Find the polar equation of the tine tangent to the polar curve $r=\cos \theta+\sin \theta$ at the origin, and then find the slope of this tangent line.

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line.

$y=2 x-x^3$ at $(1,1)$

Find the equation of the line tangent to the polar curve $r=2+3 \sin \theta$ at $\theta=\pi$.

State the slope and the y-intercept of the graph of each equation. $y=x+9$

For each of the transfer functions shown below, find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on).

$$\text { a. } T(s)=\frac{2}{s+2}$$

$$\text { b. } T(s)=\frac{5}{(s+3)(s+6)}$$

$$\text { c. } T(s)=\frac{10(s+7)}{(s+10)(s+20)}$$

$$\text { d. } T(s)=\frac{20}{s^{2}+6 s+144}$$

$$\text { e. } T(s)=\frac{s+2}{s^{2}+9}$$

$$\text { f. } T(s)=\frac{(s+5)}{(s+10)^{2}}$$

Find the limit of the following sequence or determine that the sequence diverges. $\left\{\int_{1}^{n} x^{-2} d x\right\}$

Find the slope of the tangent line to the polar curve at the given point. $r=\ln \theta$ at $\theta=e$

Find the slope of the tangent line to the polar curve for the given value of θ. r=4-3sin θ; θ=π

Sketch the region that is outside the circle r=2 and inside the lemniscate $r^2=8 \cos 2 \theta$, and find its area.

Find the area of the following region. The region inside the lemniscate $r^{2}=2 \sin 2 \theta$ and outside the circle r = 1

the graphs suggest that the limaçon r = 1 + c sinθ has an inner loop when |c| > 1. Prove that this is true, and find the values of θ that correspond to the inner loop.

(a) You may have observed that water waves advancing toward shore have their wavefronts almost always parallel to the shoreline independent of the direction of the wind? Noting the fact that the velocity of waves in water decreases as the depth of the water decreases, use Huygens' Principle to explain this phenomenon. (b) To make the analysis of (a) more specific, assume that waves, initially traveling in the x direction, enter a region in which their speed v has a systematic variation with the distance y perpendicular to the direction of travel. (For example, x could be the direction parallel to the shore, and y would then be the direction perpendicular to the shore.) Show that the direction of the waves will begin to follow the arc of a circle of radius R, such that $R=\frac{v}{d v / d y}$

Find the trigonometric Fourier series for a periodic function f(t) that is equal to $t^{2}$ over the period from t=0 to t=2.

List the minor matrix $M_{ij}$, and calculate the cofactor $A_{i j}=(-1)^{i+j} \operatorname{det}\left(M_{i j}\right)$ for the matrix A given by

$$A=\left[\begin{array}{rrrr} 2 & -1 & 3 & 1 \\ 4 & 1 & 3 & -1 \\ 6 & 2 & 4 & 1 \\ 2 & 2 & 0 & -2 \end{array}\right]$$

$M_{11}$