Find the orthogonal trajectories. First guess from the plot of the given curves. Plot or sketch some curves and trajectories. (Show the details of your work.) $y=c x^{3 / 2}$

Drug effectiveness decreases over time. If, each hour, a drug is only 90% as effective as the previous hour, at some point the patient will not be receiving enough medication and must receive another dose. If the initial dose was 200 mg and the drug was administered 3 hr ago, the expression $200(0.90)^{3}$, which equals 145.8, represents the amount of effective medication still in the system. (The exponent is equal to the number of hours since the drug was administered.) The amount of medication still available in the system is given by the function $f(t)=200(0.90)^{t}$. In this model, t is in hours and f(t) is in milligrams. How long will it take for this initial dose to reach the dangerously low level of 50 mg? Round the answer to the nearest tenth.

At what points do the curves $r=1-\cos \theta \text { and } r=1-\sin \theta$ intersect? At what angle do the curves intersect at each of these points?

Sketch the region bounded by the given lines and curves. Then express the region’s area as an iterated double integral and evaluate the integral. The curves y = ln x and y = 2 ln x and the line x = e, in the first quadrant.

Name the bone regions and curves in the vertebral column.

Find the area bounded by the curves $y=x^3$ and $y=x^4$.

In this exercise, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth?

$$x=-\sqrt{4-e^{2 t}}$$

$$y=e^t$$

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$$y=x \cos \left(x^2\right), \quad y=x^3$$

What are the customary units of solubility on solubility curves?

Sketch several level curves of the function.

$$f(x, y)=e^x+y$$

a. Find all the intersection points of the following curves. b. Find the area of the entire region that lies within both curves. $r=3 \sin \theta$ and $r=3 \cos \theta$

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approxi mately) the area of the region bounded by the curves.

$$y=3x^2-2x, y=x^3-3x+4$$

Find the area bounded by the curves $y=x^n$ and $y=x^{n-1}$ (for $n>1$ ).

For the transformation $x=u \cos v, y=u \sin v$, sketch the u-curves and v-curves for the grid $\{(u, v):(u=0,1,2,3$ and $0 \leq v \leq 2 \pi)$ or $(v=0, \pi, 2 \pi$ and $0 \leq u \leq 3)\}$.