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Tuesday, April 21, 2009

Mini Test: The Calculus Strikes Back


I really need to stop with these late night scribe posts.


For the record, Joseph is scribe, and the list REALLY needs updating.


Question (1)



In this question, the answer should be straight forward. If you approach X from the left (low), then the largest integer value is 1. This is demonstrated in the first line. If you approach from the right (high), the largest integer value is 2. These are also known as roof and floor values. Because the right approach (floor) is not equal to the low approach (roof) then the value cannot exist, meaning the answer is (E).



Question (2)



For this question, we need to absolutely remember what the definition of a derivative is. Because this is it! Written right there. 1 is the value of f(x), which is also equal to sin(π/2). You will never actually see this, but you need to recognize it. Once you see it, this question becomes easy. The derivative of sin(π/2) is 0, because at sin(π/2) you are at the top of the curve and the tangent line (derivative) will be perfectly horizontal. Thus, the value must be zero, or (C).



Question (3)


Just keep deriving. You’ll get there eventually. (C)


Question (4)


A painfully simple solution to this question. To find a line, you need to have the slope and a point. You are given the point. Therefore, logically you should find the slope. Derive the equation of your curve, and you know that equation also equals the line with y = 1 and x = –1 (the equation for any line is y – y1 = m(x – x1). Solve for m with algebra. Once done that, do some rearranging with algebra, but the answer should become obvious as being (A).


Question (5)

sldies_Page_7 Speed is the scalar version of velocity, which has vector (direction). When velocity is negative, you are moving backwards. Because speed is scalar and has no direction, it is the absolute value of velocity, so in terms of speed even if you are moving “backwards”, your speed will be positive because you are moving period. So if you re plot the absolute value of the graph, you will see the greatest value is at t = 8 which is answer (E).


Question (6)

sldies_Page_8 If at t = 3 our object is at the origin, and its velocity is positive, it means the object is moving away from the origin in the positive direction for 3 units (the triangle formed by the graph from 3 to 6 is base 3 and height 2, so (1/2)2(3) = 3). So, we want to find where the object moves in the negative direction for 3 units, Since the value of the triangle created by the graph from 6 to 7 is base 1 and height 2, the area and total distance changed is 1 in the negative direction, meaning we havent gone backwards enough to be back at the origin. If we find the area from 6 to 8 then the base is and the height is 4, so by t = 8 you’ve moved 4 units negatively, which is past the origin, but we can logically deduce that at some point between t = 7 and t =8 we had moved 3 units negatively and thus returned to the origin. So, logically, the answer is (E).


Question (7) Free Response


For (a), you know S(0) = 6 so use that to solve for C. Quick math determines that C = 6 because e ^(0k) = 1 so 1C = 6. You also know that consumption doubles every 5 years, so S(5) = 2(6) and 2(6) = 6e^(5k) so e^5k = 2. Take the ln of both sides, so ln2 = 5k, and solve for k.


In (b), you want the integral from 3 to 13 (remember that your function counts years from 1980 so 1983 – 1980 = 3 as your starting value, and you want a 10 year period so 10+3 = 13 as your other limit) of the function. Calculator please. Really, you dont want to integrate by hand, but be my guest.


(d): “The integral of S(t)dt from 3 to 7 gives the total amount of cola consumed in the United States in billions of gallons per year during the time period from January 1, 1983 to January 1, 1987.”



Remember guys to start making your significant edits to the wiki solutions manual, and to at least try 3 exam questions a night. This is crunch time!



I dont really have a new funny/interesting youtube video, so here’s Intergalactic.


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