Understanding growth rates in functions is crucial for analyzing changes in output relative to input. This concept is pivotal in calculus, with applications in biology, economics, and more. It involves calculating average and instantaneous growth rates using differences and derivatives, comparing long-term behavior, and exploring the remarkable growth rates of exponential functions. Practical examples demonstrate how to compare growth rates effectively.
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1
Growth Rate Definition
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2
Application of Growth Rates
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3
Linear vs Non-linear Growth
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4
To calculate the average rate of change for a function within a certain range, one uses the ______, represented as G = (f(x2) - f(x1)) / (x2 - x1).
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5
Derivative of g(x) = x^3 + 2x^2 - 1 using power rule
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6
Interpretation of negative growth rate
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7
Graphical depiction of derivative function
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8
L'Hôpital's Rule is used to resolve ______ forms encountered while evaluating limits, by differentiating the ______ and ______ until the limit becomes determinate.
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9
Characteristic property of e^x regarding differentiation.
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10
Comparison of growth rates: exponential vs polynomial.
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11
Applying L'Hôpital's Rule to e^x/P(x) limit.
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12
To show that the function ______ grows more rapidly than ______ as x becomes very large, one evaluates the limit of their quotient after differentiation.
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