Calculus I (AB): Miscellaneous Integrating Techniques

All of these integrals look the same, but they're not. Trust me. Try and integrate them to see why.

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(a)    This probelm requires you to complete the square in the denominator. You usually do this with an unfactorable quadratic in the denominator and an empty numerator.Once you complete the square, you can integrate to get inverse tangent (arctan). (b)    Here, you'll have to start with u-substitution. You'll have to get more devious as the problem continues by adding and subtracting the same thing in the numerator. (c)    Since the degree of the numerator is greater than or equal to that of the denominator, begin by long dividing.

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Calculus II (BC): Slope fields and Euler's Method

Given the differential equation ,
    (a)    Draw the slope field for dy/dx on the below grid, marking the correct slope segments at the indicated points.     (b)    If the solution, f,  to dy/dx contains the point (0,2), approximate f (1) using Euler's Method, with three steps of equal size.

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Solution: (a)    Your slope field should resemble the below picture (although yours won't quite be as accurate). This picture was generated by Greg Hoerst's Differential Equations program. Download a copy for your TI-83+ here. (b)

The differential equations program mentioned in part (a) also solved Euler's Method problems. Go ahead and download it! If you don't have a TI Graph Link, a copy of the problem is in Mr. Kelley's book for you to type in.