💾 Archived View for rawtext.club › ~jmq › teaching › calculus2 › index.gmi captured on 2024-06-16 at 12:33:12. Gemini links have been rewritten to link to archived content
⬅️ Previous capture (2023-01-29)
-=-=-=-=-=-=-
Starting from the definition of the Riemann integral, this course explores some practical applications of integration, as well as the techniques commonly employed to find closed-form antiderivatives. The focus shifts briefly to countably infinite sequences and recursion equations, thereby providing the background for a multifaceted treatment of differential equations (graphically, verbally, symbolically, and numerically).
The practice of Standards-Based Grading offers students a more granular
picture of where they stand in relation to the learning outcomes.
Everybody's individual progress will be measured using a set of 54 standards
(29 core, 25 advanced), and the gradebook will be updated after each weekly
assessment.
Here are the possible scores on each standard, and their interpretations.
Each standard will appear again the week after its inital assessment, giving students a second chance to earn a 3 or 4. Failing to earn a 3 on the second appearance will push that standard onto the individual's final exam. The goal should be to master every standard on its first or second appearance. Otherwise, the individualized final exam might contain more questions than are doable in a two-hour window.
First character: C=core, A=advanced
C 5.5SR Evaluate indefinite and definite integrals via application of the substitution rule only.
C 5.6IBP Evaluate indefinite and definite integrals via a single application of integration by parts only.
A 5.6BPS Evaluate indefinite and definite integrals via multiple applications of integration by parts or integration by parts and the substitution rule.
C 5.7TI Evaluate indefinite and definite integrals involving trigonometric functions by making appropriate use of the Substitution Rule and relevant trigonometric identities.
A 5.7TS Evaluate indefinite and definite integrals by making a trigonometric substitution.
A 5.7PF Evaluate indefinite and definite integrals by a partial fraction decomposition of the integrand.
A 5.8NI Recognize when an elementary function does not have an elementary antiderivative and cannot be evaluated using the techniques of integration we have learned.
C 5.10RI Recognize improper integrals and rewrite them as limits.
A 5.10CI Determine whether an improper integral converges or diverges.
C 5.9CA Compute left endpoint, right endpoint, midpoint rule, trapezoidal rule, and Simpson's rule approximations of definite integrals.
A 5.9AE Compute error bounds and number of intervals needed for error bound with midpoint, trapezoidal, and Simpson's rule approximations.
C 6.1AF Set up integrals representing areas bounded by functions, both with respect to x and with respect to y. (Evaluation of the integral may be done using technology.)
A 6.1AP Set up integrals representing areas bounded by parametric curves. (Evaluation of the integral may be done using technology.)
A 6.2VS Set up integrals representing volumes by the general slicing method. (Evaluation of the integral may be done using technology.)
C 6.2VR Set up integrals representing volumes of solids of revolution by the washer or disk method. (Evaluation of the integral may be done using technology.)
A 6.3CS Set up integrals representing volumes of solids of revolution by the cylindrical shell method. (Evaluation of the integral may be done using technology.)
C 6.4AC Set up integrals representing arc length of functions with respect to x and with respect to y. (Evaluation of the integral may be done using technology.)
A 6.4LP Set up integrals representing arc length of parametric curves. (Evaluation of the integral may be done using technology.)
C H.G Graph a polar function.
C H.A Set up integrals representing areas of polar regions. (Evaluation of the integral may be done using technology)
C 6.5AV Set up integrals representing average value of functions. (Evaluation of the integral may be done using technology.)
A 6.5MVT Apply the Mean Value Theorem of Integrals appropriately.
C 6.6WK Set up integrals representing work. (Evaluation of the integral may be done using technology.)
A 6.6CM Compute the center of mass for regions in the xy-plane.
C 6.8PB Compute the probability of various outcomes given a probability density function.
A 6.8MM Compute the mean and median for a probability density function.
C 8.1SQ Determine whether a sequence converges with justification, and find the limit of the sequence if possible.
A 8.1RS Compute the limit of a recursive sequence.
C 8.2TD Apply the Test for Divergence appropriately.
C 8.2GS Analyze geometric series for convergence or divergence.
A 8.2GA Applications of geometric series.
C 8.8TP Generate Taylor polynomials from the definition.
A 8.8ER Compute error bounds for Taylor polynomial approximations of functions.
A 8.7TS Generate Taylor series for functions from the definition.
C 8.7MS Memorize the Maclaurin series for important functions.
C 8.4RT Apply the Ratio Test appropriately.
C 8.5RI Determine the radius and interval of convergence of power series.
C 8.3IT Apply the Integral Test appropriately.
A 8.3RE Use the Remainder Estimate for the Integral Test for error bounding in approximating series by partial sums.
C 8.3CT Apply the Comparison and Limit Comparison Tests appropriately.
C 8.4AST Apply the Alternating Series Test appropriately.
A 8.4ASE Use the Alternating Series Estimation Theorem for error bounding in approximating series by partial sums.
C 8.6PS Generate power series for new functions from power series of known functions.
C 7.1DE Testing possible solutions to differential equations.
C 7.2DF Relate differential equations to direction fields, and sketch solutions curves in direction fields.
C 7.2EM Apply Euler's Method to find approximate solutions of Initial Value Problems.
C 7.3SE Solve separable differential equations.
A 7.3OT Find the orthogonal trajectories of families of curves.
A 7.3MP Solve a mixing problem.
C 7.4EX Generate exponential growth and decay models and analyze their solution curves.
A 7.4CH Solve a Newton's law of cooling or heating problem.
A 7.5LG Generate logistic growth models and analyze their solution curves.
By keeping track of your performance on each standard in
CSV (comma-separated values) format,
you can compute your course grade by running the following awk script with the CSV file as its argument.
#!/usr/bin/awk -f
BEGIN {
FS=","
CoreFail=0
CorePass=0
AdvFail=0
}
{
best=0
if ($2 == "C") {
for (i=3; i<=NF; i++) {
best = (best > $i ? best : $i)
}
CoreSum += best
if (best < 3) {
CoreFail += 1
unmasteredC[CoreFail] = $1
}
else {
CorePass += 1
}
} else if ($2 == "A") {
for (i=3; i<=NF; i++) {
best = (best > $i ? best : $i)
}
AdvSum += best
if (best < 3) {
AdvFail += 1
unmasteredA[AdvFail] = $1
}
}
}
END {
AdvMean = AdvSum/23
printf("Number of Core Standards Mastered: %2i\n",CorePass);
printf("Average on Advanced Standards: %1.3f\n",AdvMean);
if (CorePass >= 29) {
if (AdvMean < 2) {
mygrade="C"
} else if (AdvMean < 3) {
mygrade="B"
} else {
mygrade="A"
}
} else {
if (CoreSum/29 < 2) {
mygrade="F"
} else {
mygrade="D"
}
}
print "Core standards still unmastered: "
for (x in unmasteredC) {
printf("%6s\t",unmasteredC[x]);
}
print "\nAdvanced standards still unmastered: "
for (x in unmasteredA) {
printf("%6s\t",unmasteredA[x]);
}
printf("\nOverall course grade: %1s\n",mygrade);
}