Calculus I (AB): Antiderivatives and Acceleration I have designed a new automobile called the Hoopty Harbinger. It is very fast and, inexplicably, shaped like a gall bladder. It is able to accelerate from 12 mph to 55 mph in precisely 35 seconds. Assuming constant acceleration, find the following:
(a) The acceleration in meters per second per second
(b) The distance traveled by the car during those extreme 35 seconds.Difficulty Rating: 
(a) This problem requires information be given about acceleration in meters, and the original information is given in miles. Further, it's miles per hour and we want meters per second, so there is some conversion to be done. We will say 12mph occurs at time t = 0 seconds, while 55mph occurs at time t = 35 seconds to begin. Now we need to convert the speeds. Knowing that 1 mile = 1609.347 meters (and you know that now if you didn't before!), and that there are 3600 seconds in an hour, 12mph = 5.36449 m/s and 55mph = 24.58725 m/s. It's not too hard to figure out why, so if it's not obvious, think about it.
Solution:
Because acceleration is constant, we will call it A. We have so much information about velocity, let's jump to it by taking the antiderivative of acceleration. Once we do that, we can plug in our given information--v(0) = 5.36449 and v(35)=24.58725-- to find the constant of integration C and (as a result) A:(b) To find the distance traveled during that time, we need to antidifferentiate again, to get the position function. Assuming at t = 0 we start at position 0 (just to make life easier), we need to find out our position at t = 35:
Calculus II (BC): Homogeneous Functions Determine whether the function is homogeneous, and if it is, determine its degree:
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Difficulty Rating:All we must do is plug in tx for x and ty for y to see if we get a t multiple of the original function:
Solution:Therefore, we have a homogeneous function of degree 1, since t = t1.
