Math 10B Name: Winter 2016 Practice Midterm 1 Time Limit: Minutes Exam Preparer: MATH 10B Teaching Assistants This practice exam contains 4 pages (including this cover page) and 14 questions. Total of points is 1017304. The questions presented in this practice exam should reflect the material presented in lecture. If anything, this exam might be harder than the practice exams. This is in the hope of overpreparing you so that given any function, you can calculate the anti-derivative. The points do not matter - Drew Carey, Former Host of Whose Line Is It Anyways? Grade Table (for your convenience) Question Points Score 1 1000000 2 10 3 10 4 1000 5 2000 6 6 7 1200 8 1554 9 1500 10 2000 11 5000 12 1500 13 500 14 1024 Total: 1017304 1. (1000000 points) Carefully read and complete the instructions at the top of this exam sheet (see below) and any additional instructions written on the chalkboard during the exam. No calculators or other electronic devices are allowed during this exam. You may use one page of notes, but no books or other assistance during this exam. Write your
Math 10B Practice Midterm 1 - Page 2 of 4 Name, PID, and Section on the front of your Blue Book. Write the Version of your exam at the top of the page on the front of your Blue Book. Write your solutions clearly in your Blue Book (a) Carefully indicate the number and letter of each question and question part. (b) Present your answers in the same order they appear in the exam. (c) Start each question on a new side of a page. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. d sin(e x 2. (10 points) 2 ) dx ln(5x) e dt t 6 +t 3. (10 points) I throw a tomato with an acceleration of 10 m s 2. Assuming it travels straight, how fast will it hit the target 70 meters away? 4. Fill in the blanks. (a) (100 points) When looking at the graph of a function, the definite integral finds the of the region between the graph and the x-axis. (b) (100 points) Because we consider the portion of a graph above the x-axis to have positive area and the portion of a graph below the x-axis to have negative area, we can say the definite integral finds area. (c) (100 points) By the Fundamental Theorem of Calculus, we can say that, if f(x) = F (x) on [a, b], for some function F, then the definite integral finds the change from F (a) to F (b). (d) (100 points) For some functions F and f, if f(x) = F (x) then we can f is of F, and we call F an of f. (e) (100 points) The definite integral gives us gives us.. The indefinite integral (f) (100 points) An equation of the form y = f(x) or dy = f(x) is called a. dx The general solution of this type of equation is. If this equation also has a starting value y 0, we call this a problem. (g) (100 points) We can approximate the area under a curve using. (h) (100 points) If we compute a definite integral over an interval, then divide this value by the length of the interval, we are really finding. (i) (100 points) Computing the definite integral of the constant function f(x) = 1 over an interval gives the of the interval. (j) (100 points) When doing an indefinite integral, it is necessary to include a in the expression. 5. (2000 points) Find the solution of the initial value problem. dy dt = 2 cos(t) + 4 ( π ) π, y = 5 2
Math 10B Practice Midterm 1 - Page 3 of 4 6. (6 points) dy = 4 sin(t) + t, y (0) = 5 dt (a) (3 points) Find the solution of the initial value problem. (b) (3 points) Is the function y(t) increasing or decreasing when t > 4? Explain your reasoning. 7. (600 points) Suppose Professor Eggers tossed a juggling pin on fire vertically in the air from the ground and the height of the pin is represented by the following function: h(t) = 16t 2 + 14t + 6.083 where t represents the time in seconds and h(t) represents the height of the ball in feet at time t. Assume that air resistance is constant and we have accounted for Professor Eggers injured arm. (a) (200 points) What time will it take for the juggling pin to hit Professor Eggers head if Professor Eggers is 6 feet 1 inch (6.083 feet) tall? What should he do so that he doesn t get hurt? (b) (200 points) At what time will the pin hit the maximum height? (c) (200 points) Find how fast the juggling pin is moving at t =.25 seconds. 8. Suppose that Professor Eggers is taking a stroll on the moon while eating a blueberry muffin. He fails to notice that someone left a banana peel lying on the ground, and slips on it, sending his blueberry muffin flying. Assume that the acceleration due to gravity on the moon is 6ft/s 2. The blueberry muffin leaves Professor Eggerss hand with an initial upward velocity of v(0) = 4ft/s, at a height of 4ft from the ground. (In this problem, ft = feet and s = second.) (a) (777 points) What is the maximum height of the blueberry muffin? (b) (777 points) How long does it take for the blueberry muffin to hit the ground? 9. Let π (sin 2 (x) + 2)dx = 5π, π (a) (500 points) What is π 0 sin 2 (x)dx? (b) What is the average value of the function f(x) = sin 2 (x) over (a) (500 points) the interval π x π? (b) (500 points) the interval 0 x π? 10. Dr. Eggers recently watched a television segment featuring one of his colleagues lecturing during the La Jolla Storm of 2016. Wanting to be famous, Dr. Eggers awaits for the next storm which happens tomorrow. During the torrential downpour, Dr. Eggers classroom begins to flood. He valiantly continues to lecture, but the water level in the room begins to rise. The rate at which the water level is rising can be modeled by the function = (1/8)x 4 (1/7)x 2 + (1/5)x 15/2 in feet per minute. dw dt
Math 10B Practice Midterm 1 - Page 4 of 4 (a) (1000 points) If at t = 1 minute, the water level is one foot high, how high is the water level after 4 minutes? (b) (1000 points) Is this deep enough for Dr. Eggers to become famous on Reddit, Facebook, Buzzfeed, and social media in general? 11. (5000 points) Let f(t) = x t 0 1234et2 sin(ln( 3 5678 tan(t) ))t4 cos( 101t). Find the derivative of f. NOTE: The problems related to finding Riemann Sums from Wiley Plus as well as being able to calculate the areas given any graph are highly suggested. They are often put on midterms so you should be familiar with basic calculations of area under a curve (i.e. area of a triangle, rectangles, dealing with negative areas, etc.) 12. Suppose the following graph depicts f(x). Assume the following: from [0, 1], f(x) is linear; from [1, 3]; f(x) is linearly horizontal; from [3, 5], f(x) is a perfect quarter of a circle (f(x, y) = 2 = (x 3) 2 + y 2 ); from [5, 6], f(x) is linear Calculate the following: (a) (300 points) 6 x=5 f(x)dx (b) (300 points) 6 f(x)dx x=0 (c) (300 points) The Left Hand Riemann Sum using 3 rectangles. f(x, y). Solve for y so that you can get the height) (d) (300 points) The Right Hand Riemann Sum using 3 rectangles (You will need (e) (300 points) List in order from least to greatest of the approximations and the exact. 13. (500 points) Determine f(x) given that f (x) = 12x 2 + 24x + 2 + e 2x, and f(0) = 1, f(1) = 6
Math 10B Practice Midterm 1 - Page 5 of 4 14. (1024 points) Find the total area bounded by f(x) = x 2 2 and g(x) = x, the y = 0, and y = 2. Hint: find the intersection of the functions, and use the property that integral is the signed area under a graph. NOTE: Even if we do not say the word, justify, you are expected to justify all work. Think of an exam as a court room. You are trying to give as much evidence to convince us (the TAs and Instructors), that you are innocent of a failing grade (which deductively means you are worthy of a good grade). The points do not matter.