Final answer:
To obtain an upper bound for the error of the approximation of cos(0.7), we can use Taylor's Theorem. By taking the second degree Taylor polynomial of cos(x) centered at x=0 and evaluating the expression [(0.7^3)/(3!)] * 1, we can obtain an upper bound for the error to be approximately 0.033.
Step-by-step explanation:
To obtain an upper bound for the error of the approximation, we can use Taylor's Theorem. Taylor's Theorem states that for a function f(x) that is differentiable n+1 times on an interval containing x=a, we can approximate the value of f(x) near a using a Taylor polynomial of degree n. The error of the approximation is given by the remainder term which can be upper bounded by the (n+1)th derivative of f evaluated at some point c between a and x, multiplied by [(x-a)^(n+1)]/(n+1)!. In this case, we want to approximate cos(0.7) using a Taylor polynomial centered at x=0. The Taylor series expansion of cos(x) at x=0 is 1 - (x^2)/2!, and if we take the second degree (n=2) Taylor polynomial, the error is given by [(0.7-0)^3]/(3!) * cos(c) for some c between 0 and 0.7. Since cos(c) is always between -1 and 1, the error can be upper bounded by [(0.7^3)/(3!)] * 1. Evaluating this expression, we get an upper bound for the error of the approximation of cos(0.7) to be approximately 0.033.