October 22, 2013 Name: Key
You must SHOW ALL WORK to receive credit!
Be sure to include units in answers wherever appropriate.
Questions that ask you to interpret should be answered in complete English sentences.
Find the derivative of each of the following functions. You do not need to simplify your answers.
5. a. Find by implicit differentiation:
b. Find the slope of the line tangent to the curve at the point (1,1)
6. The value of an investment is after t years. Compute and interpret A’(4)
4 years after this investment is made, its value is growing at a rate of $2754.26 dollars/year.
Testing this: After 4 years, the value is
While this is not exactly the same as what we estimated, it is in the same ballpark, which is encouraging. If we looked at a smaller time interval, the estimate would be much better. For example, one day (1/365 of a year) later, the derivative indicates that the account should grow by $2754.26/365 = $7.55. In reality, it grows to
7. ACME’s cost of producing x widgets is C(x) = 1000+8x - 0.01x2 dollars.
a. Evaluate C(100) and interpret what this value means to ACME.
. This is ACME’s total cost if it produces 100 widgets.
b. Evaluate C’(100), and interpret what this value means to ACME.
. This is ACME’s marginal cost. If ACME is currently producing 100 widgets, each additional would cost an additional $6 to produce.
8. ACME (continuing from the previous problem) has price demand function p(x) = 20-.04x
a. Find ACME’s revenue function, R(x).
b. How many widgets should ACME produce to maximize its profit? Recall that profit = revenue – cost
When P’(x) is positive, ie when x<200, the profit increases when production increases. So if ACME is making fewer than 200 widgets, it should increase production.
When P’(x) is negative, ie when x>200,the profit decreases when production increases. So if ACME is making more than 200 widgets, it should decrease production.
Putting these two statements together, the profit must be maximized if ACME produces exactly 200 widgets.
At this production level, its profit is
9. If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if
a. the interest is compounded quarterly?
b. the interest is compounded continuously?
10. The half-life of radioactive cesius-137 is 30 years. Suppose you have a 100-mg sample.
a. Write a formula that gives the mass that remains after t years.
After 30 years, 50 g remains so using
b. After how long will only 1 mg remain?