Unit 3 check

Unit 3 check: Differentiation: Composite, Implicit, and Inverse Functions

10 questions from 6 lessons in Unit 3. 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 3.1 · Match

No calculator. Each function is a composition $f (g (x))$ . Tap a function, then its inner function $g (x)$ .

Question 2 · from 3.2 · Multiple choice

No calculator. The equation $y^{2} - x y = 4$ defines $y$ as a differentiable function of $x$ . Which expression is $\frac{d y}{d x}$ ?

Question 3 · from 3.3 · Decide and justify

No calculator. Let $f (x) = x^{3} - 3 x^{2} + 3 x$ . Then $f^{'} (x) = 3 (x - 1)^{2}$ , $f$ is increasing, and $f (1) = 1$ , so $f$ has an inverse function $g$ with $g (1) = 1$ . A classmate says $g^{'} (1) = 0$ . Is the classmate right?

Which reason supports your answer?

Question 4 · from 3.4 · Multiple choice

No calculator. Let $g (x) = x \arctan x$ . What is $g^{'} (1)$ ?

Question 5 · from 3.5 · Multiple choice

No calculator. For $- 1 < x < 1$ , what is $\frac{d}{d x} \arcsin (x^{2})$ ?

Setup for question 6

A function $g$ is twice differentiable for all real numbers. The table gives values of $g$ , its derivative $g^{'}$ and its second derivative $g^{''}$ at three inputs.

Values of g and its first two derivatives
$x$	$g (x)$	$g^{'} (x)$	$g^{''} (x)$
1	4	$- 2$	3
2	1	$- 3$	5
3	$- 2$	2	$- 1$

Question 6 · from 3.6 · Calculate

No calculator. Let $k (x) = x^{2} g (x)$ . What is $k^{''} (2)$ ?

Your answer

Question 7 · from 3.1 · Multiple choice

No calculator. Which expression is the derivative of $\sin (3 x^{2})$ ?

Question 8 · from 3.2 · Calculate

No calculator. The point $(2, 1)$ lies on the curve $x^{3} + y^{3} = 9$ . What is the slope of the curve at $(2, 1)$ ?

Your answer

Question 9 · from 3.3 · Match

No calculator. The function $r$ is differentiable and increasing, so it has an inverse function $r^{- 1}$ . Some values of $r$ and $r^{'}$ are in the table.

\begin{matrix} x & 1 & 2 & 3 & 4 \\ r (x) & 2 & 5 & 9 & 14 \\ r^{'} (x) & 3 & 4 & 5 & 6 \end{matrix}

Tap an expression, then its value.

Question 10 · from 3.4 · Sort it

No calculator. Tap a function, then the sign of its derivative wherever the derivative exists.

Derivative positive

Derivative negative

0 of 10 answered.