Question

(a) As an alternative to the linear Taylor polynomial, construct a linear polynomialq(x), satisfying q(a)=f(a),q(b)=f(b) for given points a and b. (b) Apply this to f(x)=eˣ with a=0 and b=1. For 0≤x≤1, numerically compare q(x) with the linear Taylor polynomial of this section.

Answer 1

To construct a linear polynomial that satisfies given conditions, use the formula q(x) = f(a) + (x-a) * [(f(b)-f(a))/(b-a)]. For f(x) = e^x, the linear polynomial q(x) is 1 + x(e-1). Compare q(x) with the linear Taylor polynomial within the given range.

Step-by-step explanation:

To construct a linear polynomial q(x) that satisfies q(a)=f(a) and q(b)=f(b), we can use the formula:

q(x) = f(a) + (x-a) * [(f(b)-f(a))/(b-a)]

For the function f(x) = e^x, with a=0 and b=1, the linear polynomial q(x) can be written as:

q(x) = f(0) + (x-0) * [(f(1)-f(0))/(1-0)]

Simplifying this equation gives:

q(x) = 1 + x(e-1)

To numerically compare q(x) with the linear Taylor polynomial, we would need the expression for the linear Taylor polynomial of f(x) = e^x. However, since the question only asks for a comparison and not the exact Taylor polynomial, we can compare the values of q(x) and f(x) within the range 0≤x≤1.