Math 122 Test 2. October 15, 2013

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1 SI: Math 1 Test October 15, 013 EF: Total Name Directions: 1. No books, notes or Government shut-downs. You may use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone.. You should show your work, and explain how you arrived at your answers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. 3. You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. 4. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. 5. This test has 7 problems.

2 1. (0 points) (a) Find the arc length of the curve y = ln(cos x) for 0 x π 4. (b) Determine the force on the following triangle plate that is submerged in water (ρ =9810 N/m 3 ) as shown (distances are in meters).!

3 . (0 points) (a) Find the center of mass of the region bounded by y = cos(x) and y = 0 on the interval [ π 4, π ] 4 (b) Below is the graph of y = f(x) Which of the following could be the second degree Taylor polynomial for f(x) at a = 1. i. T (x) = (x 1) + 3(x 1) ii. T (x) = (x 1) 3(x 1) iii. T (x) = + (x 1) + 3(x 1) iv. T (x) = + (x 1) 3(x 1)

4 3. (10 points) For the differential equation: dy dx = y + x (a) Fill-in the missing numbers in the chart below and sketch the slope field for x =, 1, 0, 1, and y =, 1, 0, 1, x y (b) Use Euler s method with h = 1 to find y() if y(0) =

5 4. (15 points) Find the solution for the following differential equations. (a) y = 3x y ; y(0) = 1 (b) dy dx + 3x y = 6x

6 5. (10 points) A mug of hot chocolate with initial temperature 65 o C is in a room held at 0 o C. After 30 minutes, the temperature of the hot chocolate is 55 o C. What is the temperature of the hot chocolate after another 10 minutes? 6. (10 points) A deer population for a certain area is initially y 0. After 6 years, the population increases to 50. After another 4 years, the population increases to 300. Assuming logistic growth with a carrying capacity of 400, what is y 0?

7 7. (15 points) Chris needs to make some pancake batter for his morning 10 sit-ups, 10 push-ups, 10 algebra problems and 10 pancakes ritual. In a 300 gallon tank, he initially has 00 gallons of batter containing 5 lbs of chocolate chips. Batter containing 1 lb of chocolate chips per gallon enters the tank at the rate of 4 gal/min, and the well-mixed brew in the tank flows out at the rate of gal/min. (a) Write a differential equation for the amount of chocolate chips in the tank (with initial condition). (b) Solve the differential equation from part (a). (c) How many pounds of chocolate chips will be in the tank after 0 minutes?

8 FORMULA PAGE sin θ + cos θ = 1 tan θ + 1 = sec θ 1 + cot θ = csc θ sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β tan(α + β) = tan α + tan β 1 tan α tan β sin 1 cos x x = cos 1 + cos x x = sin x = sin x cos x (sin x) = cos x (cos x) = sin x (tan x) = sec x (sec x) = sec x tan x (csc x) = csc x cot x (cot x) = csc x (e x ) = e x (sinh x) = cosh x (cosh x) = sinh x (arcsinh x) 1 = x + 1 (arccosh x) 1 = x 1 (arctanh x) = 1 1 x x n+1 = x n f(x n) f (x n ) f (c) = f(b) f(a) b a f(c) = 1 b f(x)dx b a a sec x dx = ln sec x + tan x + C sec 3 x dx = 1 [sec x tan x + ln sec x + tan x ]+C csc x dx = ln csc x cot x + C y = A C = y 0 1 e kt y 0 A C = (ln x) = 1 x (arcsin x) 1 = 1 x (arctan x) = x (arcsec x) = 1 x x 1 sinh x = ex e x cosh x = ex + e x cosh x sinh x = 1

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