Indeterminate Forms Show that the indeterminate forms $0^0$, $\infty^0$, and $1^{\infty}$ do not always have a value of 1 by evaluating each limit.

$$\lim\limits_{x \rightarrow 0}(x+1)^{(\ln 2) / x}$$

Use a graphing utility to graph the function $y=e^{-x^2}$.

In this exercise, find the present value $P$ of a continuous income flow of $c(t)$ dollars per year if

$$P=\int_0^{t_1}c(t)e^{-rt}dt$$

where $t_1$ is the time in years and $r$ is the annual interest rate compounded continuously.

$$c(t)=10,000+4000t, r=5\%,t_1=10$$

A model for the ability $M$ of a child to memorize, measured on a scale from $0$ to $10$, is given by $M=1+1.6t\ln t,\quad 0<t\leq4$, where $t$ is the child's age in years. Find the average value of this model

between the child's third and fourth birthdays.

A model for the ability $M$ of a child to memorize, measured on a scale from $0$ to $10$, is given by $M=1+1.6t\ln t,\quad 0<t\leq4$, where $t$ is the child's age in years. Find the average value of this model

between the child's first and second birthdays.

Prove the following generalization of the Mean Value Theorem. If $f$ is twice differentiable on the closed interval $[a, b]$, then

$$f(b)-f(a)=f^{\prime}(a)(b-a)-\int_a^b f^{\prime \prime}(t)(t-b) d t.$$

In this task, rewrite the improper integral as a proper integral using the given $u$-substitution. Then use the Trapezoidal Rule with $n=5$ to approximate the integral.

108. $\displaystyle\int_0^1 \frac{\cos x}{\sqrt{1-x}} d x, \quad u=\sqrt{1-x}$

A damping force affects the vibration of a spring so that the displacement of the spring is given by $y=e^{-4t}(\cos2t+5\sin2t)$. Find the average value of $y$ on the interval from $t=0$ to $t=\pi$.

In this task, rewrite the improper integral as a proper integral using the given $u$-substitution. Then use the Trapezoidal Rule with $n=5$ to approximate the integral.

$$\displaystyle\int_0^1 \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x}$$

Prove that if $f(x) \geq 0, \lim\limits_{x \rightarrow a} f(x)=0$, and $\lim\limits_{x \rightarrow a} g(x)=-\infty$, then $\lim\limits_{x \rightarrow a} f(x)^{g(x)}=\infty$.

Prove that if $f(x) \geq 0, \lim\limits_{x \rightarrow a} f(x)=0$, and $\lim\limits_{x \rightarrow a} g(x)=\infty$, then $\lim\limits_{x \rightarrow a} f(x)^{g(x)}=0$.

Find the centroid of the region bounded by the graphs of $f(x)=x^2, g(x)=2^x, x=2$, and $x=4$.

Find the volume of the solid generated by revolving the unbounded region lying between $y=-\ln x$ and the $y$-axis $(y \geq 0)$ about the $x$-axis.

Consider the limit $\lim\limits_{x \rightarrow 0^{+}}(-x \ln x)$.

Use a graphing utility to verify the result of previous part.