University of Regina Department of Mathematics and Statistics Math 111 All Sections (Winter 2013) Final Exam April 25, 2013

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1 University of Regina Department of Mathematics and Statistics Math 111 All Sections (Winter 013) Final Exam April 5, 013 Name: Student Number: Please Check Off Your Instructor: Dr. R. McIntosh (001) Dr. S. M c Cann (00 & 003) Dr. P. Semukhin (004) Instructions: It is strongly advised to read over the entire exam before starting. You will have 180 minutes to complete this exam. For each question, place all of your work in the given work space. The last page of this exam is the formula list and the second last page has been left for scrap work. ***** Only calculators not capable of electronic communication, programming, Good Luck! integration, or differentiation are permitted. ***** Questions 1 & /8 Question 3 /16 Question 4 /1 Question 5 /1 Question 6 /18 Question 7 /6 Question 8 /10 Question 9 /10 Question 10 /8 Final Exam Grade: /100

2 Page of 1 1. (4 points) Find the inverse, f 1 (x), of f(x) = 3x+1 x+1.. (4 points) Find (f 1 ) (5), where f(x) = x 3 + 4x + 5. Observe that f(x) is an increasing (hence, one-to-one) function since f (x) > 0.

3 Page 3 of 1 3. (16 points) Find the derivative of the following functions. You do not have to simplify your answers, however, expanding the function before differentiating maybe helpful. (a) f(x) = e sin x sin e x (b) h(t) = ln( (t 4)e t t+1 ) (c) g(u) = u arctan(u) (d) f(x) = x x

4 Page 4 of 1 4. (1 points) Evaluate the following limits. sin(πx) (a) lim x 0 ln(x + 1) (b) lim x ( x x + 1 ) x (c) lim x (ln(3x + 1) ln(x + 1))

5 Page 5 of 1 5. (1 points) Evaluate the integral (a) A substitution u = g(x) : x 3 dx by the following methods: 1 x (b) A trigonometric substitution: (c) Long (polynomial) division followed by partial fractions:

6 Page 6 of 1 6. (18 points) Solve THREE of the following FOUR integrals π (a) x 1 sin x dx (c) 0 (x + 1) (b) tan 3 x sec 4 x dx (d) x 1 dx 3/ dx

7 Page 7 of 1 7. (6 points) Find the arclength of the curve f(x) = x x = 4 1 ln x from x = 1 to

8 Page 8 of 1 8. (10 points) Consider the solid of revolution obtained by rotating about y = the region bounded by the curves y = x and y = x (a) Set up (but do not evaluate) the volume integral by using the washer method. (b) Set up (but do not evaluate) the volume integral by using the cylindrical shell method. (c) Evaluate one of the above integrals to find the volume of the solid. You may wish to confirm your answer by doing the other integral on scrap paper.

9 Page 9 of 1 9. (10 points) Solve the following differential equations: (a) dy dx = x y y + 1 ; y(3) = (b) (e y y cos xy)dx + (y + xe y x cos xy)dy = 0

10 Page 10 of (8 points) Suppose that a large mixing tank initially holds 100 gallons of water in which 50 pounds of salt has been dissolved. Another brine with a concentration of 10 pounds per gallon is pumped into the tank at a rate of 6 gallons per minute. The solution in the tank is kept uniformly mixed and is drained from the tank at same the rate of 6 gallons per minute. (a) Construct a differential equation for the amount y(t) of salt in the tank at time t. (b) Solve the differential equation. (c) Calculate the limiting (at t ) concentration of salt (pounds per gallon) in the tank. How does this compare with the incoming concentration?

11 Page 11 of 1 Extra Page for Scrap Work

12 Page 1 of 1 Formula List: sin x + cos x = 1 tan x + 1 = sec x 1 + cot x = csc x sin A cos B = 1 [sin(a B) + sin(a + B)] sin A sin B = 1 [cos(a B) cos(a + B)] cos A cos B = 1 [cos(a B) + cos(a + B)] sin x = sin x cos x cos x = cos x sin x sin x dx = cos x + C cos x dx = sin x + C tan x dx = ln sec x + C sec x dx = ln sec x + tan x + C sec x dx = tan x + C sec x tan x dx = sec x + C csc x dx = ln csc x cot x + C csc x dx = cot x + C csc x cot x dx = csc x + C cot x dx = ln sin x + C sin x = cos x = 1 cos x 1+cos x 1 dx = ln x + C x a x dx = ax + C, a > 0, a 1 ln a ln x dx = x ln x x + C 1 a x dx = arcsin( x a ) + C 1 a +x dx = 1 a arctan( x a ) + C 1 x dx = 1arcsec ( x) + C x a a a L = b 1 + [ dy a dx ] dx V Washer = b a π[r(x) r(x) ] dx V Cyl = b πr(x)[h(x) h(x)] dx a sec 3 x dx = 1 [sec x tan x + ln sec x + tan x ] + C (Linear Integrating Factor) µ(x) = e P (x) dx ; y = µ(x) 1 [ µ(x)q(x) dx + C] (Mixing Problems) da = input rate - output rate dt (Newton s Law of Cooling) dt = k(t T dt m) (Exponential Growth) dp = kp dt (Logistic Equation) dp = kp (1 P ) dt M

Have a Safe Winter Break

Have a Safe Winter Break SI: Math 122 Final December 8, 2015 EF: Name 1-2 /20 3-4 /20 5-6 /20 7-8 /20 9-10 /20 11-12 /20 13-14 /20 15-16 /20 17-18 /20 19-20 /20 Directions: Total / 200 1. No books, notes or Keshara in any word