Find a polar equation for the circle with rectangular equation $$x^2+y^2=25$$
By substituting $$r^2=x^2+y^2$$ we get $$r^2=25$$. Then we take the square root of both side resulting in $$r=5, -5$$ and since 5 and -5 result in the same circle, we are left just with the equation $$r=5$$
What is $$sin(\pi/4)$$?
If you know you're unit circle, you know this is $$\sqrt(2)/2$$, but it is also very important to know where this comes from on the unit circle. First we need to convert this from radians to degrees. $$\pi/4 \times 180/\pi = 45$$ degrees Given a triangle, we know $$sin(\theta) = opposite/hypotenuse$$. Then knowing the sides of a 45-45-90 triangle to be x-x-$$\sqrt2x$$ we get $$sin(45) = x/(\sqrt2x)$$ which reduces to $$1/\sqrt2$$. Then multiplying the top and bottom by $$\sqrt2$$ we result in our original answer $$\sqrt(2)/2$$
You are on a highway with a speed limit of 55 mph driving. A cop clocks you driving at 50 mph. Then exactly 60 minutes later, 60 miles down the road another cop clocks you driving 50 mph. The second cops pulls you over for speeding even though neither clocked you going over the speed limit. How did the cops know you were speeding? What What assumptions did they have to make?
The cops know you were speeding because of the Mean Value Theorem. They find you're average speed by taking the difference in positions of cop B (mile 60) and cop A (mile 0) divided by the time (1 hour), which results in 60 mph. Then using the Mean Value Theorem, the cops know there must be a point in between A and B where you were going over 60 mph. The assumptions they must make are that your driving was continuous and differentiable, in other words that you did not time warp at all during your drive, which is a pretty safe assumption to make.