From c914eb9437b0ba4cb9a182e9919d206ed5045a2d Mon Sep 17 00:00:00 2001 From: Oscar Levin Date: Thu, 18 Sep 2025 09:23:55 -0600 Subject: [PATCH 1/3] clean up chapter 1 --- source/bookinfo.xml | 3 +- source/chap-1.xml | 8 +- source/sec-1-4-derivative-fxn.xml | 187 +------------- source/sec-1-5-units.xml | 331 +------------------------ source/sec-1-6-second-d.xml | 358 +-------------------------- source/sec-1-7-lim-cont-diff.xml | 375 +---------------------------- source/sec-1-8-tan-line-approx.xml | 262 ++------------------ 7 files changed, 65 insertions(+), 1459 deletions(-) diff --git a/source/bookinfo.xml b/source/bookinfo.xml index 07269b7..e8e7ecd 100644 --- a/source/bookinfo.xml +++ b/source/bookinfo.xml @@ -1,8 +1,7 @@ - - + \usepackage{pgfplots} \definecolor{blue}{HTML}{0000FF} diff --git a/source/chap-1.xml b/source/chap-1.xml index 44245be..e1d3b46 100644 --- a/source/chap-1.xml +++ b/source/chap-1.xml @@ -5,11 +5,11 @@ - - - + + diff --git a/source/sec-1-4-derivative-fxn.xml b/source/sec-1-4-derivative-fxn.xml index 48e4fd1..9d1e6e2 100644 --- a/source/sec-1-4-derivative-fxn.xml +++ b/source/sec-1-4-derivative-fxn.xml @@ -3,7 +3,7 @@
The derivative function - + + + + Additional Practice + + + + + Foundations - +
diff --git a/source/sec-1-5-units.xml b/source/sec-1-5-units.xml index 1096605..c6cfbfe 100644 --- a/source/sec-1-5-units.xml +++ b/source/sec-1-5-units.xml @@ -5,7 +5,7 @@
Interpreting, estimating, and using the derivative - + - - -
- Plot of the temperature function, P(t) = 212-148e^{-0.05t}. - - - - f(x)=212-148*exp(-0.05*x) - - - - - - - - - - -
- -
- Plot of the temperature function's derivative, P'(t). - - - - f(x)=148*0.05*exp(-0.05*x) - a = 1 - - - - - - - - - - -
- - - - - - - - - Summary

@@ -382,7 +65,15 @@

- + --> + + + Additional Practice + + + + + Foundations - +
diff --git a/source/sec-1-6-second-d.xml b/source/sec-1-6-second-d.xml index 94bf33e..0005203 100644 --- a/source/sec-1-6-second-d.xml +++ b/source/sec-1-6-second-d.xml @@ -2,7 +2,7 @@
The second derivative - + - -

- In calculus, we describe these possible behaviors through the language of concavity. - concavity - When a curve opens upward on a given interval, - such as the parabola y = x^2 or the natural exponential function y = e^x, - we say that the curve is concave up on that interval. - Likewise, when a curve opens down, - such as the parabola y = -x^2 or the opposite of the exponential function y = -e^{x}, - we say that the function is concave down. - Concavity is linked to both the first and second derivatives of the function. -

- -

- In Figure, - we see two functions and a sequence of tangent lines to each. - On the lefthand plot, where the function is concave up, - observe that the tangent lines always lie below the curve itself, and - the slopes of the tangent lines are increasing as we move from left to right. - In other words, - the function f is concave up on the interval shown because its derivative, - f', is increasing on that interval. - Similarly, on the righthand plot in Figure, - where the function shown is concave down, - we see that the tangent lines always lie above the curve, - and the slopes of the tangent lines are decreasing as we move from left to right. - The fact that its derivative, - f', is decreasing makes f concave down on the interval. -

- -
- At left, a function that is concave up; at right, one that is concave down. - -
- -

- We state these most recent observations formally as the definitions of the terms - concave up and concave down. -

- - - -

- Let f be a differentiable function on an interval (a,b). - Then f is concave up - concave up - on (a,b) if and only if f' is increasing on (a,b); - f is concave down - concave down - on (a,b) if and only if f' is decreasing on (a,b). -

- - - -

- Returning to Figure with the three decreasing functions, we can now say that the leftmost function is decreasing and concave up, while the function on the right is decreasing and concave down. Applying the language of concavity to the options for increasing functions shown in Figure, the leftmost function is increasing and concave up, while the function on the right is increasing and concave down. -

