Find the exact length of the curve $y=\ln (1-x^2)$ for $0\le x\le \frac{1}{3}$.

Which of the following best describes the remainder estimate for the integral test when determining the convergence of a series $\sum \underset{\infty }{n=1}{a}_n$ with $a_n=f\left(n\right)$, where $f$ is positive, continuous, and decreasing for $x\ge 1$?

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $f\left(x\right)$ at a point $x=a$?

Approximate the sum of the series $\sum _{n=1}\infty \frac{{-}^{n}6n}{n!}$ correct to four decimal places.

Given the double integral $\int {\int }_{y=0}{y=2}^{}{\int }_{x=y^2}{x=4}^{}f(x,y) dx dy$, which of the following represents the correct limits of integration after changing the order of integration?

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.

limₕ→₀ (1 + 2h)^{1/h}

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.

limₕ→₀ (1 + 3h)^{2/h}

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)0}-4x+2$