Consider a function f and a point a . Suppose there is a number L such that the linear function g g ( x ) = f ( a ) + L ( x − a ) is a good approximation to f . By good approximation, we mean that lim x → a E L ( x ) x − a = 0. where E L ( x ) is the approximation error defined by f ( x ) = g ( x ) + E L ( x ) = f ( a ) + L ( x − a ) + E L ( x ) . Show that f is differentiable at x = a and that f ′( a ) = L . Thus the tangent line approximation is the only good linear approximation.
Consider a function f and a point a . Suppose there is a number L such that the linear function g g ( x ) = f ( a ) + L ( x − a ) is a good approximation to f . By good approximation, we mean that lim x → a E L ( x ) x − a = 0. where E L ( x ) is the approximation error defined by f ( x ) = g ( x ) + E L ( x ) = f ( a ) + L ( x − a ) + E L ( x ) . Show that f is differentiable at x = a and that f ′( a ) = L . Thus the tangent line approximation is the only good linear approximation.
Consider a function f and a point a. Suppose there is a number L such that the linear function g
g(x) = f(a) + L(x − a)
is a good approximation to f. By good approximation, we mean that
lim
x
→
a
E
L
(
x
)
x
−
a
=
0.
where EL(x) is the approximation error defined by
f(x) = g(x) + EL(x) = f(a) + L(x − a) + EL(x).
Show that f is differentiable at x = a and that f′(a) = L. Thus the tangent line approximation is the only good linear approximation.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Assume that functions ƒ and g are differentiable with ƒ(2) = 3, ƒ′(2) = -1, g(2) = -4, and g′(2) = 1. Find an equation of the line perpendicular to the graph of F(x) = (ƒ(x) + 3) /(x - g(x)) at x = 2.
Consider the graph of the function f given below.
f(x)
-2
-4
B) If h(x) =
2
0
2
X
4
A) If g(x) = xƒ(x), calculate g' (3). [Select]
x²
ƒ(x)'
C) If i(x) = x² ƒ(x), calculate ¿" (−1). [Select]
calculate h' (−1). [Select]
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