Find the sum of the given series, or show that the series diverges (possibly to infinity or negative infinity). $\sum_{n=1}^{\infty} \frac{2}{n+1}$

Find dy/dx by implicit differentiation. then find the slope of the graph at the given point. $x=y^{3}-7 y^{2}+2$, (-4, 1)

(a) $\lim _{x \rightarrow 0^{-}} \frac{1}{x}=$ _____. (b) $\lim _{x \rightarrow 0^{+}} \frac{1}{x}=$ _____. (c) $\lim _{x \rightarrow 0^{+}} \ln x=$ _____.

Find the modulus $r=|z|$ and the principal argument $\theta=\operatorname{Arg}(z)$ of each given complex number z, and express z in terms of r and $\theta$. $z=3 i$.

Calculate $L\left(f, P_{n}\right) \text { and } U\left(f, P_{n}\right)$ for the given function f over the given interval $[a, b]$, where $P_{n}$ is the partition of the interval into n subintervals of equal length $\Delta x=(b-a) / n$. Show that $\lim _{n \rightarrow \infty} L\left(f, P_{n}\right)=\lim _{n \rightarrow \infty} U\left(f, P_{n}\right)$. Hence, f is integrable on [a, b]. (Why?) What is $\int_{a}^{b} f(x) d x$? $f(x)=e^{x},[a, b]=[0,3]$

Use integration by parts to prove the formula. ∫e^ax sin bx dx = e^ax(a sin bx - b cos bx) / a²+b² + C

Find examples to show that if the series

$$\sum u _ { k }$$

and

$$\sum v _ { k }$$

both diverge, then the series

$$\sum \left( u _ { k } + v _ { k } \right)$$

and

$$\sum \left( u _ { k } - v _ { k } \right)$$

may either converge or diverge.

Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. $f(x)=-x+\cos 3 \pi x$, $\left[0, \frac{\pi}{6}\right]$

Use integration by parts to prove the formula. Assume that n is a positive integer.) ∫ e^ax cos bx dx = e^ax(a cos bx + b sin bx) / a² + b² + C

Find the work done by the given vector field F in moving a particle from the point A to the point B whose coordinates are as indicated. $\mathbf{F}=2 x y \cos z \mathbf{i}+x^{2} \cos z \mathbf{j}-x^{2} y \sin z \mathbf{k}; A(1,1, \pi / 2), B(2,3,0)$

Write each system of linear differential equations in matrix notation. $d x / d t=x-2, \quad d y / d t=2 y+3 x-1$

Show that the function $f(x)=\frac{a x+b}{a x+d}$ has no critical points and no inflection points.

Find an equation of the tangent to the given curve at the given point. $2 x+y-\sqrt{2} \sin (x y)=\pi / 2 \text { at }\left(\frac{\pi}{4}, 1\right)$

Each limit represents the derivative of a function f at some number c. Determine f and c in each case. $\lim _{x \rightarrow 0} \frac{3 x^{3}-2 x}{x}$