Question

1. evaluate the following limit:

\\( \lim _ { x \
ightarrow \infty } \left( e ^ { 2 x } + x \
ight) ^ { \frac { 1 } { x } } \\)
(a) 2
(b) 1
(c) \\( \infty \\)
(d) e
(e) \\( e ^ { 2 } \\)

1. a spherical snowball is melting such that its radius decreases at a rate of \\( \frac { 1 } { 2 } \mathrm { cm } / \mathrm { min } \\). at what rate is the

surface area decreasing when the radius is 3 cm?
(a) the surface area is decreasing at the rate of \\( 3 \pi \\) square centimeters per minute.
(b) the surface area is decreasing at the rate of \\( 6 \pi \\) square centimeters per minute.
(c) the surface area is decreasing at the rate of \\( \frac { 1 } { 6 \pi } \\) square centimeters per minute.
(d) the surface area is decreasing at the rate of \\( \frac { 1 } { 12 \pi } \\) square centimeters per minute.
(e) the surface area is decreasing at the rate of \\( 12 \pi \\) square centimeters per minute.

Explanation:

Step1: Recall the formula for the surface area of a sphere

The surface area formula of a sphere is \(S = 4\pi r^{2}\).

Step2: Differentiate the surface - area formula with respect to time \(t\)

Using the chain rule \(\frac{dS}{dt}=\frac{dS}{dr}\cdot\frac{dr}{dt}\).
Since \(\frac{dS}{dr} = 8\pi r\) and \(\frac{dr}{dt}=-\frac{1}{2}\) (negative because the radius is decreasing).

Step3: Substitute \(r = 3\) into the derivative formula

\(\frac{dS}{dt}=8\pi r\cdot\frac{dr}{dt}\).
Substitute \(r = 3\) and \(\frac{dr}{dt}=-\frac{1}{2}\) into the formula: \(\frac{dS}{dt}=8\pi\times3\times(-\frac{1}{2})\).
Calculate \(8\pi\times3\times(-\frac{1}{2})=- 12\pi\). The negative sign indicates that the surface area is decreasing.

Answer:

e. The surface area is decreasing at the rate of \(12\pi\) square centimeters per minute.