- -

- In Activity, we revisit Preview Activity to use these latest ideas involving the second derivative and the concavity of a function. -

- - - -

- Exploring the context of position, velocity, - and acceleration is a good way to understand how a function, - its first derivative, - and its second derivative are related to one another. - In Activity, - we can replace s, v, - and a with an arbitrary function f and its derivatives f' and f'', - and essentially all the same observations hold. - In particular, note that the following are equivalent: - on an interval where the graph of f is concave up, - f' is increasing and f'' is positive; and where the graph of f is concave down, - f' is decreasing and f'' is negative. -

- - - -

- In Subsection, we learned that for a given function f(x), the units on the first derivative function, f'(x) are units of f per unit of x. Because f''(x) = [f'(x)]', it follows that the units on f''(x) are units of f' per unit of x. In other words, the units on f''(x) are: -

-

- (units of f per unit of x) per unit of x. -

-
- As we observed in Note, it's best not to simplify the units. For example, in Activity, where s(t) represents the position of a car driving on a straight road, with t measured in minutes and s(t) in thousands of feet, the units on s'(t) are thousands of feet per minute, which represents the car's instantaneous velocity at time t. In addition, the units on s''(t) are thousands of feet per minute per minute, which represents the car's acceleration at time t. Notice that a square minute is not a meaningful quantity when thinking about motion, so it's much better to use per minute per minute. A similar observation holds from Activity, where F(t) represents the temperature of a potato in degrees Fahrenheit at time t in minutes; in that setting, the units on F''(t) are degrees Fahrenheit per minute per minute, as the second derivative measures the instantaneous rate of change of the first derivative. -

- - - - Summary @@ -424,7 +80,15 @@

- + --> + + + Additional Practice + + + + + Foundations - +
diff --git a/source/sec-1-7-lim-cont-diff.xml b/source/sec-1-7-lim-cont-diff.xml index 283991b..bce2f0c 100644 --- a/source/sec-1-7-lim-cont-diff.xml +++ b/source/sec-1-7-lim-cont-diff.xml @@ -3,7 +3,7 @@
Limits, continuity, and differentiability - + + + + Additional Practice + + + + + Foundations - +
diff --git a/source/sec-1-8-tan-line-approx.xml b/source/sec-1-8-tan-line-approx.xml index d493178..066c60a 100644 --- a/source/sec-1-8-tan-line-approx.xml +++ b/source/sec-1-8-tan-line-approx.xml @@ -3,7 +3,7 @@
The tangent line approximation - + - Introduction -

- Among all functions, linear functions are simplest. - One of the powerful consequences of a function - y = f(x) being differentiable at a point (a,f(a)) is that, - up close, - the function y = f(x) is locally linear and looks like its tangent line at that point. - In certain circumstances, - this allows us to approximate the original function f with a simpler function L that is linear: - this can be advantageous when we have limited information about f or when f is computationally or algebraically complicated. - We will explore all of these situations in what follows. -

- -

- It is essential to recall that when f is differentiable at x = a, - the value of f'(a) provides the slope of the tangent line to - y = f(x) at the point (a,f(a)). - If we know both a point on the line and the slope of the line we can find the equation of the tangent line and write the equation in point-slope form - Recall that a line with slope m that passes through - (x_0,y_0) has equation y - y_0 = m(x - x_0), - and this is the point-slope form of the equation. - . -

- - - - - - - The tangent line - tangent lineequation - -

- Given a function f that is differentiable at x = a, - we know that we can determine the slope of the tangent line to y = f(x) at - (a,f(a)) by computing f'(a). - The equation of the resulting tangent line is given in point-slope form by - - y - f(a) = f'(a)(x-a) \ \ \text{or} \ \ y = f'(a)(x-a) + f(a) - . - Note well: there is a major difference between f(a) and f(x) in this context. - The former is a constant that results from using the given fixed value of a, - while the latter is the general expression for the rule that defines the function. - The same is true for f'(a) and f'(x): - we must carefully distinguish between these expressions. - Each time we find the tangent line, - we need to evaluate the function and its derivative at a fixed a-value. -

- -
- A function y = f(x) and its tangent line at the point (a,f(a)): at left, from a distance, and at right, up close. At right, we label the tangent line function by y = L(x) and observe that for x near a, f(x) \approx L(x). - -
- -

