Chapter 4 | Applications of Derivatives
501
L ( x ) = f ( a )+ f ′( a )( x − a )
• A differential dy = f ′( x ) dx . KEY CONCEPTS 4.1 Related Rates • To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time. • In terms of the quantities, state the information given and the rate to be found. • Find an equation relating the quantities. • Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates. • Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.
4.2 Linear Approximations and Differentials • A differentiable function y = f ( x ) can be approximated at a by the linear function L ( x ) = f ( a )+ f ′( a )( x − a ). • For a function y = f ( x ), if x changes from a to a + dx , then dy = f ′( x ) dx
is an approximation for the change in y . The actual change in y is Δ y = f ( a + dx )− f ( a ). • A measurement error dx can lead to an error in a calculated quantity f ( x ). The error in the calculated quantity is known as the propagated error . The propagated error can be estimated by dy ≈ f ′( x ) dx . • To estimate the relative error of a particular quantity q , we estimate Δ q q . 4.3 Maxima and Minima • A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. • If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point. • A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint. 4.4 The Mean Value Theorem • If f is continuous over [ a , b ] and differentiable over ( a , b ) and f ( a ) =0= f ( b ), then there exists a point c ∈ ( a , b ) such that f ′( c ) =0. This is Rolle’s theorem. • If f is continuous over [ a , b ] and differentiable over ( a , b ), then there exists a point c ∈ ( a , b ) such that f ′( c ) = f ( b )− f ( a ) b − a .
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