Unit 6 check

Unit 6 check: Integration and Accumulation of Change

11 questions from 11 lessons in Unit 6. 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 6.1 · Match

No calculator. Each rate is graphed against its input. Tap a rate, then what the area between its graph and the horizontal axis gives.

Question 2 · from 6.2 · Calculate

No calculator. A function $g$ has $g (0) = 5$ , $g (1) = 8$ , $g (3) = 6$ , $g (4) = 9$ and $g (7) = 4$ . Use a right Riemann sum with the subintervals $[0, 1]$ , $[1, 3]$ , $[3, 4]$ and $[4, 7]$ to approximate $\int_{0}^{7} g (x) d x$ .

Your answer

Question 3 · from 6.3 · Match

No calculator. Each limit equals one of the definite integrals. Tap a limit, then its integral.

Question 4 · from 6.4 · Decide and justify

No calculator. Let $f (t) = 6 - 3 t$ and $g (x) = \int_{0}^{x} f (t) d t$ . A classmate notices that $f (2) = 0$ and says that $g (2) = 0$ too. Is the classmate right?

Which reason supports your answer?

Question 5 · from 6.5 · Sketch the graph

No calculator. Let $f (t) = 4 - t$ , and let $g (x) = \int_{0}^{x} f (t) d t$ for $0 \leq x \leq 4$ . Sketch the shape of the graph of $g$ : drag the points so it rises or falls, and bends, the way $g$ does. The first point is fixed at $g (0) = 0$ . The exact heights are up to you.

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

Question 6 · from 6.6 · Sort it

No calculator. Here $f$ and $g$ are continuous on an interval that contains $a$ , $b$ and $c$ . Sort each statement: is it true for every such $f$ and $g$ , or not true in general? Tap a card, then a bin.

True for every $f$ and $g$

Not true in general

Setup for question 7

The function $f$ is continuous for all real numbers, and $F$ is an antiderivative of $f$ . Some values of $F$ are in the table.

Values of an antiderivative F of f
$x$	$F (x)$
0	4
1	1
2	$- 3$
3	2
4	6
5	5

Question 7 · from 6.7 · Multiple choice

No calculator. Let $G (x) = \int_{2}^{x} f (t) d t$ . What is $G (5)$ ?

Setup for question 8

A student made this table of derivatives while studying. Every row is correct. Use it to answer the questions.

A student's table of derivatives
$F (x)$	$F^{'} (x)$
$\tan x$	$\sec^{2} x$
$x e^{x}$	$(x + 1) e^{x}$
$\sin (3 x)$	$3 \cos (3 x)$
$\ln (x^{2} + 1)$	$\frac{2 x}{x^{2} + 1}$

Question 8 · from 6.8 · Decide and justify

No calculator. A classmate reads the table's second row and writes $\int x e^{x} d x = x e^{x} + C$ . Is the classmate right?

Which reason supports your answer?

Question 9 · from 6.9 · Fill the statement

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

To find $\int e^{5 x} d x$ , let $u =$ . Then $d u = 5 d x$ , so $d x =$ , and $\int e^{5 x} d x =$ .

Question 10 · from 6.10 · Calculate

No calculator. Evaluate $\int_{0}^{1} 2 x (x^{2} + 1)^{3} d x$ . Give an exact decimal.

Your answer

Question 11 · from 6.14 · Multiple choice

No calculator. What is $\frac{d}{d x} \sin (x^{3})$ ?

0 of 11 answered.