- In Figure, - we see the graph of a function f and its tangent line at the point (a,f(a)). - Notice how when we zoom in we see the local linearity of f more clearly highlighted. - The function and its tangent line are nearly indistinguishable up close. - Local linearity can also be seen dynamically in this interactive graphic. -

- - - - - - The local linearization - local linearization - -

- A slight change in perspective and notation will enable us to be more precise in discussing how the tangent line approximates f near x = a. - By solving for y, - we can write the equation for the tangent line as - - y = f'(a)(x-a) + f(a) - - This line is itself a function of x. - Replacing the variable y with the expression L(x), we call - - L(x) = f'(a)(x-a) + f(a) - - the local linearization of f - at the point (a,f(a)). - In this notation, - L(x) is nothing more than a new name for the tangent line. - As we saw above, for x close to a, - f(x) \approx L(x). -

- - - -

- Suppose that a function - y = f(x) has its tangent line approximation given by - L(x) = 3 - 2(x-1) at the point (1,3), - but we do not know anything else about the function f. How can we estimate the value of f(x) for x near 1? -

- - -

- To estimate a value of f(x) for x near 1, such as f(1.2), - we can use the fact that f(1.2) \approx L(1.2) and hence - - f(1.2) \approx L(1.2) = 3 - 2(1.2-1) = 3 - 2(0.2) = 2.6 - . -

- - - -

- We emphasize that - y = L(x) is simply a new name for the tangent line function. - Using this new notation and our observation - that - L(x) \approx f(x) for x near a, - it follows that we can write - - f(x) \approx f(a) + f'(a)(x-a) \ \text{for} \ x \ \text{near} \ a - . -

- - - -

- From Activity, we see that - the local linearization y = L(x) is a linear function that shares two important values with the function - y = f(x) that it is derived from. - In particular, - -

    -
  • - because L(x) = f(a) + f'(a)(x-a), - it follows that L(a) = f(a); and -
    • - -
  • - because L is a linear function, - its derivative is its slope. -
    • -
- - Hence, L'(x) = f'(a) for every value of x, - and specifically L'(a) = f'(a). - Therefore, we see that L is a linear function that has both the same value and the same slope as the function f at the point (a,f(a)). -

-

- Thus, if we know the linear approximation y = L(x) for a function, - we know the original function's value and its slope at the point of tangency. - What remains unknown, however, - is the shape of the function f at the point of tangency. - There are essentially four possibilities, - as shown in Figure. -

- -
- Four possible graphs for a nonlinear differentiable function and how it can be situated relative to its tangent line at a point. - -
- -

- These possible shapes result from the fact that there are three options for the value of the second derivative: - either f''(a) \lt 0, - f''(a) = 0, or f''(a) \gt 0. - -

    -
  • - If f''(a) \gt 0, - then we know the graph of f is concave up, - and we see the first possibility on the left, - where the tangent line lies entirely below the curve. -
    • - -
  • - If f''(a) \lt 0, - then f is concave down and the tangent line lies above the curve, - as shown in the second figure. -
    • - -
  • - If f''(a) = 0 and f'' changes sign at x = a, - the concavity of the graph will change, - and we will see either the third or fourth figure. - It is possible that - f''(a) = 0 but f'' does not change sign at x = a, - in which case the graph will look like one of the first two options. - -
    • - -
  • - A fifth option (which is not very interesting) - can occur if the function f itself is linear, - so that f(x) = L(x) for all values of x. -
    • -
-

- -

- The plots in Figure - highlight yet another important thing that we can learn from the concavity of the graph near the point of tangency: - whether the tangent line lies above or below the curve itself. - This is key because it tells us whether or not the tangent line approximation's values will be too large or too small in comparison to the true value of f. - For instance, - in the first situation in the leftmost plot in Figure where f''(a) > 0, - because the tangent line falls below the curve, - we know that L(x) \le f(x) for all values of x near a. -

