Rule of 70 Bankers use the Rule of 70, which says that if an account increases at a fixed rate of p%/yr, its doubling time is approximately 70/p. Use linear approximation to explain why and when this is true.

Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 75) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?

Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy xy < 1/2 in order for the area condition to be met. Then argue that the required probability is: 1/2 + ∫[1/2 to 1] (dx / 2x) and evaluate the integral.

37–56. Integrals Evaluate each integral.

∫ cosh 2x dx

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?