Difference between revisions of "MAT1223"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
 
(5 intermediate revisions by the same user not shown)
Line 33: Line 33:
  
  
|Week 2
+
|Week 1/2
  
 
||
 
||
Line 59: Line 59:
  
  
|Week 2  
+
|Week 2/3
  
 
||
 
||
Line 87: Line 87:
  
  
|Week 3  
+
|Week 3
  
 
||
 
||
Line 108: Line 108:
 
* Calculate the volume of a solid of revolution by using the method of cylindrical shells.
 
* Calculate the volume of a solid of revolution by using the method of cylindrical shells.
 
* Compare the different methods for calculating a volume of revolution.
 
* Compare the different methods for calculating a volume of revolution.
 
|-
 
 
 
|Week 3
 
||
 
 
<div style="text-align: center;">2.4</div>
 
 
||
 
 
 
[[Arc Length and Surface Area]]
 
 
||
 
 
* [[Differentiation Rules]] <!-- 1214-3.3 -->
 
* [[The Fundamental Theorem of Calculus]] <!-- 1214-5.3 -->
 
* [[Integration by Substitution]] <!-- 1224-1.5 -->
 
 
||
 
 
* Determine the length of a plane curve between two points.
 
* Find the surface area of a solid of revolution.
 
  
 
|-
 
|-
Line 155: Line 132:
 
||
 
||
  
* Determine the mass of a one-dimensional object from its linear density function.
 
* Determine the mass of a two-dimensional circular object from its radial density function.
 
 
* Calculate the work done by a variable force acting along a line.
 
* Calculate the work done by a variable force acting along a line.
 
* Calculate the work done in stretching/compressing a spring.
 
* Calculate the work done in stretching/compressing a spring.
 
* Calculate the work done in lifting a rope/cable.
 
* Calculate the work done in lifting a rope/cable.
 
* Calculate the work done in pumping a liquid from one height to another.
 
* Calculate the work done in pumping a liquid from one height to another.
* Find the hydrostatic force against a submerged vertical plate.
 
  
  
Line 167: Line 141:
  
  
|Week&nbsp;5-6
+
|Week&nbsp;5
  
 
||
 
||
Line 218: Line 192:
 
* Evaluate integrals involving products and powers of sin(x) and cos(x).
 
* Evaluate integrals involving products and powers of sin(x) and cos(x).
 
* Evaluate integrals involving products and powers of sec(x) and tan(x).
 
* Evaluate integrals involving products and powers of sec(x) and tan(x).
* Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx).
 
 
* Solve problems involving applications of integration using trigonometric integrals.
 
* Solve problems involving applications of integration using trigonometric integrals.
  
Line 225: Line 198:
  
  
|Week&nbsp;6-7  
+
|Week&nbsp;6/7
  
 
||
 
||
Line 246: Line 219:
 
||
 
||
  
 +
* Recognize when to use trigonometric substitution.
 
* Evaluate integrals involving the square root of a sum or difference of two squares.
 
* Evaluate integrals involving the square root of a sum or difference of two squares.
 
* Solve problems involving applications of integration using trigonometric substitution.
 
* Solve problems involving applications of integration using trigonometric substitution.
Line 253: Line 227:
  
  
|Week&nbsp;7
+
|Week&nbsp;7/8
  
 
||
 
||
Line 274: Line 248:
 
||
 
||
  
 +
* Recognize when to use partial fraction decomposition.
 
* Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
 
* Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
 
* Recognize distinct linear factors in a rational function.
 
* Recognize distinct linear factors in a rational function.
Line 338: Line 313:
 
* Find a recursive definition of a sequence.
 
* Find a recursive definition of a sequence.
 
* Determine the convergence or divergence of a given sequence.
 
* Determine the convergence or divergence of a given sequence.
* Find the limit of a convergent sequence.  
+
* Find the limit of a convergent sequence.
 
* Determine whether a sequence is bounded and/or monotone.
 
* Determine whether a sequence is bounded and/or monotone.
 
* Apply the Monotone Convergence Theorem.
 
* Apply the Monotone Convergence Theorem.
Line 370: Line 345:
 
* Find the sum of a convergent geometric series.
 
* Find the sum of a convergent geometric series.
 
* Identify a telescoping series.
 
* Identify a telescoping series.
* Find the sum of a telescoping series.
+
* Find the sum of a convergent telescoping series.
  
 
|-
 
|-
  
  
|Week&nbsp;10-11
+
|Week&nbsp;10/11
  
 
||
 
||
Line 399: Line 374:
 
* Use the Integral Test to determine whether a series converges or diverges.
 
* Use the Integral Test to determine whether a series converges or diverges.
 
* Use the p-Series Test to determine whether a series converges or diverges.
 
* Use the p-Series Test to determine whether a series converges or diverges.
* Estimate the sum of a series by finding bounds on its remainder term.
 
  
 
|-
 
|-
  
  
|Week&nbsp;11  
+
|Week&nbsp;11
  
 
||
 
||
Line 452: Line 426:
  
 
* Use the Alternating Series Test to determine the convergence of an alternating series.
 
* Use the Alternating Series Test to determine the convergence of an alternating series.
* Estimate the sum of an alternating series.
 
 
* Explain the meaning of absolute convergence and conditional convergence.
 
* Explain the meaning of absolute convergence and conditional convergence.
 
  
 
|-
 
|-
Line 529: Line 501:
 
||
 
||
  
* Combine power series by addition or subtraction.
 
* Multiply two power series together.
 
 
* Differentiate and integrate power series term-by-term.
 
* Differentiate and integrate power series term-by-term.
 