- - - -

- The idea that a differentiable function looks linear and can be well-approximated by a linear function is an important one that finds wide application in calculus. - For example, by approximating a function with its local linearization, - it is possible to develop an effective algorithm to estimate the zeroes of a function. - Local linearity also helps us to make further sense of certain challenging limits. - For instance, we have seen that the limit - - \lim_{x \to 0} \frac{\sin(x)}{x} - - is indeterminate, because both its numerator and denominator tend to 0. - While there is no algebra that we can do to simplify \frac{\sin(x)}{x}, - it is straightforward to show that the linearization of - f(x) = \sin(x) at the point (0,0) is given by L(x) = x. - Hence, for values of x near 0, \sin(x) \approx x, - and therefore - - \frac{\sin(x)}{x} \approx \frac{x}{x} = 1 - , - which makes plausible the fact that - - \lim_{x \to 0} \frac{\sin(x)}{x} = 1 - . -

- -

- Another important question one that we study in depth in second semester calculus regards how accurate the tangent line approximation remains as we move away from the point of tangency. We explore some related ideas in the next activity. -

- - - - - - + + + + Additional Practice + + + + + + Foundations + + + -
From 10ae427474302f826bec5bb611ce60b03b73852a Mon Sep 17 00:00:00 2001 From: Oscar Levin Date: Thu, 18 Sep 2025 10:53:13 -0600 Subject: [PATCH 2/3] add exercises to 1.8 --- source/sec-1-8-tan-line-approx.xml | 72 ++++++++++++++++++++++++++- webworkfiles/sec_1-8_prob1.pg | 78 ++++++++++++++++++++++++++++++ webworkfiles/sec_1-8_prob2.pg | 66 +++++++++++++++++++++++++ 3 files changed, 215 insertions(+), 1 deletion(-) create mode 100644 webworkfiles/sec_1-8_prob1.pg create mode 100644 webworkfiles/sec_1-8_prob2.pg diff --git a/source/sec-1-8-tan-line-approx.xml b/source/sec-1-8-tan-line-approx.xml index 066c60a..6959cf4 100644 --- a/source/sec-1-8-tan-line-approx.xml +++ b/source/sec-1-8-tan-line-approx.xml @@ -75,13 +75,83 @@ Additional Practice + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Foundations - + +

