If a function m is defined as ,
    (a)    Find .
    (b)    Evaluate .
    (c)    Plot a rough graph of m and m'.

(a)    Plug  into m in place of x to get the answer (b)    To find the derivative, use the Fundamental Theorem, Part 2, which means you should plug the top boundary into the function and multiply by its derivative: (c)    You already know m' from part (b) above; to plot m, just integrate m' (if you know u-substitution). Otherwise, plug numerous points (like /2, /4, etc) into m to get a feel for the graph.

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Calculus II (BC): Zit Growth

Franklin is currently employed at the local fast food establishment (The Grease Feast), and because of his stellar job record (and the fact that he has seniority having worked there for two whole months), he was recently promoted to French Fry Overseer. All jobs have work-related hazards, and Franklin's Overseer position at the Feast is no different. In fact, he found that the rate of pimple growth (in zits/day) on his nose is proportional to the difference between 10 and his current number of zits. If he has 25 pimples when he gets the promotion (t = 0), and 35 zits two days later, how many total days will it take to achieve the presitigious 100-zit mark?

Difficulty Rating: 

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Solution:

Since the difference between 10 and the number of zits, z, is proportional to the rate of zit growth, dz/dt, you can write the differential equation dz/dt = k (10 z). Use separation of variables to solve the equation: Now, you almost have the formula for z(t); you just have to find k. Do so by using the next piece of given information: z(2) = 25. At this point, you have the model. When do you reach 100 zits? Set the model equal to 100 and solve.