Midterm 1 Practice Problems

Transcription

1 Here are some things that you need to know how to do: Build functions in the context of a model and interpret things about them. Compute average and percentage rate of change. Find the equation of a line. Sketch the graph of a linear function. Solve equations like ax 2 + bx + c = 0. Sketch quadratic functions. Find the vertex of a quadratic function. Find and interpret domains of functions (both mathematical and practical). Build and understand piecewise-defined functions, functions defined by a graph, and functions defined by a table. Understand when a function is increasing and decreasing. The problems that follow are some modeling problems of varying difficulty. I can t honestly call them all good exam problems but they are good for practicing those skills. 1. Alice, a stock analyst, predicts that the price of a particular stock will rise quickly over the coming months. She uses a function P to model the price: P(t) is the price of the stock (in dollars) t months from today and P(t) = t 100. Alice estimates that this model will be accurate for half of one year. (a) What is the mathematical domain of P? Answer: (, 1) ( 1, ) (b) What is the practical domain of P? Answer: (0, 6) (c) Alice wants to sell all of her shares in this stock when its price reaches $170. How long will she have to wait? Answer: About 2.33 months (d) Katie, a different analyst, wants to wait until the price reaches $225. What can Katie deduce from Alice s model? Answer: P(t) = 225 when t = 5 which isn t in the practical domain of P. If Alice s model is accurate then the stock price will not reach $225 in the next six months. 2. Vadim and Nick are skiing on a mountain. They both start at the top of a mountain which is at an altitude of 5000 ft and they ski to the bottom of the mountain which is at an altitue of 2000 ft. Vadim starts down the slope and skis at a constant rate. It takes him ten minutes to get to the base of the mountain. Nick leaves five minutes later. Hoping to catch Vadim, Nick skis as fast as possible; he is able to ski at a rate such that he loses 11 ft of altitude every second. Note: I know nothing about how fast people can ski. I apologize if these numbers don t make sense. Hint: Be careful with units in this problem. (a) Let t be the number of seconds after Vadim starts his descent. Define V such that Vadim is at an altitude of V(t) feet after he has been skiing for t seconds. i. Find an equation for V(t). Answer: V(t) = 5t ii. What is the practical domain of V? Answer: [0, 600] iii. What was the rate of Vadim s descent? That is, how many feet of altitude did he lose every second? Answer: He lost 5 ft of altitude every second. Last Updated: November 7, 2015 Page 1 of 5

2 (b) Define N such that Nick is at an altitude of N(t) feet after Vadim has been skiing for t seconds. (Note that t = 300 when Nick starts his descent.) i. Find an equation for N(t). Answer: N(t) = 11t ii. How long does it take for Nick to reach the base of the mountain? Answer: It took Nick approximately 4.55 min (from when he started skiing) to reach the base of the mountain. iii. What is the practical domain of N? Answer: [300, ] (c) Will Nick catch up with Vadim before either of them reach the bottom of the mountain? If so, how long does it take? Answer: Yes, Nick will catch up to Vadim after Vadim has been skiing for approximately 9.17 min. 3. A softball team estimates that if they score 1 run in a game then they have a 10 % chance of winning the game and if they score 10 runs in a game then they have a 90 % chance of winning the game. Assuming that the relationship between runs scored and win probability is linear, find the probability that they win a game if they score 3 runs. Answer: Approximately % 4. Stuart bought his house ten years ago and today it is worth $ An appraiser tells Stuart that he should expect the value of his house to decline by $ every five years. (a) Find a function, V, such that V(t) is the value of Stuart s house after he has owned it for t years. Answer: V(t) = 6000t (b) How much will Stuart s house be worth ten years from today? Answer: $ (c) Find the y-intercept of V and interpret your answer. Answer: The y-intercept is (0, ). This tells us that the house was worth % when Stuart bought it. (d) Find the x-intercept of V and interpret your answer. Answer: The x-intercept is (35, 0). This tells us that the house will be worth nothing after 35 years. 5. Alicia throws a football and her coach tracks the ball s flight. He finds that it is at a height of h(d) = 0.02d d yards after it has travelled a horizontal distance of d yards down the field. (Round all of your answer to two decimal places if necessary.) (a) What was the height of the ball when Alicia released it? Answer: 1.8 yd (b) How far horizontally did the ball travel before it hit the ground? Answer: yd (c) What was the maximum height that the ball reached? Answer: yd { 4 x 2 if 0 x 2 6. Let f(x) = 12 4x if 2 < x 4. (a) What is the domain of f? Answer: [0, 4] (b) What is f(1)? Answer: f(1) = 3 Last Updated: November 7, 2015 Page 2 of 5

