2013 Mid-Year Exam Jan. 28, 2013

Transcription

1 Moses Brown School Honors Precalculus Name Honors Precalculus 2013 Mid-Year Exam Jan. 28, 2013 Jeff Cruzan, Ben Young 150 pts. possible note special instructions on number 14 (last problem) This exam should include 17 pages, including (this) page of formulae. Point values are given for all problems. Please draw a box around your final answer for each problem. Take a deep breath and dig in. You'll be fine. Good luck! Formulae Simple interest Compound interest Half life Continuous growth A(t) = A o (1 + r) t A t = A o 1 r n nt A t = A o 1 2 t k A(t) = A o e rt Summary of discrete probabilities A P A [0,1] Probability must be between zero and one. not A Subtraction rule A or B Addition rule A and B Multiplication Rule A given B Bayes' theorem P! A = 1 P A P A B = P A P B P A B P A B = P A P B If A and B are mutually exclusive P A B = P A B P B P A B = P A P B P A B = P A B P B P A B = P B A P A P B Binomial coefficients: N r = N! r! N r! If A and B are independent Connects P(B A) and P(A B) Honors Precalculus Midyear Exam January 28, 2013 page 1 of 17

2 1. (30 pts., 5 pts. each) Solve the following equations. Give exact answers only. (a) x 3 x 2 16 x = 16 (b) ln x = ln33 ln x 8 (c) log 3 x log 3 x 5 = log 3 24 Honors Precalculus Midyear Exam January 28, 2013 page 2 of 17

3 (d) 2 e 2x 3 e x = 2 (e) x 12 x = 35 (f) x 3 x 2 9 = 7 x Honors Precalculus Midyear Exam January 28, 2013 page 3 of 17

4 2. (10 pts.) In 1996, a fish count in Lake Piscacotawataquoddymoggin counted only 24 adult Northern Pike. In 2006, the count was repeated and 1037 Pike were counted. Develop an equation based on continuous growth to model the Pike population in this lake as a function of number of years since Honors Precalculus Midyear Exam January 28, 2013 page 4 of 17

5 3. (10 pts.) Given that (x + i) is a zero of f(x) = x 5 6x 3-6x 2 7x 6, find all other zeros of f(x). Express your answers exactly. Honors Precalculus Midyear Exam January 28, 2013 page 5 of 17

6 4. (10 pts.) What minimum interest rate must be earned on an investment compounded monthly if the investor wishes to triple his/her initial investment in 20 years? Please write your answer as a percent to the thousandths place. Honors Precalculus Midyear Exam January 28, 2013 page 6 of 17

7 5. (5 pts.) Find x 5 5 x 4 5 x 3 25 x 2 4 x 20 divided by (x + 5). Honors Precalculus Midyear Exam January 28, 2013 page 7 of 17

8 6. (10 pts.) Sketch a graph of f(x). Find any vertical, horizontal or slanted asymptotes, or holes and report them in exact terms. Make sure to completely label your graph and keep it neat. f x = 2 x 2 9 x 35 x 2 6 Honors Precalculus Midyear Exam January 28, 2013 page 8 of 17

9 7. (10 pts.) Sketch a graph of f(x). Find any vertical, horizontal or slanted asymptotes, or holes and report them in exact terms. Make sure to completely label your graph and keep it neat. f x = x 3 x 2 4x 4 x 2 x 2 Honors Precalculus Midyear Exam January 28, 2013 page 9 of 17

10 8. (10 pts.) Let's say it costs $5 million to re-tool a factory to produce a new tablet computer, and $180 per unit to manufacture them. How many tablets must be manufactured to keep the cost per unit below $260? Honors Precalculus Midyear Exam January 28, 2013 page 10 of 17

11 9. (10 pts.) A logistic function like this one is used to model real populations, which cannot grow arbitrarily large. For example, the number of rabbits in a pasture is limited by the amount of food available (ignore predators). P t = 10, e 0.02t (a) Calculate the population at time t = 0., P = population and t = time in years. (b) What is the maximum population that can be reached using this equation as a model? (c) After how many years will the population exceed 5000? Honors Precalculus Midyear Exam January 28, 2013 page 11 of 17

12 10.(10 pts.) A concrete planter with the proportions shown is to be constructed on a street corner. It needs to have an internal volume of 64 cubic feet, which will be filled with soil for plants, and the walls must be one foot thick so that it can be used as a sitting bench. The planter has no bottom; you're calculating the dimensions of the rectangular concrete ring only. Honors Precalculus Midyear Exam January 28, 2013 page 12 of 17

13 11.(10 pts.) Find the inverse of the function f(x) = x 2 2x - 8. Write the domain and range of both f(x) and f -1 (x) in bracket notation: [ ] ( ). Sketch f(x), its inverse and the line y = x. Honors Precalculus Midyear Exam January 28, 2013 page 13 of 17

14 12.(10 pts.) At Barack Obama s second inauguration in Washington D.C., security was very tight. Any bag brought into the ceremony went through an X-ray scanner. Bags that were flagged by the personnel monitoring the X-ray machine were searched by hand. From a research study, it has been found that 98.3% of bags that contain a banned substance will be flagged and searched by hand and 15.4% of bags that do not contain a banned substance will also be flagged and searched by hand. Only 0.4% of all bags contain a banned substance. What is the probability that a bag that is currently being searched by hand actually contains a banned substance? Honors Precalculus Midyear Exam January 28, 2013 page 14 of 17

15 13.(5 pts.) Take the parent function of all cubic functions and apply the following transformations in the order shown. After applying the transformations, express the function in standard form, f(x) = Ax 3 + Bx 2 + Cx + D. i. reflect across the y-axis ii. then reflect the result across the x-axis iii. then shift the resulting function 2 units to the left. Honors Precalculus Midyear Exam January 28, 2013 page 15 of 17

16 For problem 14, do either version A or version B, your choice. If you have time to do both, you will receive some extra credit for the second version. 14.(A 10 pts.) A tank in the shape of a triangular prism has the dimensions shown at right. The tank is filled with liquid via the fill port a rate of 2 liters/min (1 m 3 = 1000 liters). Find a formula for the height of liquid in the tank (measured from the bottom) as a function of time. Honors Precalculus Midyear Exam January 28, 2013 page 16 of 17

17 (B 10 pts.) The figure shows a square inscribed in a circle (each of its four vertices lies on the circle). Find a formula for the area of the square as a function of the circumference of the circle. Honors Precalculus Midyear Exam January 28, 2013 page 17 of 17