Exercise I: Random Stuff (6 points) (a) Let f (z) = x3 + 2:5. Calculate (f '1)' (3) without calculating f '1(a3) (b) Suppose that :5 represents the quantity supplied in thousands for a price p in hundreds of dollars. .7) and p are related by 62\"\" 3p = m. What is the sales price when production is at 600 units? How fast is the supply changing (with respect to the price) at this level? You can use a calculator for approximations. (c) Use logarithmic differentiation to nd the derivative of y = (32m + 1)'/5. No need to simplify fully. Exercise II: Implicit Differentiation (6 points) Assume that a factory can produce a quantity 25::3 + 33y + y4 of goods, when cc is the number of skilled workers, and y the number of unskilled workers. 1. Right now, there are 10 skilled and 8 unskilled workers. What is the quantity produced? 2. Assume that the quantity of goods produced stays unchanged. For the number of workers in part (a), d nd dy in terms of m and y, and evaluate it at this level. I 3. The management of the factory is considering replacing a skilled worker (paid $40/h) by unskilled workers (paid $15/h), while maintaining the production at the same level. Is this a good move? Explain precisely using the previous question, namely the value of 3/ you found and what it means. Exercise III: Linear Approximations (6 points) Use linear approximation to compute the following quantity. Explain what you are doing, for instance by writing down what function you are using, or what approximation you are making. Show your computations. 1. Approximate 1 8.481/3' Do not use a calculator for any step, and write down every computation. 2. Assume a:y3 2g + 5x = 2. Approximate y(1.1). Exercise IV: Elasticity (6 points) Assume that the demand for broccoli is given by D(p) = 5000 500172, where D(p) is in tons and p is the price of a pound of broccoli. 1. If the current price of broccoli is $2 per pound, how much broccoli will be sold? 2. Compute the elasticity of the demand as a function of price. 3. Is the demand at $2 elastic or inelastic? Is it more accurate to say \"People want broccoli and will buy it no matter what the price\" or \"Broccoli is a luxury item and people will stop buying it if the price gets too high\"? Why? 4. Find p such that the demand at p is unit elastic. Gives the exact value and an approximation. What does this price mean in terms of revenue? 1We will later learn tools to verify that this is a bijective function, no need to verify