Calculus Answers

2. Y = e^-2x . Sin 3x show that

d^2y/dx^2 = -13y - 4 dy/dx

Ali is trying to find the limit of a function which is expressed as L = lim _x→0+ (1 + x)^1/x . From his understanding, the quantity (1 + x), must be greater than 1 for x > 0. Furthermore, the power1 / x is going to infinity as ? approaches 0 from the right. So L is the result of taking a number greater than 1 to higher power, therefore L = ∞. On the other hand, he sees that (1 + x) is approaching 1 as x approaches 0, and 1 taken to any power whatever is 1. Ali concluded, L = 1. Help Ali by pointing out to him the error of his ways.

A right circular cone is held with its apex facing downwards and is inscribed in a sphere with fixed radius 𝑘 cm within the interval 1 cm ≤ 𝑘 ≤ 5 cm and the distance from the base of the cone to the center of the sphere is 𝑥 cm.

a. Evaluate the maximum volume of the cone by assigning 𝑘 to a value within the given interval. Discuss the difference of use between the first derivative test and second derivative test to optimize the volume.

b. Water is poured into the cone at a rate of 10 m3s−1. Find the rate at which the water level is rising when the depth of the water is (𝑥 + 𝑘) − 3.

Let f be the function f (x) = x^2 − lnx^8 where x > 1

(a) Use the sign pattern for f '(x) to determine the intervals where f rises and where f

falls

(b) Determine the coordinates of the local extreme point(s).

(c) Find f ''(x) and determine where the graph of f is concave up and where it is concave

down.