500
Chapter 4 | Applications of Derivatives
initial value problem
a problem that requires finding a function y that satisfies the differential equation dy dx = f ( x )
together with the initial condition y ( x 0 ) = y 0
process in which a list of numbers x 0 , x 1 , x 2 , x 3 … is generated by starting with a number x 0 and
iterative process
defining x n = F ( x n −1 ) for n ≥1
the limiting value, if it exists, of a function as x →∞ or x →−∞ the linear function L ( x ) = f ( a )+ f ′( a )( x − a ) is the linear approximation of f at x = a if f has a local maximum or local minimum at c , we say f has a local extremum at c if there exists an interval I such that f ( c ) ≥ f ( x ) for all x ∈ I , we say f has a local maximum at
limit at infinity linear approximation
local extremum local maximum
c
if there exists an interval I such that f ( c ) ≤ f ( x ) for all x ∈ I , we say f has a local minimum at c if f and g are differentiable functions over an interval a , except possibly at a , and
local minimum L’Hôpital’s rule
f ( x ) g ( x ) = lim x → a
f ′( x ) g ′( x ) ,
f ( x ) =0= lim x → a
g ( x ) or lim x → a
f ( x ) and lim x → a
g ( x ) are infinite, then lim x → a
lim x → a
assuming the
limit on the right exists or is ∞ or −∞
if f is continuous over [ a , b ] and differentiable over ( a , b ), then there exists c ∈ ( a , b ) such
mean value theorem
that
f ′( c ) = f ( b )− f ( a ) b − a method for approximating roots of f ( x ) =0; using an initial guess x 0 ; each subsequent f ( x n −1 ) f ′( x n −1 ) the line y = mx + b if f ( x ) approaches it as x →∞ or x →−∞ problems that are solved by finding the maximum or minimum value of a function the relative error expressed as a percentage
Newton’s method
approximation is defined by the equation x n = x n −1 −
oblique asymptote optimization problems
percentage error propagated error
related rates relative error the error that results in a calculated quantity f ( x ) resulting from a measurement error dx are rates of change associated with two or more related quantities that are changing over time given an absolute error Δ q for a particular quantity, Δ q q is the relative error. if f is continuous over [ a , b ] and differentiable over ( a , b ), and if f ( a ) = f ( b ), then there exists c ∈ ( a , b ) such that f ′( c ) =0 suppose f ′( c ) =0 and f ″ is continuous over an interval containing c ; if f ″( c ) >0, then f has a local minimum at c ; if f ″( c ) <0, then f has a local maximum at c ; if f ″( c ) =0, then the test is inconclusive since the linear approximation of f at x = a is defined using the equation of the tangent line, the linear approximation of f at x = a is also known as the tangent line approximation to f at x = a KEY EQUATIONS • Linear approximation rolle’s theorem second derivative test tangent line approximation (linearization)
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