Describe what is happening at each of the following plate boundaries and identify an example in each case:

(a) Convergent plate boundary:__________________________________________________________________________________________________________________________

(b) Divergent plate boundary: __________________________________________________________________________________________________________________________

(c) Transform plate boundary: __________________________________________________________________________________________________________________________

Transform each equation to a form without an $xy$-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.

$$3x^2 + 4xy = 4$$

In given exercise, use trigonometric identities to transform one side of the equation into the other $(0<\theta<\pi / 2)$.

$$\tan \theta \cos \theta=\sin \theta$$

Writing Which side of the equation below should you transform to verify the identity? Explain. ((cos² θ + tan²θ - 1)/(sin² θ)) = tan² θ

Use Definition $1$ to determine the Laplace transform of the given function.

$$f(t)= \begin{cases}1-t, & 0<t<1 \\ 0, & 1<t\end{cases}$$

Tell how to transform the first inequality to obtain the second one.

$$\begin{array}{l}\frac{r}{-3} \leq-5 \\ r \geq 15\end{array}$$

Use linearity and Table of the Laplace transforms to determine the Laplace transform of the function $f(t)=t^{3} e^{-3 t}+4 e^{-t} \cos 3 t$.

Transform to canonical form (to principal axes). Express [x1 x2]^T in terms of the new variables [y1 y2]^T.

$$3.7x_1^2+3.2x_1x_2+1.3x_2^2=4.5$$

Derive the Fourier transform for the following functions: (a) $f(t)=e^{-2 t} \cos 4 t u(t)$. (b) $f(t)=e^{-2 t} \sin 4 t u(t)$

Transform $\frac{1}{1+\sin C}+\frac{1}{1-\sin C}$ into $2 \sec ^2 C$. What values of $C$ are excluded from the domain?

Apply the Gram-Schmidt orthonormalization process to transform the set of vectors into an orthonormal basis for $R^3$.

$$\{(2, 0, 0), (1, 1, 1), (0, 1, 2)\}$$

Find the inverse Laplace transform of each function. Create a piecewise definition for your solution that doesn't use the Heaviside function. $F(s)=\frac{e^{-s}}{s-2}$

Find the exponential Fourier transform of the given f(x) and write f(x) as a Fourier integral.

$$f(x)=\left\{\begin{array}{ll} 1, & 0<x<1 \\ 0, & \text { otherwise } \end{array}\right.$$

Find the Laplace transform of the solution of each of the following initialvalue problems. $y^{\prime \prime}+3 y^{\prime}+7 y=\cos t ; \quad y(0)=0, y^{\prime}(0)-2$