Unit 2 check

Unit 2 check: Differentiation: Definition and Fundamental Properties

10 questions from 10 lessons in Unit 2. 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 2.1 · Calculate

No calculator. Let $f (x) = x^{3} - 2 x$ . What is the average rate of change of $f$ over the interval $[1, 3]$ ?

Your answer

Question 2 · from 2.2 · Fill the statement

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

A block of ice melts on a counter. It weighs $y = M (t)$ grams $t$ minutes after it is set out. Its derivative can be written $M^{'} (t)$ , $\frac{d y}{d t}$ or , and it is measured in .

Question 3 · from 2.3 · Calculate

Graphing calculator required. In radian mode, let $f (x) = x \cos x$ . Use the calculator's numerical derivative to find $f^{'} (2)$ . Give your answer to three decimal places.

Your answer

Question 4 · from 2.4 · Calculate

No calculator. Let $f (x) = x^{2} + k$ for $x \leq 2$ and $f (x) = 4 x - 1$ for $x > 2$ , where $k$ is a constant. What value of $k$ makes $f$ differentiable at $x = 2$ ?

Your answer

Question 5 · from 2.5 · Decide and justify

No calculator. A classmate applies the power rule to $f (x) = 2^{x}$ and writes $f^{'} (x) = x \cdot 2^{x - 1}$ . Is the classmate right?

Which reason supports your answer?

Setup for question 6

The functions $f$ and $g$ are differentiable for all real numbers. The table gives values of each function and its derivative at three inputs.

Values of f, g and their derivatives
$x$	$f (x)$	$f^{'} (x)$	$g (x)$	$g^{'} (x)$
1	4	$- 2$	3	5
2	7	3	$- 1$	4
3	$- 2$	6	0	$- 3$

Question 6 · from 2.6 · Calculate

No calculator. Let $h (x) = 2 f (x) - x^{3}$ . What is the slope of the line tangent to the graph of $h$ at $x = 2$ ?

Your answer

Setup for question 7

A car on a Ferris wheel passes its lowest point at $t = 0$ . Its height above the ground, in meters, $t$ minutes later is $H (t) = 12 - 10 \cos t$ . Some values are in the table.

Height of the car above the ground
$t$ (minutes)	$H (t)$ (meters)
$0$	2
$\frac{π}{2}$	12
$π$	22
$\frac{3 π}{2}$	12

Question 7 · from 2.7 · Multiple choice

No calculator. Which statement describes the car at $t = \frac{3 π}{2}$ minutes?

Question 8 · from 2.8 · Spot the error

No calculator. A student finds the derivative of $f (x) = (x^{3} + 2 x) \ln x$ . Tap the first line with a mistake.

Question 9 · from 2.9 · Match

No calculator. Tap a function, then its derivative.

Question 10 · from 2.10 · Calculate

No calculator. A function $p$ is the quotient $p (x) = \frac{g (x)}{h (x)}$ , where $g$ and $h$ are differentiable, $g (2) = 6$ , $g^{'} (2) = 7$ , $h (2) = 2$ and $h^{'} (2) = - 1$ . Find $p^{'} (2)$ .

Your answer

0 of 10 answered.