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


c. ln (x + y) = ln x + ln y

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

d. Plot a graph of V(t) for 0 ≤ t ≤ 15. What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin?

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:

c. (exp(x))ᵖ = exp(px), p rational

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln(1 + √2) = −ln(−1 + √2)

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

c. csch⁻¹ 5

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.

c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.