Problems

In Problems 1-4, find the derivative.

In Problems 9-12 use the first and second derivatives to sketch the curve.

In Problems 13-20 evaluate the integral.

13           

21             Prove the identity tanh² x + sech² x = 1.

22            Find the length of the curve y = cosh x, - 1 ≤ x ≤ 1.

23            Find the volume of the solid formed by rotating the curve y = cosh x, 0 ≤ x ≤ 1, about (a) the x-axis, (b) the y-axis.

24            Find the surface area generated by rotating the curve y = coshx,0 ≤ x ≤ 1, about (a) the x-axis, (b) the y-axis.

25            Money is received at the constant rate of 5000 dollars per year and earns interest at the annual rate of 10%. How much is accumulated in 20 years?

26            Money is received at the rate of 20 - 2t dollars per year and earns interest at the annual rate of 8%. How much capital is accumulated between times t = 0 and t = 10?

27            A firm initially loses (and borrows) money but later makes a profit, and its net rate of profit is

f(t) = 10⁶(t - 1)

dollars per year. All interest rates are at 10%. Starting at t = 0, find the net capital accumulated after (a) 2 years, (b) 3 years.

28            A firm in a fluctuating economy receives or loses money at the rate f(t) = sin t. Find the net capital accumulated between times f = 0 and f = 2π if all interest is at 10%. ,

29           The present value of z dollars t years in the future is the quantity y = ze^-rt, where r is the interest rate. This is because y = ze^-rt dollars today will grow to ye^rt = z dollars in f years. Use the Infinite Sum Theorem to justify the following formula for the present value V of all future profits where f(f) is the profit per unit time.

V= ∫ f(t)e^-rtdt.