Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


b. ln 0 = 1

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.

b. Find the function that gives the amount of oil consumed between t=0 and any future time t.

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:

b. exp(x − y) = exp(x) / exp(y)

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

61–62. Points of intersection and area

b. Compute the area of the region described.

f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 - e⁻ᵗ/ᶜ), where a and c are positive constants, t is measured in seconds, and v has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, 26, Sep 1973)

b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

A slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4 / (t + 1) mi/hr and v(t) = 4e^(−t/2) mi/hr, respectively, for t ≥ 0.

b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?