3 - 1 © 2012 Pearson Education, Inc.. All rights reserved. Chapter 3 The Derivative
3 - 2 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.1 Limits
3 - 3 © 2012 Pearson Education, Inc.. All rights reserved.
3 - 4 © 2012 Pearson Education, Inc.. All rights reserved. Figure 2
3 - 5 © 2012 Pearson Education, Inc.. All rights reserved. Notation *from Spivak’s Calculus
3 - 6 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn Solution: since The numerator also approaches 0 as x approaches −3, and 0/0 is meaningless. For x ≠ − 3 we can, however, simplify the function by rewriting the fraction as Now
3 - 7 © 2012 Pearson Education, Inc.. All rights reserved. Left and Right
3 - 8 © 2012 Pearson Education, Inc.. All rights reserved. Left and Right What can you say about lim f(x) as x 10 if lim f(x) as x 10 - (from the left) is 5 lim f(x) as x 10 + (from the right) is 5 ?
3 - 9 © 2012 Pearson Education, Inc.. All rights reserved. infinity lim 1/x as x infinity ?
© 2012 Pearson Education, Inc.. All rights reserved. Two Tools with Limits
© 2012 Pearson Education, Inc.. All rights reserved. Your Turn 5 Suppose and find Solution:
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© 2012 Pearson Education, Inc.. All rights reserved. Your Turn 8 Solution: Here, the highest power of x is x 2, which is used to divide each term in the numerator and denominator.
© 2012 Pearson Education, Inc.. All rights reserved. Your Turn
© 2012 Pearson Education, Inc.. All rights reserved. Section 3.2 Continuity
© 2012 Pearson Education, Inc.. All rights reserved. Figure 14
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© 2012 Pearson Education, Inc.. All rights reserved. Figure
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© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved.
© 2012 Pearson Education, Inc.. All rights reserved. Your Turn 1 Find all values x = a where the function is discontinuous. Solution: This root function is discontinuous wherever the radicand is negative. There is a discontinuity when 5x + 3 < 0
© 2012 Pearson Education, Inc.. All rights reserved. Your Turn 2 Find all values of x where the piecewise function is discontinuous. Solution: Since each piece of this function is a polynomial, the only x-values where f might be discontinuous here are 0 and 3. We investigate at x = 0 first. From the left, where x-values are less than 0, From the right, where x-values are greater than 0 Continued
© 2012 Pearson Education, Inc.. All rights reserved. Your Turn 2 Continued Because the limit does not exist, so f is discontinuous at x = 0 regardless of the value of f(0). Now let us investigate at x = 3. Thus, f is continuous at x = 3.
© 2012 Pearson Education, Inc.. All rights reserved. Figure 20
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© 2012 Pearson Education, Inc.. All rights reserved. Section 3.3 Rates of Change
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© 2012 Pearson Education, Inc.. All rights reserved. Figure 23
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© 2012 Pearson Education, Inc.. All rights reserved. Figure 25
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© 2012 Pearson Education, Inc.. All rights reserved. Section 3.4 Definition of the Derivative
© 2012 Pearson Education, Inc.. All rights reserved. Figure 27
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© 2012 Pearson Education, Inc.. All rights reserved. Figure 38
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© 2012 Pearson Education, Inc.. All rights reserved. Figure 39
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© 2012 Pearson Education, Inc.. All rights reserved. Figure 40
© 2012 Pearson Education, Inc.. All rights reserved. Figure
© 2012 Pearson Education, Inc.. All rights reserved. Figure 43
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© 2012 Pearson Education, Inc.. All rights reserved. Section 3.5 Graphical Differentiation
© 2012 Pearson Education, Inc.. All rights reserved. Figure 45
© 2012 Pearson Education, Inc.. All rights reserved. Figure 46
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© 2012 Pearson Education, Inc.. All rights reserved. Chapter 3 Extended Application: A Model for Drugs Administered Intravenously
© 2012 Pearson Education, Inc.. All rights reserved. Figure 55
© 2012 Pearson Education, Inc.. All rights reserved. Figure 56
© 2012 Pearson Education, Inc.. All rights reserved. Figure 57