Find the values of p for which each integral converges. a. ∫12 d x x(lnx)p b. ∫2∞ d x x(lnx)p a. For what values of p will ∫12 d x x(lnx)p converge? Select the correct choice below and fillinthe answer boxtocomplete your choice. A. p > ◻ B. p ◻ C. p
The parametric equations and parameter interval for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = cos( 5π4 − t), y = sin( 5π4 − t),0≤ t ≤π2 The Cartesian equation for the particle is ◻ . Choose the correct graph that represents this motion. A. B. C. D.
Investigate lim x → ∞ ln( x +9) lnx and lim x → ∞ ln( x +978) lnx . Then use l'Hôpital's rule to explain what is found. What does this say about the relative rates at which the functions f(x)= ln( x + a)and g(x)= lnx grow? Investigate the following limits using a graphing utility. lim x → ∞ ln( x +9) lnx = ◻ lim x → ∞ ln( x +978) lnx = ◻ (Type an integer or a simplified fraction.) Use l'Hôpital's rule to evaluate the given limits. lim x → ∞ ln( x +9) lnx = ◻ lim x → ∞ ln( x +978) lnx = ◻ (Type an integer or a simplified fraction.) What can be said about the relative rates at which the functions f(x)= ln( x + a)and g(x)= lnx grow for any positive real number a? Choose the correct answer below. A. The function f(x)= ln( x + a)grows slower than g(x)= lnx as x → ∞ . B. The functions f(x)= ln( x + a)and g(x)= lnx grow at the same rate as x → ∞ . C. The function f(x)= ln( x + a)grows faster than g(x)= lnx as x → ∞ .
Assuming that the equation defines x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t)at the given value of t . x3+3t2 = 13,2y3
Suppose that the bacteria in a colony grow unchecked according to the Law of Exponential Change. The colony starts with 1 bacterium and triples in number every 12 minutes. How many bacteria will the colony contain at the end of 24 hours? At the end of 24 hours, the colony will contain bacteria. (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to one decimal place as needed.)