Final answer:
The question requires the calculation of a tangent's slope, the construction of a linear approximation, and the estimation of insurance costs using the linear approximation for a specific age.
Step-by-step explanation:
The student's question pertains to linearization and the estimation of costs using a given function, concepts that are common in college-level calculus and statistics. More specifically, the question involves determining the slope of a tangent line at a given point (which represents a rate of change), writing the linear approximation of a cost function at a certain value, and finally using this approximation to estimate insurance costs
For example, if the given function is c(a) = 35.5818045 - 0.19182491a and we know the y-intercept (often denoted by a in equations) is 50 and the slope (denoted by b) is 100, one could use these values to construct the linear equation of the tangent line at a = 58. By substituting this value into the function, the slope at that point would be the derivative evaluated at a = 58. The linearization would then be used to estimate the cost for the specific age of 61 years old by plugging in a = 61 into the linear equation.