CALCULUS

Prof.  G. Battaly, Westchester Community College

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CALC Homework:

Homework Help:                       Step-by-Step Procedures  
                                                     Powerpoint Lessions
Problems:
           

                                                  | Chapter 5 | Chapter 6 | Chapter 7 | Chapter 8 |

 

Chapter 5
Ch 5.3
Fundamental Theorum of Calculus
Ch 5.4Indefinite Integrals; Net Change  
Ch 5.5Integration by Substitution

Chapter 6
Ch 6.1Area between Two Curves
Ch 6.2Volume-Disk & Cross Section Methods

Ch 6.3Volume-Shell Method
Ch 6.5Average Value and Mean Value Theorum for Integrals

Chapter 7
Ch 7.1Integration by Parts
Ch 7.2Trigonometric Integrals
Ch 7.3Trig Substitution

Ch 7.4Partial Fractions
Ch 7.5Strategy and Review
Ch 7.6CAS, tables, and Patterns
Ch 7.7Approximate Integration

Ch 7.8 Improper Integrals

Chapter 8
Ch 8.1 Arc Length

Chapter 11
Ch 11.1 Sequences
Ch 11.2 Series
Ch 11.3 Integral Test
Ch 11.4 Comparison & Limit Tests
Ch 11.5 Alternating Series
Ch 11.6 Ratio and Root Tests
Ch 11.8 Power Series
Ch 11.9 Functions as Power Series
Ch 11.10 Taylor and Maclaurin Series

 
Homework Help:  Step-by-Step Procedures 
Word Problems     Related Rates    
Find Absolute Extrema on a Closed Interval [a,b]
Relative Extrema, Increasing & Decreasing Functions, & the First Derivative Test
 
**
Finding Relative Extrema given f '(x) (a gif animation) **
Relative Extrema, Concavity and the Second Derivative Test
Limits at Infinity for Rational Functions
Curve Sketching
Optimization Problems
Using the Limit Definition to Find Area
Integrating Rational Expressions

Solving a First Order Linear Differential Equation
Finding the Area of the Region Bounded by 2 or more Curves  
Volumes of Revolution - Disk Method

Volumes of Revolution - Which Method?

Volumes of Revolution - Shell Method
 


 

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How to Do Word Problems:

  1. Read the problem.
  2. Read the problem again, identifying what's given and what you need to find.
  3. Organize the information:  draw a diagram, construct a table, etc.
  4. Identify the unknown variables, including the appropriate rates, such as dy/dt or dA/dt, etc.
  5. Write an equation to relate the given and the to find.
  6. Solve the equation.  For related rates, this involves implicit differentiation.
  7. Check the solution:   Is "to find" found?
                                       Does solution make sense?
                                       Do numbers fit?

RELATED RATES:

  1. Identify: a "given" rate, a "to find" rate, other conditions
  2. Determine the relationship (equation) between the given and the to find. Use a diagram, if possible, or know formula.
  3. If possible, use the given information to reduce the number of variables.
  4. Differentiate implicitly with respect to time. This always involves the chain rule. For example,
  5. Substitute the given values and solve for the unknown rate.
  6. Check:

Integrating Rational Expressions:

  1. Is the denominator a monomial?
            If yes, divide and integrate.
            If no, proceed to #2.
  2. If the quotient is rewritten as a product, is the exponent (-1)?
            If yes, think   ln   and look for   du/u
            If no, think   un   and look for   un du

To Find Absolute Extrema on a Closed Interval [a,b]:

  1. Find relative extrema.
        a)  Find critical numbers [f '(c) = 0 or f '(c) undefined]
        b)  Find f(c) for all c.
  2. Evaulate function at endpoints:  find f(a) and f(b)
  3. Compare f values of relative extrema and endpoints and select:
        a)  the point (x,f(x)) with the largest f value is the absolute maximum
        b)  the point (x,f(x)) with the smallest f value is the absolute minimum

To Find Relative Extrema of a continuous function using intervals and the First Derivative Test:

  1. Find critical numbers [f '(c) = 0 or f '(c) undefined]
  2. Determine intervals for evaluation of f ' and begin the interval table: 
        a)  Locate the critical numbers along a number line containing the domain of the function.
        b)  Determine the intervals, using the critical numbers as endpoints.
  3. Continue the interval table by:
      
    a)  Selecting a test value for each interval.
        b)  Express f '(x) in factored form, and write each factor in the first column.
        c)  Find the sign of each factor in each interval and indicate the sign on the table.
        d)  For each interval, find the sign of f '(x) by determining the number of negative factors.
  4. Determine whether f(x), the original function, is increasing (when f '(x) >0) or decreasing (when f '(x) <0) on each interval.
  5. The critical value for which f(x) is increasing to the left and decreasing to the right is a relative max.  /    \
    The critical value for which f(x) is decreasing to the left and increasing to the right is a relative min.  \    /
  6. Find the corresponding f or y value for each critical value determined to be a relative max or min, and write the ordered pair (c,f(c)).

