. Annual growth vs. continuous growth. We say that an exponential function written in the form f(t) = aekt has a continuous growth rate k (for k > 0, decay for k < 0). Consider an initial investment of $1000 at t = 0, where t is in years.

(a) Write a formula for g(t), the value of the investment at time t, assuming that it grows at a rate of 3% per year.

(b) Write a formula for h(t), the value of the investment at time t, assuming that it grows at a continuous rate of 3% per year.

(c) Which is greater, g(1) or h(1)? (Just use a calculator. Well use calculus in the last part.)

(d) For r close to zero, e r (1 + r) is a reasonable approximation. Based on your answer to the previous part, which do you expect to be greater, e r or (1 + r)? Based on this, would you prefer an annual growth rate of r or a continuous growth rate of r for your investment?

(e) Find the linear approximation of the function F(r) = e r near r = 0. Recall the second derivative tells us about the concavity of the function, therefore whether the linear approximation is an overestimate or underestimate. Relate this to the previous part.

Can you do d and e please