Math 106 Final Exam Spring 2016

Math 106 Final Exam Spring 2016 Name: Read This First! • This is a closed-book examination. No books, notes, cell phones, communication devices of any sort, webpages, or other aids are permitted. • Please give an exact answer when possible. The answer cos(22◦ ) = .92718 is fine, while cos(45◦ ) = .707 will not get full credit. • Please read each question carefully. Show ALL work clearly in the space provided. • If additional work space for a problem, please use the back of the previous page . • In order to receive full credit on a problem, solution methods must be complete, logical and understandable • Answers must be clearly labeled in the spaces provided after each question. • The exam consists of Questions 1–12, which total to 200 points. Grading - For Instructor Use Only Question: 1 2 3 4 5 6 7 8 9 10 11 12 Total Points: 35 35 10 15 15 10 20 10 10 15 10 15 200 Score: Math 106 Final Exam 1. [35 points] Compute the following derivatives. Do not simplify unless instructed to do so.
d ln(x) sin(x) = dx  2  θ d = (b) dθ cos3 θ  Z x d 1 (c) dt = t dx 0 e + tan t (a) (d) (Simplify as much as possible) (e) d (sec x)x = dx √
 d  ln π + ln ex = dx 2. [35 points] Compute the following integrals. Z (a) tan2 x sec2 x dx = (b) Z ln 27 √ 3 ex dx = ln 8 ex dx = ex − e3 Z √π/3 3x cos(x2 ) dx = (d) 0 Z  2 1 (e) + 3 dx = x (c) Z 3. [10 points] Two numbers A and B are related by the equation A = log B. Adding 2 to A will change B. I ask two Math 111 students how B will change. Here’s what they said: • Student 1: 100 will be added to B since log 100 = 2. • Student 2: 100 will be multiplied by B since log 100 = 2. One of these students is full of baloney. Which one is wrong? Please explain why. R4 4. [15 points] Consider the integral 1 20 − x2 dx. (a) Approximate the integral using a Riemann sum with n = 3 and the left endpoint of each interval. (b) Without computing the integral, tell me whether the approximation of part (a) is bigger or smaller than the integral. Hint: Draw a picture. 5. [15 points] Consider the region bounded by the curves y = 1 + x1 , y = 2 and x = 3. If this region is rotated about the x-axis, find the volume of the resulting 3-dimensional object. 6. [10 points] Find the equation of the line tangent to the curve y = sin(ex ) at the point where the x-coordinate is ln( π4 ). 7. [20 points] Consider the region in the plane abounded by the curves y = ex , x = 1, and y = e3 . (a) Draw a picture of the region. Page 1 of 3 Math 106 Final Exam (b) Express the area using an integral with x as the independent variable. Do not evaluate the integral. (c) Express the area using an integral with y as the independent variable. Do not evaluate the integral. (d) Evaluate one of the integrals from parts (b) and (c). 8. [10 points] Find the inverse function of f (x) = 2x+1 . 9. [10 points] Simplify as much as possible without using a calculator. (a) log3 (log3 (39 )) = (b) 2x+log2 (3x) = 500t cubic feet per hour. (1 + t2 )2 Compute the amount of water that was added during the time period 0 ≤ t ≤ 7 hours. 10. [15 points] Water is added to a swimming pool at a rate of 11. [10 points] We start with 10 grams of a radioactive substance. Five days later we have 8 grams. What is the half-life of the substance? 12. [15 points] A bacteria colony grows exponentially, doubling in size every three days. Initially, there were 1,000 bacteria. (a) Give a formula for the number N of bacteria after t days. In your formula, please use the exact value of the growth rate k. (b) This problem was taken from a book where the answer in the back is given by the formula N = 1000 2t/3 . Show how the answer you gave in part (a) can be simplified to this formula. Page 2 of 3