#
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. 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. 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?

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

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- Description: collection of exercises on Taylor expansions of real functions. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, analysis, calculus, Taylor, function, derivative, integral, differential equation