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Answer :
The derivative of the consumption function, p'(t), is found by using the power rule of differentiation, giving 272t + 1,077. The annual consumption and instantaneous rate of consumption in 2026 is given by the values of p(16) and p'(16) respectively. This indicates the amount of consumption and its rate of change in that year.
To find p'(t), we'll need to employ the power rule for differentiation, which is a powerful tool in calculus. The power rule states that the derivative of t^n, where n is any real constant, is n*t^(n-1). Therefore, applying this rule to the function p(t) = 136t² + 1,077t + 14,913, the derivative p'(t) = 272t + 1,077.
For part (B), we need to find the annual consumption and the instantaneous rate of change in the year 2026. Since t=0 corresponds to the year 2010, t=16 would correspond to 2026. Substituting t=16 in the function p(t), we get p(16) which gives us the total consumption in that year. The instantaneous rate of change at that year would be given by the derivative p'(16).
A verbal interpretation would be that the annual consumption in the year 2026 is given by p(16), and the rate at which this consumption is changing in that year is given by p'(16). The latter indicates the sensitivity of the consumption with respect to time – i.e., how rapidly consumption is increasing or decreasing at that specific point in time.
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