Chapter 1. Test Tutorial

1.1 Problem Statement

$a: {2,4,6,8} $a2: eval pow($a, 2) $p1_1: eval $a2+.002 $p1_2: eval $a2+.001 $p1_3: eval $a2+.0005 $p1_4: eval $a2+.0001 $p1_1_a: eval round((sqrt($p1_1)-$a)/($p1_1-$a2),7) $p1_2_a: eval round((sqrt($p1_2)-$a)/($p1_2-$a2),7) $p1_3_a: eval round((sqrt($p1_3)-$a)/($p1_3-$a2),7) $p1_4_a: eval round((sqrt($p1_4)-$a)/($p1_4-$a2),7) $ans: eval round(1/(2*$a),4)

Estimate the limit numerically or state that the limit doesn't exist.

1.2 Step 1

To determine if a limit exists numerically for , make a table of values of f(x) for x close to c but greater than c (that is, x → c+) and a second table of values of f(x) for x close to c but less than c (that is, x → c–). If both tables indicate convergence to the same number L, we take L to be an estimate for the limit.

In the given problem, c is $a2 and f(x) is .

Question Sequence

Question 1.1

Complete the table of values of f(x) as x → $a2+. (Round your answers to seven decimal places.)

x	f(x)
$p1_1	$p1_1_a
$p1_2	4KozdQDseRa1gpJK
$p1_3	jY3eeGkeo+oqGJ6H
$p1_4	Bd1SHbUm9tYOoTh7

_max_tries:2 _feedback_incorrect_first: No. Calculate f(x) for each of the values of x given in the left column, and enter those values in the right column. Remember to round to 7 decimal places. _feedback_incorrect: Incorrect. See above for the correct answers, and try to work out where you went wrong before going on to the next question. _feedback_correct: Nice job. _question_report_text: Values of f(x) for x = c + 0.001 / c + 0.0005 / c + 0.0001

Question 1.2

As x → $a2+, f(x) approaches qjqZz1N+poo= (rounded to four decimal places)

_max_tries:2 _feedback_incorrect_first: No. What number does f(x) appear to approach as x approaches $a2 from above? Remember to round to 4 decimal places. _feedback_incorrect: Incorrect. Do you see where you went wrong? _feedback_correct: Exactly. _question_report_text: As x → c+, f(x) approaches:

1.3 Step 2

Question 1.3

Since both tables indicate convergence to the same number $ans as x approaches $a2 from the left and from the right, then $ans is an estimate for the limit and we write

= qjqZz1N+poo= .

_max_tries:2 _feedback_incorrect_first: No. What number did you find that f(x) approached as x approached $a2 from above and below? _feedback_incorrect: Incorrect. Do you see where you went wrong? _feedback_correct: Excellent work. _question_report_text: Final estimate of the limit of f(x)

1.4 Demo step

This step is included in the template to illustrate a couple of extra things.

Question Sequence

Question 1.4

BMazM8HC8L/DSeERf50Wf8pdx7RipV8eFYERvJBS0wMeCmWF/OhU6sL6MYGluISATiljoaLkvpilg7PKMFW8sZKflSeXsAqi9yU7Xoh6G7sOoOoP2PMVbRH4h8soNANdcaWSfNpUPMXDEr3+YxOlG++A1Mt+z3XiijdyikHI+HLTFwTZYQB1VQsTWkKOcTxC3U8Myg==

_max_tries:2 _feedback_correct: Correct. _feedback_hint: That's not right. Check your work. _feedback_incorrect: Incorrect.

Question 1.5

This is a simple fill-in-the-blank question. What is the capital of Oregon?

vXzYERXVdXog8cNH

_max_tries:2 _feedback_correct: Correct. _feedback_hint: That's not right. Check your work. _feedback_incorrect: Incorrect.

Question 1.6

This is a more complex question, which includes i+gXJPH7GXE=-in blanks and EqtjiwKqDdYLdXEKkBEBEAckAWaamwky menus.

_max_tries:2 _feedback_correct: Correct. _feedback_hint: That's not right. Check your work. _feedback_incorrect: Incorrect.