Use the Calculus > Taylor Series command or press Ctrl+Shift+T to find a Taylor polynomial approximation to an expression.  This command allows you to select the expansion variable, the order (maximum degree) of the approximation, and the expansion point.


Alternatively, the Taylor polynomial approximation of the expression u with respect to the expansion variable x about the expansion point a of order n can be entered by typing in an expression of the form TAYLOR(u, x, a, n).  For example,

TAYLOR(#e^x, x, 0, 5)

simplifies to

   5      4      3      2         
  x      x      x      x          
————— + ———— + ———— + ———— + x + 1
 120     24      6      2         


The order of the approximation is the number of partial derivatives used in the truncated Taylor series approximating the expression.  Some of these derivatives may simplify to 0 at the expansion point, so the number of terms and the degree of the resulting polynomial may be less than the specified order.  For example,

TAYLOR(c·SIN(x),x,0,6)

simplifies to

    5        3       
 c·x      c·x        
—————— - —————— + c·x
  120       6        


For some expressions, successive partial derivatives grow rapidly in size and hence computing time.  Thus when computing Taylor polynomial approximations, it is wise to start with a small value for the order of approximation (such as 2 or 3).  For example, compute the first, second, and third derivatives of

(x+1)/(EXP(x)+1)

and note the strongly increasing complexity and computing time for successive derivatives.


It is possible for all the derivatives of an expression to evaluate to 0 at a point, even though the expression is not a constant.  The Taylor polynomial approximation of such an expression expanded about this point is 0, no matter how large the order of approximation.  For example, verify that the fourth, fifth, and sixth order Taylor polynomial approximations of

EXP(-1/x^2)

about the point x=0 are all 0.  Then plot the above expression to see that 0 is indeed a good approximation of it near x=0.


Some expressions cannot be approximated by a Taylor polynomial beyond a certain order because the derivatives are not finite at the expansion point beyond that order.  In such cases, TAYLOR simplifies to plus-or-minus infinity, complex infinity, or ? (unknown).  For example, the 0th order Taylor polynomial approximation of SQRT(x) about the point x=0 is 0, but the first order approximation is complex infinity and the second order approximation is ? .


Functions that use Taylor series approximations to solve equations, invert functions, and solve first and second order differential equations are defined in various utility files (see Utility File LibraryG5BS2R).


Other Calculus commandsCalculus_commands 

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