A thorny chain rule problem and the fundamental theorem of calculus MC question. A thorny chain rule problem and the fundamental theorem of calculus MC question.

1. A problem ...
in a thorny chain ...
chain and thorns

2. 1. The annual interest rate on mortgages (loans for home purchases) during the
1980’s varied according to the function r(t) = 0.07 + 0.03cos (0.53t - 1.60),
where t represents the number of years since January 1, 1980. (So, for example,
r(3) ≈ 0.10 indicates a 10% mortgage interest rate on January 1, 1983.)
Bank data from the same years reveal that the number N of home mortgages
issued per year depended on the mortgage interest rate r, according to the
formula N(r) = (40 000)(0.5)r. (So, for example, N(0.05) ≈ 38 637 tells us that
when the mortgage rate was 5%, the banks issued 38 637 mortgages per year.)
(a) What was the mortgage interest rate on January 1, 1985.
(b) How many mortgages per year were issued when the interest rate was 9%?
(c) Since r (the interest rate) depends on t (time) and N (the number of
mortgages) depends on r, then N must be a function of t. Determine the rate at
which the number of mortgages was changing (with respect to time) 5 years
after January 1, 1980.

3. (c) Since r (the interest rate) depends on t (time) and N (the number of
mortgages) depends on r, then N must be a function of t. Determine the rate at
which the number of mortgages was changing (with respect to time) 5 years
after January 1, 1980.