Chapter 4 | Applications of Derivatives
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• Look for critical points to locate local extrema.
4.8 L’Hôpital’s Rule • L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form 0 0
or ∞/∞ arises.
• L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form 0 0 or ∞/∞.
• The exponential function e x grows faster than any power function x p , p >0. • The logarithmic function ln x grows more slowly than any power function x p , p >0.
4.9 Newton’s Method • Newton’s method approximates roots of f ( x ) =0 by starting with an initial approximation x 0 , then uses tangent lines to the graph of f to create a sequence of approximations x 1 , x 2 , x 3 ,…. • Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers x 0 , x 1 , x 2 ,… does not approach a finite value or it approaches a value other than the root sought. • Any process in which a list of numbers x 0 , x 1 , x 2 ,… is generated by defining an initial number x 0 and defining the subsequent numbers by the equation x n = F ( x n −1 ) for some function F is an iterative process. Newton’s method is an example of an iterative process, where the function F ( x ) = x − ⎡ ⎣ f ( x ) f ′( x ) ⎤ ⎦ for a given function f . 4.10 Antiderivatives • If F is an antiderivative of f , then every antiderivative of f is of the form F ( x )+ C for some constant C . • Solving the initial-value problem dy dx = f ( x ), y ( x 0 ) = y 0 requires us first to find the set of antiderivatives of f and then to look for the particular antiderivative that also satisfies the initial condition.
CHAPTER 4 REVIEW EXERCISES True or False ? Justify your answer with a proof or a counterexample. Assume that f ( x ) is continuous and differentiable unless stated otherwise. 525. If f (−1) =−6 and f (1) =2, then there exists at least one point x ∈ [−1, 1] such that f ′( x ) =4. 526. If f ′( c ) =0, there is a maximum or minimum at x = c . 527. There is a function such that f ( x ) <0, f ′( x ) >0, and f ″( x ) <0. (A graphical “proof” is acceptable for this answer.)
528. There is a function such that there is both an inflection point and a critical point for some value x = a .
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