Use Green's Theorem to prove the plane case of Theorem 14.3D; that is, show that $\partial N / \partial x=\partial M / \partial y$ implies that $\oint_C M d x+N d y=0$, which implies that $\mathbf{F}=M \mathbf{i}+N \mathbf{j}$ is conservative.

Expand the below expression using the binomial theorem.

$$(2 x-5)^3$$

Use the binomial theorem to expand each expression.

$$(3 a-2 b)^5$$

Expand using the Binomial Theorem.

$$\left(x-y^3\right)^6$$

Evaluate $\int_{\gamma} f(z) d z.$ All closed curves are positively oriented. These problems may involve Cauchy's theorem, the integral formulas and/or the deformation theorem. $f(z)=2i\overline{z}| z |, \gamma$ is the line segment from 1 to –i.

Use the binomial theorem to write the binomial expansion.

$$(3 x-1)^4$$

In this exercise, determine whether Rolle's Theorem can be applied to $f$ on the closed interval $[a, b]$. If Rolle's Theorem can be applied, find all values of $c$ in the open interval $(a, b)$ such that $f^{\prime}(c)=0$.

$$f(x)=x^{2 / 3}-1,[-8,8]$$

Consider the following functions on the given interval [a, b]. a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. $f(x)=|x-1| ;[-1,4]$

Use the binomial theorem to write the binomial expansion.

$$(2 x+5)^3$$

Determine whether Rolle's Theorem can be applied to $f$ on the closed interval $[a, b]$. If Rolle's Theorem can be applied, find all values of $c$ in the open interval $(a, b)$ such that $f^{\prime}(c)=0$. \

$$f(x)=(x-2)(x+3)^2, \quad[-3,2]$$

Expand $\left(x y-2 z^{3}\right)^{6}$ using the Binomial Theorem.

Use the Fundamental Theorem of Calculus given below to evaluate the integral, or explain why it does not exist.

The Fundamental Theorem: If $f$ is continuous on $[a, b]$, then

$$\int_a^b f(x) d x=F(b)-F(a)$$

where $F$ is any antiderivative of $f$, that is, a function such that $F^{\prime}=f$.

$$\int_1^2 x^{-2} d x$$

Verify the following part of Theorem $6.2 .2$ :

$$\|\mathbf{u}\| \geq 0$$

Write a formal proof of the theorem. If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.