Final answer:
The question pertains to calculating a second-degree Taylor polynomial for a function, providing a quadratic approximation centered at a given point. This entails evaluating the function, its first and second derivatives at the centering point, and constructing the polynomial to understand the function's behavior, including tangency and rate of change.
Step-by-step explanation:
The student's question involves finding Taylor polynomials of degree two, also known as second-degree approximations, for a given function centered at a specified point.
To find a second-degree Taylor polynomial, one must calculate the function's value, its first derivative (slope), and its second derivative at the centering point. The Taylor polynomial P(x) will then be:
P(x) = f(a) + f'(a)(x - a) + ½ f''(a)(x - a)²
Where a is the centering point, and f(a), f'(a), and f''(a) are the function, its first derivative, and second derivative evaluated at a, respectively. These calculations give us important information about the function such as the tangent line to the curve, the slope of the function, and the rate of change at the point of interest.