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

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

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 [,] ?

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 [,] ?

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 ?

() | '() | ''() | ^{(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

() | '() | ''() | ^{(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

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?

- is below .
- is above .
- is below at left (when x<), and above at right (when x>).
- is above at left, and below at right.

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

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

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