ has a at $x = 7$ . The graph of $f^{'}$ falls until $x = 9$ and rises after it, so the graph of $f$ has a at $x = 9$ .

Question 9 · from 5.9 · Sketch the graph

No calculator. For $0 \leq x \leq 5$ , $f^{'} (x)$ is positive and $f^{'}$ is decreasing, and $f (0) = 0$ . Drag the points to sketch a possible graph of $f$ . The first point is fixed at 0.

Drag a point up or down, or tap it and use the arrow keys. Only the shape is checked.

Question 10 · from 5.10 · Multiple choice

No calculator. The derivative of a function $f$ is $f^{'} (x) = (x - 1) (x - 4)^{2}$ . Which statement about $f$ at $x = 4$ is true?

Question 11 · from 5.11 · Multiple choice

No calculator. A tank holds $V (t)$ liters of water $t$ minutes after a pump starts. What are the units of $V^{'} (t)$ ?

Question 12 · from 5.12 · Sort it

No calculator. For the circle $(x - 1)^{2} + (y + 2)^{2} = 25$ , implicit differentiation gives $\frac{d y}{d x} = \frac{1 - x}{y + 2}$ . Tap a point, then tap where it belongs.

Critical point, $\frac{d y}{d x} = 0$

Critical point, $\frac{d y}{d x}$ does not exist

Not a critical point

0 of 12 answered.