15. EXAMPLE 4, PAGE 334 An object in rectilinear motion moves along the \( x \) - axis so that its position at time \( t \geq 1 \) (in seconds) is \( x(t)=\frac{\ln t}{t} \). At what A. \( 2 e \) seconds \( t \) is the object farthest from the origin? B. 1 second C. 2 seconds D. \( e \) seconds

A. ctangle initially has dimensions 2 cm by 7 cm . All sides begin increasing in length at a rate of \( 2 \mathrm{~cm} / \mathrm{s} \). At what rate is the area of the rectangle increasing after 16 s ? Let \( \mathrm{A}, \mathrm{b} \), and h be the area, base, and height of a rectangle, respectively. Write an equation relating \( \mathrm{A}, \mathrm{b} \), and h . \( \mathrm{A}=\mathrm{b} \times \mathrm{h} \) Differentiate both sides of the equation with respect to t . \( \mathrm{dA}= \)

Determine the intervals on which the following function is continuous. \( f(x)=\frac{x-10}{x^{2}-100} \)

Consider the function \( f(x)=\frac{2 x^{4}+4 x^{3}-16 x^{2}}{x^{4}-20 x^{2}+64} \) a. Evaluate \( \lim _{x \rightarrow \infty} f(x) \) and \( \lim _{x \rightarrow-\infty} f(x) \), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each asymptote \( x=a \), evaluate \( \lim _{x \rightarrow a^{-}} f(x) \) and \( \lim _{x \rightarrow a^{+}} f(x) \).

Calculate the limit. \[ \lim _{h \rightarrow 0} \frac{4(3+h)^{2}-36}{h}=\square \text { (Simplify your answer.) } \]

Find the derivative of the following function by first expanding the expression. \[ h(x)=\left(x^{5}+1\right)^{2} \] The expanded expression is \( h(x)=\square \) (Simplify your answer.)