MA 241 Calculus 1 Answer Key: Practice Problems for Test 2 Fall 2022 1. 2 x4/5- - 2+ +8ex 2. (a)y' = 1+2_6 14 + 25, (b) y' = 4x3 - 8x , (c) y' = -2 tan 0 sec2 0 3. (a) y' = x(2Inx +1) , (b) y' = 0(2 tan 0 + 0 sec2 0) , (c) y' = x2e *(3 -x) , (d) y' = cos2 x - sin2 x 4. (a) f'(x) = 2(1-x) 3 secx 3ex , (b) f' (x) = (* tanx) , (c) f' (x) = , (d) f'(x) = - csc2 x 5. (a) y' = -- 5 . y" -25 250 3750 4-5x (4-5x) 2 y'll = y(3) = - (4-5x)3 y4)= - (4-5x)" (b) y' = 2 cos(2x) , y" = -4sin(2x) , y'' = y(3) = -8 cos(2x) , y(4) = 16 sin(2x) (c) y' = -2e-2x , y" = 4e-2x , y'll = y(3) = -8e-2x , y(4) = 16e-2x (e) y' = 3x2 - 2 , y" = 6x , y"' = y(3) = 6, y(4) = 0 6. (a) 15x4 sec2(3x5) , (b) 7(18x5 - 8)(3x6 - 8x)6 , (c) - csco cote ecsce , (d) -2 tan(2t) (e) - - 30x2 (5x3_1)s , (f) 45x +1 (In 4) 10x , (g) . (h) ax = x242 2x 2(3x+1)] \x(3x+1) x2 +2 7. (a) dy _ 3-3x ty (b) b) dy _ 1-y sin(xy) 4-x dx 1+x sin(xy) ' (C) -=- 1-2y2+xy (d) dy -10yz+y dx x+y? seczy 8. (a) y = 2x - 8, (b)y = 4x + 1 , (c) y = -x+4 9. 72mr miles?/hour 10. 1/32n m/sec 11. 8/3 ft/sec 12. 46.86 mph 13. (a) = [In(cosx) - x tan x](cosx)* , (b) a = [In(x) + 1]xx , (c) dy = [cos(x) In(x) + sinx] xsinx [In(0)+2]tNIH. Lil]. bILLIIU: A 7. 8. 9. . . - d? Use implicit differentiation to find 5' 4y+x3 Jul'31: " ]"U)#y2+xy=0 a_ s A 5. _. tan : 101' b. y costxy) = x d' y 0') Find the tangent line to the curve: _ 4 . a. y = 21n(x 3) at the point (4,0) b. y e " at the point (0,1) c. y 2 i at the point [3.1) A search and rescue airplane is ying in a circle with radius expanding at a rate of 3 mph. At what rate is the search area expanding when the radius is 12 miles? {Answers should be exact.) 10. A solid spherical ball is heated and begins to expand. At what rate is the radius increasing when the radius is 4 m and the volume is increasing at a rate of 2 m'j'sec? [Answers should be exact.) {Volume of a sphere is V x 'i a 3m.) 11. A 1?-foot ladder is leaning against a wall when it begins to slide away from the wall. At what rate is the ladder sliding down the wall when the top of the ladder is at a height of 15 feet and the base of the ladder is slipping away from the wall at a rate of Bfo'sec? 12. Two trains leave a station, one headed due west and the other headed due south. How fast the is southern bound train moving if the west bound train is 26 miles from the station traveling at a constant speed of 28 mph, the distance between the trains is 25 miles, and the distance between them is increasing at a rate of 40 mph. [Round answer to nearest hundredth.) 13. Find 1}: for the following equations. a. y = (cos x)' b. y = x" c. y = xsm" d. y t'E