8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
503
Taylor Polynomials
Preview Activity 8.5 illustrates the first steps in the process of approximating complicated functions with polynomials. Using this process we can approximate trigonometric,
exponential, logarithmic, and other nonpolynomial functions as closely as we like (for
certain values of x) with polynomials. This is extraordinarily useful in that it allows us to
calculate values of these functions to whatever precision we like using only the operations
of addition, subtraction, multiplication, and division, which are operations that can be
easily programmed in a computer.
We next extend the approach in Preview Activity 8.5 to arbitrary functions at arbitrary
points. Let f be a function that has as many derivatives at a point x = a as we need.
Since first learning it in Section 1.8, we have regularly used the linear approximation P 1 (x)
to f at x = a, which in one sense is the best linear approximation to f near a. Recall that
P 1 (x) is the tangent line to f at (a, f (a)) and is given by the formula
P 1 (x) = f (a) + f
′ (a)(x − a).
If we proceed as in Preview Activity 8.5, we then want to find the best quadratic approximation
P 2 (x) = P 1 (x) + c 2 (x − a)
2
so that P 2 (x) more closely models f (x) near x = a. Consider the following calculations of
the values and derivatives of P 2 (x):
P 2 (x) = P 1 (x) + c 2 (x − a)
2
P
′
2 (x) = P
′
1 (x) + 2c 2 (x − a)
P
′′
2 (x) = 2c 2
P 2 (a) = P 1 (a) = f (a)
P
′
2 (a) = P
′
1 (a) = f
′ (a)
P
′′
2 (a) = 2c 2 .
To make P 2 (x) fit f (x) better than P 1 (x), we want P 2 (x) and f (x) to have the same
concavity at x = a. That is, we want to have
P
′′
2 (a) = f
′′ (a).
This implies that
2c 2 = f
′′ (a)
503
Taylor Polynomials
Preview Activity 8.5 illustrates the first steps in the process of approximating complicated functions with polynomials. Using this process we can approximate trigonometric,
exponential, logarithmic, and other nonpolynomial functions as closely as we like (for
certain values of x) with polynomials. This is extraordinarily useful in that it allows us to
calculate values of these functions to whatever precision we like using only the operations
of addition, subtraction, multiplication, and division, which are operations that can be
easily programmed in a computer.
We next extend the approach in Preview Activity 8.5 to arbitrary functions at arbitrary
points. Let f be a function that has as many derivatives at a point x = a as we need.
Since first learning it in Section 1.8, we have regularly used the linear approximation P 1 (x)
to f at x = a, which in one sense is the best linear approximation to f near a. Recall that
P 1 (x) is the tangent line to f at (a, f (a)) and is given by the formula
P 1 (x) = f (a) + f
′ (a)(x − a).
If we proceed as in Preview Activity 8.5, we then want to find the best quadratic approximation
P 2 (x) = P 1 (x) + c 2 (x − a)
2
so that P 2 (x) more closely models f (x) near x = a. Consider the following calculations of
the values and derivatives of P 2 (x):
P 2 (x) = P 1 (x) + c 2 (x − a)
2
P
′
2 (x) = P
′
1 (x) + 2c 2 (x − a)
P
′′
2 (x) = 2c 2
P 2 (a) = P 1 (a) = f (a)
P
′
2 (a) = P
′
1 (a) = f
′ (a)
P
′′
2 (a) = 2c 2 .
To make P 2 (x) fit f (x) better than P 1 (x), we want P 2 (x) and f (x) to have the same
concavity at x = a. That is, we want to have
P
′′
2 (a) = f
′′ (a).
This implies that
2c 2 = f
′′ (a)