CALCULUS

Prof.  G. Battaly, Westchester Community College

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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?

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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:

• Did you include units?
• Does the answer make sense?
• If there were angles involved, did you use radians?

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

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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
   

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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)).

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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)

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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.

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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 c_i 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:
   • Find the points of intersection of the curves.
   • If the curves are close and orientation is difficult to determine, substitute values between those of the points of intersection to determine which is above (or to the right of) the other.
2. Use the sketch to determine which integral to use:

   • If each curve passes the vertical line test in the bounded region, use vertical rectangles, the x variable, and the integral:  
      

   • If a curve fails the vertical line test but passes the horizontal line test in the bounded region, use horizontal rectangles, the variable y, and the integral:
     
     

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:
   • Use the coordinates of the points of intersection.
   • If x = k₁  or y = k₂  is given this may be one of the limits.

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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.        

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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 horizontal, 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:  November 17, 2008