+ To be able to find the equation of a tangent line, you must know how to find the equation of a line given a point and a slope. Here are some practice exercises for this foundational skill. +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/webworkfiles/sec_1-8_prob1.pg b/webworkfiles/sec_1-8_prob1.pg new file mode 100644 index 0000000..e8483d4 --- /dev/null +++ b/webworkfiles/sec_1-8_prob1.pg @@ -0,0 +1,78 @@ +## DESCRIPTION +## College Algebra, use substitution to reduce an equation to a quadratic form +## ENDDESCRIPTION + + +## DBsubject(Calculus - single variable) +## DBchapter(Applications of differentiation) +## DBsection(Linear approximation and differentials) +## Date(05/11/2018) +## Institution(Colorado Community College System) +## Author(Eric Fleming) +## MO(1) +## KEYWORDS('Linear approximation','reciprocal') + + +########################### +# Initialization + +DOCUMENT(); + +loadMacros( +"PGstandard.pl", +"MathObjects.pl", +"AnswerFormatHelp.pl", +"PGML.pl", +"PGcourse.pl", +"parserAssignment.pl", +); + +TEXT(beginproblem()); +$showPartialCorrectAnswers = 1; + +########################### +# Setup + +Context("Numeric"); +Context()->variables->add(y => 'Real'); +parser::Assignment->Allow; +parser::Assignment->Function("f"); + +$a=non_zero_random(-10,10,1); +$b=non_zero_random(-10,10,1); +$c=non_zero_random(-10,10,1); +$d=random(2,10,1); + +$f=Formula("1/x"); +$feval=$f->eval(x=>$a); +$fprime=Formula("-1/x^2"); +$m=$fprime->eval(x=>$a); + + +$ans=Formula("$m (x-$a) +$feval"); + +########################### +# Main text + +BEGIN_PGML +It turns out that the function [`f(x) = [$f]`] has derivative [`f'(x) = [$fprime]`]. + +Find the linear approximation [`L(x)`] to [`y=[$f]`] at [`x=[$a]`]. + +[`L(x)=`][_______________________________]{$ans} + + + +END_PGML + + +############################ +# Solution + +#BEGIN_PGML_SOLUTION +#Solution explanation goes here. +#END_PGML_SOLUTION + +COMMENT('MathObject version. Uses PGML.'); + +ENDDOCUMENT(); \ No newline at end of file diff --git a/webworkfiles/sec_1-8_prob2.pg b/webworkfiles/sec_1-8_prob2.pg new file mode 100644 index 0000000..6635ca1 --- /dev/null +++ b/webworkfiles/sec_1-8_prob2.pg @@ -0,0 +1,66 @@ +#DESCRIPTION +## Calculus: Linear approximation +##ENDDESCRIPTION + + +## DBsubject(Calculus - single variable) +## DBchapter(Applications of differentiation) +## DBsection(Linear approximation and differentials) +## Date(11/17/2010) +## Institution(University of Minnesota) +## Author(Justin Sukiennik) +## Level(3) +## MO(1) +## TitleText1('Calculus: Concepts and Contexts') +## AuthorText1('Stewart') +## EditionText1('4 Custom UMTYMP Ed.') +## Section1('3.9') +## Problem1('5') +## KEYWORDS('calculus', 'derivative', 'linear approximations') + +##################################################################### +DOCUMENT(); # This should be the first executable line in the problem. + +loadMacros( + "PGstandard.pl", + "MathObjects.pl", + "PGgraphmacros.pl", + "PGcourse.pl" +); +##################################################################### + +TEXT(beginproblem()); + +$showPartialCorrectAnswers = 1; + +##################################################################### + +Context("Numeric"); + +$a = non_zero_random(-2,2,1); +$b = non_zero_random(-3,3,1); +$c = non_zero_random(-2,5,1); + +$f = Formula("x^4-$b*x^2+$c")->reduce; +$df = Formula("4x^3-2*$b*x")->reduce; +$m = Compute("4*$a^3-2*$b*$a"); +$b = Compute("($a)^4-$b*($a)^2+$c"); + +$ans = Formula("$m*(x-$a)+$b"); + +##################################################################### +Context()->texStrings; +BEGIN_TEXT +The function \(f(x) = $f\) has derivative \(f'(x) = $df\). + +Find the local linearization \(L(x)\) of \(f\) at \(x=$a.\) +$PAR +\( L(x) = \) \{ans_rule(40)\} +END_TEXT +Context()->normalStrings; + +##################################################################### + +ANS($ans->cmp()); + +ENDDOCUMENT(); # This should be the last executable line in the problem. From febdfa2b94ed84070f4a0a15b4b6ffe551d42834 Mon Sep 17 00:00:00 2001 From: Oscar Levin Date: Thu, 18 Sep 2025 11:02:23 -0600 Subject: [PATCH 3/3] prep for rs --- project.ptx | 1 + publication/publication-rs.ptx | 216 +++++++++++++++++++++++++++++++++ publication/publication.ptx | 4 +- source/bibinfo.xml | 2 +- source/bookinfo.xml | 6 +- 5 files changed, 223 insertions(+), 6 deletions(-) create mode 100644 publication/publication-rs.ptx diff --git a/project.ptx b/project.ptx index 517288c..f566c11 100644 --- a/project.ptx +++ b/project.ptx @@ -6,6 +6,7 @@ + diff --git a/publication/publication-rs.ptx b/publication/publication-rs.ptx new file mode 100644 index 0000000..773cb85 --- /dev/null +++ b/publication/publication-rs.ptx @@ -0,0 +1,216 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/publication/publication.ptx b/publication/publication.ptx index 9724d71..b9a5228 100644 --- a/publication/publication.ptx +++ b/publication/publication.ptx @@ -15,7 +15,7 @@ - + @@ -47,7 +47,7 @@ - + - Second + 0 diff --git a/source/bookinfo.xml b/source/bookinfo.xml index e8e7ecd..23567b3 100644 --- a/source/bookinfo.xml +++ b/source/bookinfo.xml @@ -44,10 +44,10 @@ AC - ac-single - active-calc-proteus + ac-companion - Active Calculus Single Variable supports an active learning approach in the first two semesters of calculus. Every section of Active Calculus Single Variable offers engaging activities for students to complete before and during class; additional exercises that challenge students to connect and assimilate core concepts; interactive WeBWorK exercises; opportunities for students to develop conceptual understanding and improve their skills at communicating mathematical idea. The text is free and open-source, available in HTML, PDF, and print formats. Ancillary materials for instructors are also available. + The Active Calculus Companion is intended to give students using Active Calculus (Single Variable) additional practice exercises, as well as help them sure up foundational skills. + Note: this book is still in early draft form. Preview Activity