OEF Taylor --- Introduction ---

This module actually contains 10 exercises on Taylor expansions of functions of one real variable.

Derivative I

We have a function f having a Taylor expansion

f(x) =

near . What is the derivative of order of f at the point  ?

Derivative II

Let be a real function, and suppose that we can write

(x) = .

This determines the derivative of on a certain point . What is  , and what is () ?

Estimating error I

We have a function f with a Taylor expansion

f(x)=

near 0. Given that f is differentiable to order 4 in the interval [,], and that |f(4)(x)|<, what is the maximal error if we replace f(x) by in [,] ?

Estimating error II

We have a function f with a Taylor expansion

f(x)=

near . Given that f is differentiable to order in the interval [,], and that |f()(x)|<, what is the maximal error if we replace f(x) by

in [,] ?

Estimating error III

We have a function f with a Taylor expansion

f(x)=

near 0. Given that f is differentiable to order 4, and that |f(4)(x)|<, what is the maximal value of r such that one can replace f(x) by in the interval [-r,r], while being sure that the error introduced by this replacement does not exceed  ?

Table 2

Let be a real function, with the following derivative table.

()'()''() (3)()
-
0

What is the principal part of the Taylor expansion of of order 2 near , that is, the polynomial P(x) in the Taylor expansion

(x) = P(x) + o(()2) ?

Table 3

Let be a real function, with the following derivative table.

()'()''() (3)()
-
0

What is the principal part of the Taylor expansion of of order 3 near , that is, the polynomial P(x) in the Taylor expansion

(x) = P(x) + o(()3) ?

Tangent

We have a function having a Taylor expansion

(x) =

near . Consider the position of the curve of with respect to its tangent at the point (,()). For x very close to , which one of the following 4 situations is good?

  1. is below .
  2. is above .
  3. is below at left (when x<), and above at right (when x>).
  4. is above at left, and below at right.

Value

Let be a real function, and suppose that we can write

(x) = .

This determines the value of on a certain point . What is  , and what is () ?

Value II

Let be a real function, and suppose that we can write

(x) = .

This determines the value of on a certain point . What is  , and what is () ?

Other exercises on:
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