* Use differentiation and integration of power series to represent certain functions as power series.
 
* Use differentiation and integration of power series to represent certain functions as power series.
Line 537: Line 507:
  
  
|Week&nbsp;15   
+
|Week&nbsp;14/15   
  
 
||
 
||
Line 557: Line 527:
 
* Find a Taylor or Maclaurin series representation of a function.
 
* Find a Taylor or Maclaurin series representation of a function.
 
* Find the radius of convergence of a Taylor Series or Maclaurin series.
 
* Find the radius of convergence of a Taylor Series or Maclaurin series.
* Finding a Taylor polynomial of a given order for a function.
 
* Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function.
 
  
 
|-
 
|-
  
 
|}
 
|}

Latest revision as of 09:34, 11 June 2024

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
1.5

Integration by Substitution

  • Recognize when to use integration by substitution.
  • Use substitution to evaluate indefinite integrals.
  • Use substitution to evaluate definite integrals.
Week 1/2
2.1

Area between Curves

  • Determine the area of a region between two curves by integrating with respect to the independent variable.
  • Find the area of a compound region.
  • Determine the area of a region between two curves by integrating with respect to the dependent variable.
Week 2/3
2.2

Determining Volumes by Slicing

  • Determine the volume of a solid by integrating a cross-section (the slicing method).
  • Find the volume of a solid of revolution using the disk method.
  • Find the volume of a solid of revolution with a cavity using the washer method.


Week 3
2.3

Volumes of Revolution, Cylindrical Shells

  • Calculate the volume of a solid of revolution by using the method of cylindrical shells.
  • Compare the different methods for calculating a volume of revolution.
Week 4
2.5

Physical Applications

  • Calculate the work done by a variable force acting along a line.
  • Calculate the work done in stretching/compressing a spring.
  • Calculate the work done in lifting a rope/cable.
  • Calculate the work done in pumping a liquid from one height to another.


Week 5
3.1

Integration by Parts

  • Recognize when to use integration by parts.
  • Use the integration-by-parts formula to evaluate indefinite integrals.
  • Use the integration-by-parts formula to evaluate definite integrals.
  • Use the tabular method to perform integration by parts.
  • Solve problems involving applications of integration using integration by parts.


Week 6
3.2

Trigonometric Integrals

  • Evaluate integrals involving products and powers of sin(x) and cos(x).
  • Evaluate integrals involving products and powers of sec(x) and tan(x).
  • Solve problems involving applications of integration using trigonometric integrals.


Week 6/7
3.3

Trigonometric Substitution

  • Recognize when to use trigonometric substitution.
  • Evaluate integrals involving the square root of a sum or difference of two squares.
  • Solve problems involving applications of integration using trigonometric substitution.


Week 7/8
3.4

Partial Fractions

  • Recognize when to use partial fraction decomposition.
  • Integrate a rational function whose denominator is a product of linear and quadratic polynomials.
  • Recognize distinct linear factors in a rational function.
  • Recognize repeated linear factors in a rational function.
  • Recognize distinct irreducible quadratic factors in a rational function.
  • Recognize repeated irreducible quadratic factors in a rational function.
  • Solve problems involving applications of integration using partial fractions.
Week 8
3.7

Improper Integrals

  • Recognize improper integrals and determine their convergence or divergence.
  • Evaluate an integral over an infinite interval.
  • Evaluate an integral over a closed interval with an infinite discontinuity within the interval.
  • Use the comparison theorem to determine whether an improper integral is convergent or divergent.
Week 9
5.1

Sequences

  • Find a formula for the general term of a sequence.
  • Find a recursive definition of a sequence.
  • Determine the convergence or divergence of a given sequence.
  • Find the limit of a convergent sequence.
  • Determine whether a sequence is bounded and/or monotone.
  • Apply the Monotone Convergence Theorem.
Week 10
5.2

Infinite Series

  • Write an infinite series using sigma notation.
  • Find the nth partial sum of an infinite series.
  • Define the convergence or divergence of an infinite series.
  • Identify a geometric series.
  • Apply the Geometric Series Test.
  • Find the sum of a convergent geometric series.
  • Identify a telescoping series.
  • Find the sum of a convergent telescoping series.
Week 10/11
5.3

The Divergence and Integral Tests

  • Use the Divergence Test to determine whether a series diverges.
  • Use the Integral Test to determine whether a series converges or diverges.
  • Use the p-Series Test to determine whether a series converges or diverges.
Week 11
5.4

Comparison Tests

  • Use the Direct Comparison Test to determine whether a series converges or diverges.
  • Use the Limit Comparison Test to determine whether a series converges or diverges.
Week 12
5.5

Alternating Series

  • Use the Alternating Series Test to determine the convergence of an alternating series.
  • Explain the meaning of absolute convergence and conditional convergence.
Week 12
5.6

Ratio and Root Tests

  • Use the Ratio Test to determine absolute convergence or divergence of a series.
  • Use the Root Test to determine absolute convergence or divergence of a series.
  • Describe a strategy for testing the convergence or divergence of a series.
Week 13
6.1

Power Series and Functions

  • Identify a power series.
  • Determine the interval of convergence and radius of convergence of a power series.
  • Use a power series to represent certain functions.
Week 14
6.2

Properties of Power Series

  • Differentiate and integrate power series term-by-term.
  • Use differentiation and integration of power series to represent certain functions as power series.
Week 14/15
6.3

Taylor and Maclaurin Series

  • Find a Taylor or Maclaurin series representation of a function.
  • Find the radius of convergence of a Taylor Series or Maclaurin series.