From f636ce062ce837e5c4b508b11641d2cd52db87e2 Mon Sep 17 00:00:00 2001 From: Chrissy Date: Thu, 20 Nov 2025 18:37:40 +0000 Subject: [PATCH] 4-3 Calculus Practice --- source/sec-4-3-definite-integral.xml | 107 ++++++++++++++++++++++++--- webworkfiles/defintproperty.pg | 57 ++++++++++++++ 2 files changed, 153 insertions(+), 11 deletions(-) create mode 100644 webworkfiles/defintproperty.pg diff --git a/source/sec-4-3-definite-integral.xml b/source/sec-4-3-definite-integral.xml index df07919..58f0e25 100644 --- a/source/sec-4-3-definite-integral.xml +++ b/source/sec-4-3-definite-integral.xml @@ -3,28 +3,28 @@
The definite integral -

  • - What is the definition of the definite integral of a function f over the interval [a,b]? + Understand the definite integral as the exact net-signed area between a function and the + x-axis, and be able to use known area formulas to calculate \int_a^b f(x)dx.

  • - What does the definite integral measure exactly, - and what are some of the key properties of the definite integral? + Be able to use known properties of the definite integral that arise from the fact that the definite + integral measures the exact net-signed area.

    • - --> + Foundations

    @@ -34,10 +34,95 @@ Calculus Practice - + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral and average value + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using areas to calculate the definite integral + + + + Using properties of the definite integral + + + + Using properties of the definite integral + + + + Using properties of the definite integral + + + + + Using properties of the definite integral + + + + Using properties of the definite integral + + + + Using properties of the definite integral + + + + Approximating the average value of a continuous function + + + + Meaning of the definite integral + + + + Meaning of the definite integral + + + + Meaning of the definite integral + + + + Meaning of the definite integral + + + + Meaning of the definite integral + + +

    diff --git a/webworkfiles/defintproperty.pg b/webworkfiles/defintproperty.pg new file mode 100644 index 0000000..c625af5 --- /dev/null +++ b/webworkfiles/defintproperty.pg @@ -0,0 +1,57 @@ +## DESCRIPTION +## Calculus +## ENDDESCRIPTION + +## Tagged by tda2d + +## DBsubject(Calculus - single variable) +## DBchapter(Integrals) +## DBsection(Conceptual understanding of integration) +## Institution(Rochester) +## MLT(integral_concept_partition_ab) +## MLTleader(1) +## Level(2) +## TitleText1('Calculus: Early Transcendentals') +## AuthorText1('Rogawski') +## EditionText1('1') +## Section1('5.2') +## Problem1('61') +## KEYWORDS('integral') + +DOCUMENT(); # This should be the first executable line in the problem. + +loadMacros('PGstandard.pl', 'PGML.pl', 'PGchoicemacros.pl', 'PGcourse.pl'); + +$showPartialCorrectAnswers = 1; + +$a = random(1, 10, 1); +$add1 = random(1, 10, 1); +$add2 = random(1, 10, 1); +$b = $a + $add1; +$c = $b + $add2; + +BEGIN_PGML +[``` \int_{[$a]}^{[$c]} f(x) \,dx - \int_{[$a]}^{[$b]} f(x) \,dx = \int_{a}^{b} f(x) \,dx```] + + + where [` a= `] [_]{$b} +and [` b= `] [_]{$c}. +END_PGML + +BEGIN_PGML_SOLUTION + +First recall that + [``` \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx = \int_{a}^{c} f(x) \,dx ```] +and therefore we can rearrange it to look like this: + [``` \int_{a}^{c} f(x) \,dx - \int_{a}^{b} f(x) \,dx = \int_{b}^{c} f(x) \,dx ```] + + +Applying that in this specific context, we find: + [``` + \int_{[$a]}^{[$c]} f(x) \,dx - \int_{[$a]}^{[$b]} f(x) \,dx = \int_{[$b]}^{[$c]} f(x) \,dx + ```] +Thus we find [` a=[$b]`] and [`b=[$c]`]. + +END_PGML_SOLUTION + +ENDDOCUMENT(); # This should be the last executable line in the problem.