3 (c) What is f( 5 2 )? Answer: f( 5 2 ) = 2 (d) What is the y-intercept of f? Answer: (0, 4) (e) What are the x-intercepts of f? (This is a tough one.) Answer: (2, 0) and (3, 0)) (f) Sketch the graph of y = f(x). (This is even tougher.) Answer: y t A farmer is fencing in a large rectangular play-pen in which his dog can play. Three of the walls are made of inexpensive chain-link fencing and cost $2 per foot. The third wall, however, faces the road and must be made of a special reinforced material which costs $5 per foot. Assume that the farmer only has $500 to spend on the fence. (a) Let l be the length of the side which faces the road and let w be the length of the adjacent side. (Draw a picture if it helps.) Find an equation which relates l and w. (Be sure to solve for w.) Answer: w = 7 4 l (b) Define a function A such that A(l) is the area of the garden when the side facing the road has a length of l. Answer: A(l) = 7 4 l l (c) What is the practical domain of A? Answer: (0, ) (d) What are the dimensions of the play pen which gives the maximum possible area? Answer: The side facing the road will have a length of approximately ft and the adjacent sides will have a length of 62.5 ft. (e) What is the maximum possible area of the play pen? Answer: Approximately ft 2 8. A Norman window is shaped like a half-circle on top of a rectangle as shown in the image below. Last Updated: November 7, 2015 Page 3 of 5

4 h w Assume that a particular Norman window must have a fixed perimeter of 10 ft. Find the width that will maximize the window s area. You may find it helpful to know that if a circle has a radius of r then its perimeter is 2πr and its area is πr 2. Round your answer to two decimal places. Answer: Approximately 7 ft 2 The last problems work off of the following information. They are very involved and they are much too long for fair exam problems but they are good practice as they make use a lot of skills. Round all of your answers to two decimal places even if you normally wouldn t do so. For example, if you find that profit is maximized when pianos are sold then you should leave it as a decimal; assume that such numbers represents a monthly average. Dave is the president of a company which manufactures musical instruments. He is reviewing the price of their upright pianos and has accumulated information from various people within the company. The salesman in charge of these pianos found that they currently sell 50 pianos every month and they currently charge $4500 per piano. The marketing department estimated that the demand equation is linear and that if they reduce the sales price by $1000 then they can sell five more pianos every month. The building supervisor reported that piano sales are responsible for a fixed cost of $5000 every month due to the building s lease and the upkeep of the equipment used. The head of manufacturing calculated that each piano costs the company $2250 in parts and labor. In the questions that follow, let x be the sales price of each piano and let q be the number of pianos that the company sells every month. (Note that question 9 and question 10 are not related; you can do either without having done the other.) 9. In this exercise we are going to try and investigate what happens when the sales price, x, is used as a variable. (a) Find a demand equation relating x and q. Be sure to solve for q. Answer: q = 0.005x (b) Find a function, R, for revenue in terms of price, x. Answer: R(x) = 0.005x x (c) Find a function, C, for cost in terms of price, x. Answer: C(x) = 11.25x (d) Find a function, P, for profit in terms of price, x. Answer: P(x) = 0.005x x (e) Find any sales prices at which the company breaks even. (You may get more than one.) Answer: $ and $ (f) Find the price which maximizes revenue. Answer: $7250 (g) Find the maximum possible revenue. Answer: $ (h) Find the price which maximizes profit. Answer: $8375 (i) Find the maximum possible profit. Answer: $ (j) Use the demand function from part (a) to find the quantity at which profit is maximized. Answer: pianos Last Updated: November 7, 2015 Page 4 of 5

5 10. In this exercise we are going to try and investigate what happens when the quantity, q, is used as a variable. (a) Find a demand equation relating x and q. Be sure to solve for x. Answer: x = 200q (b) Find a function, R, for revenue in terms of quantity, q. Answer: R(q) = 200q q (c) Find a function, Ĉ, for cost in terms of quantity, q. Answer: Ĉ(q) = q (d) Find a function, P, for profit in terms of quantity, q. Answer: P(q) = 200q q 5000 (e) Find any quantities at which the company breaks even. (You may get more than one.) Answer: 0.41 pianos and pianos (f) Find the quantity which maximizes revenue. Answer: pianos (g) Find the maximum possible revenue. Answer: $ (h) Find the quantity which maximizes profit. Answer: pianos (i) Find the maximum possible profit. Answer: $ (j) Use the demand function from part (a) to find the sales price at which profit is maximized. Answer: $ You should be able to graph any of y = R(x), y = C(x), y = P(x), y = R(q), y = Ĉ(q), and y = P(q). The numbers get large, but don t let that scare you. Answer: Nahhhh... Last Updated: November 7, 2015 Page 5 of 5