Calc 2
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Calculus
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Calc 2
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Calc 2
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LIMITS AND CONTINUITY
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Limits and Continuity Brief Review Limit ? intended height (y-value) of the function. Properties: add, subtract, divide, multiply, multiply constant and raise to any power. Techniques to Evaluation: Direct Substitution ? plug the x-value in?if you get a number you are done?if you get an indeterminate form?. 1.) Try to factor the expression. Cancel common factors and try direct substitution again. 2.) Try tables or graphs?.try plugging in a number close to the x-value to the right and the left. 2. 3. 4. 5. 6 7. 8.
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Ch9 SG
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372 CHAPTER 9 Mathematical Modeling with Differential Equations EXERCISE SET 9.1 1. y? = 2x2ex 3/3 = x2y and y(0) = 2 by inspection. 2. y? = x3 ? 2 sinx, y(0) = 3 by inspection. 3. (a) ?rst order; dy dx = c; (1 + x) dy dx = (1 + x)c = y (b) second order; y? = c1 cos t? c2 sin t, y?? + y = ?c1 sin t? c2 cos t+ (c1 sin t+ c2 cos t) = 0 4. (a) ?rst order; 2 dy dx + y = 2 ( ? c 2 e?x/2 + 1 ) + ce?x/2 + x? 3 = x? 1 (b) second order; y? = c1et ? c2e?t, y?? ? y = c1et + c2e?t ? ( c1et + c2e?t ) = 0 5. 1 y dy dx = x dy dx + y, dy dx (1? xy) = y2, dy dx = y2 1? xy 6. 2x+ y2 + 2xy dy dx = 0, by inspection. 7. (a) IF: ? = e3 ? dx = e3x, d dx [ ye3x ] = 0, ye3x = C, y = Ce?3x separation of variables: dy y = ?3dx, ln |y| = ?3x+ C1, y = ?e?3xeC1 = Ce?3x
Multi-Variable Calculus summary
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Series
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CALCULUS BC SERIES SERIES: PARTIAL SUMS: If (that it, the sequence of partial sums converges), then is the sum of the series so . In this case is CONVERGENT and has sum s. Thus a series converges if its sequence of partial sums converges. [Sec 11.2: p 2] EXAMPLES Does the series converge or diverge? 1. Partial sums: so, and thus the series converges and has sum = 1 2. Harmonic series: Partial sums: Thus the series diverges. [Sec 11.2: p3] 3. Here, (using partial fractions) Partial sum: Thus the series converges and has sum = 5. This is an example of a TELESCOPING SERIES.
Trig functions
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Review : Trig Functions The intent of this section is to remind you of some of the more important (from a Calculus standpoint?) topics from a trig class. One of the most important (but not the first) of these topics will be how to use the unit circle. We will actually leave the most important topic to the next section. First let?s start with the six trig functions and how they relate to each other. Recall as well that all the trig functions can be defined in terms of a right triangle. From this right triangle we get the following definitions of the six trig functions. Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion.
Continuity
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Function continuity at x=a (bridges) F(x) is continuous at x=a if must exist f(a) must be defined f(a)= Ex. Continuous: f(a) is defined f(a)= Ex. Discontinuous: f(a) is defined f(a)
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