Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Find the limits in Exercises 1–6.

3. lim(x→0⁺) (cox(√x))^(1/x)

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

Find the limits in Exercises 1–6.

5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

In Exercises 9 and 10, use implicit differentiation to find dy/dx.

9. y^e^x = x^y + 1