211-craciun-07faex01 - Math 211 ~ MIDTERM EXAM 1 —~ Fall...

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Unformatted text preview: Math 211 ~ MIDTERM EXAM 1 —~ Fall 2007 YOUR NAME: Circle your TA’S name: Daniela Sarb David Seal Holly Chen Jingwei Hu Li Wang Weldong Yin Yingzhe Shi Show all your work. Answers without showing all work receive zero points. 1. (.70 points) Recall that if an ancient sample was dead for t years, then the amount of the radioactive 014 isotope it contains is given by an exponential decay formula y(t) : A64“. The half-life of 014 is 5,568 years. A sample of wood found in an ancient pyramid is known to be 4,800 years old. The rate of radioactive decay in this sample was measured and compared with the radioactive decay rate measured in fresh samples. What should be the ratio of these two measurements? 2.(10 points) Draw the graph of a function defined on the interval [3, 10] that is discontinuous at exactly three points and has no limit at exactly two points. \ 3.(10 points) Suppose that $5000 is put in an account paying 7% interest compounded‘monthly. (7;) Write a formula for the amount in the account at the end of t years. (ii) How much will be in the account at the end of 10 years? (iii) How much will be in the account at the end of 10 years if the 7% interest is compounded continuously instead of monthly? 4.{15 points) Two cars are moving in the same (positive) direction along a straight road. Let A(t) and B (t) be the distances (in yards) of cars A and B from the starting point at the end of 15 seconds. Translate the following statements into a formula involving the functions A(t) and B (t) and/ or their derivatives: , (i) At the end of 45 seconds, car A is exactly 70 yards ahead of car B. (ii) At the end of 10 seconds, car A is traveling exactly 15 feet per second faster than car B. (M) At the end of 12 seconds, car A has exactly three times higher acceleration than car B. 5. (20 points] (1') Compute the derivative of f (£1!) = 6‘231nx. (ii) Compute the derivative of f (1;) = Compute the derivative of f ($) = 1n(3 + 6””). (iv) Find the equation of the tangent line to the curve y = Vac + 5 at the point Where as = 2. 6.(15 points ) Compute the following: Vt ~ 2 t2 (a) lim t—>2+ (b) 2:3 — 1 (c) Consider f (x) = l and compute f’(2) using the definition of the derivative. (12 7. (10 points) Use linear approximation to estimate \3/ 998. 8. {10 points ) The revenue from the sale of 20,000 units is $12,500, and the marginal revenue at that level is 81 cents. Use linear approximation to estimate the revenue if the sales level drops to 19,000 units. ...
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