To Find Relative Extrema of a continuous function using Concavity and the Second Derivative Test:

  1. Find critical numbers [f '(c) = 0 or f '(c) undefined]
  2. Find f ''(x). 
  3. Find f ''(c) for all critical numbers.
  4. Determine the relative extrema using the Second Derivative Test:
    a)  If f ''(c) > 0, then f is concave up and f(c) is a relative min
    b)  If f ''(c) < 0, then f is concave down and f(c) is a relative max
    c)  If f ''(c) = 0, then the test fails.  (consider an Inflection Point - a point where concavity changes)

To Find the Limit at Infinity for a Rational Function, f(x) = g(x)/h(x):

  1. If degree of numerator < degree of denominator, then limit is 0.
  2. If degree of numerator = degree of denominator, then limit is ratio of leading coefficients.
  3. If degree of numerator > degree of denominator, then limit does not exist.
  4. For the first 2 cases, where a limit k exists, then y = k is a horizontal asymptote.  Be sure to consider the limit approaching both positive infinity and negative infinity.

How to Do Optimization Problems:

1.  Read the problem carefully, and identify what's given and what you need to find.
2.  Organize the info: draw a diagram, construct a table, etc.
3.  Identify the unknown variables; add to diagram or table.
4.  Write an equation to relate the given and the to find.
5.  Reduce the number of variables to 2.
6.  Find the derivative and Critical Numbers.
7.  Test the critical numbers for max or min, using 1st derivative or 2nd derivative test, and state solution.
8.  Check the solution: Is "to find" found? Does solution make sense? Do numbers fit? 


 

Using the Limit Definition to Find Area:

  1. Sketch the function f and indicate the region on the interval [a,b].
  2. Is f continuous and nonnegative on the interval?  If yes, then:

   5.   Substitute ci for x in the function and proceed with the sums, using Summation Formulas, Theorum 4.2.
   6.   Evaluate the limits at infinity.


Curve Sketching:        DRIve A Car NEXt TRIP

D:  Domain,        R:  Range,        I:  Intercepts,       A:  Asymptotes,       
CN:  Critical Numbers,         EXTR:  Extrema,         IP:  Inflection Points


Solving a First Order Linear Differential Equation:

 


Finding the Area of the Region Bounded by 2 or More Curves:

  1. Sketch the curves:
  2. Use the sketch to determine which integral to use:
  3. If the bounded area contains more than one distinct region, write the area as the sum of the areas of each distinct region.

  4. Limits of integration:

Volumes of Revolution - Disk Method

  1. Sketch the curves and identify the region, using the points of intersection.
  2. Locate the axis of revolution on the sketch.
  3. Decide whether to use a horizontal or vertical rectangle.  The rectangle should be perpendicular to the axis of revolution.
  4. Sketch the rectangle and determine the variable of integration.
       
         a)  If the rectangle is horizontal, then integrate with respect to y (use dy).  The integrand must be in terms of y.
            b)  If the rectangle is vertical, then integrate with respect to x (use dx).  The integrand must be in terms of x.
  5. Determine the integrand:      R2,    or    R2 - r2    ?
            a)  If the rectangle touches the axis of revolution, identify R as the length of the rectangle.  Find R in terms of the appropriate variable (see above), and use R2 as the integrand.
            b)  If the rectangle does not touch the axis of revolution, identify R as the distance of the furthest end of the rectangle from the axis of revolution and r as the distance of the closest end of the rectangle from the axis of revolution.  Use  R2 - r2  as the integrand.        

Volumes of Revolution - Which Method?

  1. Sketch the curves and identify the region, using the points of intersection.
  2. Locate the axis of revolution on the sketch.
  3. Decide whether to use a horizontal or vertical rectangle.  Select the orientation that requires the least number of separate sections.
  4. Decide whether to use the Disc Method or the Shell Method:
        
    a)  If the rectange is perpendicular to the axis of revolution, use the Disc Method.
         b)  If the rectangle is parallel to the axis of revolution, use the Shell Method.

Volumes of Revolution - Shell Method

  1. Complete Steps 1 to 4 in Volumes of Revolution, which Method? noted above.
  2. Be sure that your rectangle is parallel to the axis of revolution.
  3. Determine the variable of integration:
            a)  If the rectangle is horizontal, then integrate with respect to y (use dy).  The integrand must be in terms of y.
            b)  If the rectangle is vertical, then integrate with respect to x (use dx).  The integrand must be in terms of x.
  4. Determine the integrand:    p(x)h(x) or p(y)h(y) ?
     
      a)  If the rectangle is horizontal, identify p(y), the distance of the rectangle from the axis of revolution, and h(y), the length of the rectangle.  Use: 
       b)   If the rectangle is vertical, identify p(x), the distance of the rectangle from the axis of revolution, and h(x), the length of the rectangle.  Use:    

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Last updated: 
April 22, 2017