Hello! these are the answers to my hw. Could you please re write these answers in your own way? i would appreciate your help. Thank you !

1. ($2.1-2.2, 5pts each) Use algebra and the limits properties to find the following limits without graphing. X lim- (x+ 2)2 - 1im ( )=ET(X+2) 13 - 49 x-+2 12 -4 b. lim - 147 1- 4x - 21 not graded = lim x+2 (*+7)(x-71 - - ONE *+1 x- 2 247 ( xF)(x + 3 ) lim *+2 lim *+7 3 - 60 7+7 Small my x-#7 * +3 7+3 Value lim x+2 4 * + 00 * - Z Small pox. x - 8x3 + 16 Im (*7-4) ( x 2 - 4) c. lim 14-2 (1 + 2)2 *-2 (x+2)[x+2) lim ( *T)( x - 2) (*7)( x-2) * 4-2 ( x7 ) ( *+ ) lim ( x-2)(x-2) = (-2-2)6-2-2) = 16 2. ($2.3, 5pts each) Without graphing, find the interval(s) for which the following functions are continuous. 1 a. f (1) = - 2x - 7 Var + 8 b. f(0) = = 3*+$20 and 3x+[#6 y=e" is never Zero, so e"to for any -valve . 32+6>0 So, of is continuous everywhere 3 * 7-8 (-80, DO) or R3. ($2.4) Use the four-step process to find the derivative f'(x) for each of the following functions. a. (10pts) f(3) = 1. fleth ) = A. flath ) - f(x] = 2 ( # H ) - 1 2x+24- 2* +24-1 3. fixth ) - f (x ) (2 +24-1)(2x-1) Zx+7h-1 (2*+24-1)(2*-1) (2 FTW ) (2x-1) - - (ZX + 2 4-1)(2#T) h (2*+24-1)(2x-D) (2x-1)-(2*+24-1) 25x-2x-24+1 - 24 h(2x+24-1) (2x-) 4(Z*+24-1)(2x-) K (2x+24-13(2x-1) - 2 - 2 4 lim -2 (2x+24-1)(2x-1) no (2x+ 26-1)(2x1) (2x-132 b. (15pts) f(1) =9 - VF. 1. f( x th) = 9-1xth 2 . f ( x th) - f ( x) = 9- Fxth - ( 9 - Vx) = A - Jxth - 1+Vx = Vx-vxth ( vX + 5xth) ( J X + V xth ) ( 5X ) - ( rxth ) * - (*+h) * - x -h = h (fx + vx+h ) " (vx + Vxth) h ( vi + vath) - K K( X + V Xth) i + Vath 4. lim - 1 VX + 1x1. ($2.7) An energy company knows that the total cost (in dollars) of producing a gallons of synthetic oil is given by C(x) = 12500 + 202" for x 2 0. Use this function to answer the following parts. a. (15pts) Find the average cost function of producing a gallons of synthetic oil. Then find the marginal average cost function of producing a gallons. C( x) = 12500 + 203 11500 20x* = -= 125002 + 2ox C'(x) = [12500x" +20*"] = 12500-(-1)x*+ 20.2x = - 12500 * * + 40x b. (5pts) Use your answers in part a to compute and interpret C(100) and C'(100). Round your final answers to two decimal places if necessary. C (180) = 12500 (100)+ 20(100) = $200125 - the avg cost per gallon to produce 100 gallons C'1 100) = - 12500 ( 100)+ 40 ( 190) = $3798.75 - the rate of change of avg cast per gallon it increating by $1598-75 per gallon! c. (5pts) Use your answers in part b to estimate the exact cost producing the 101st gallon of synthetic oil. Then estimate the average cost per gallon if 101 gallons are the production level. Recall C (x ) 2 exact cost of producing x+1 unit C' ( x ) = 12500 + 20*3 = 0+ 20.3x2 = 60x2 6' (100) = 60 ( 12012 = $60Dood = exact cast to produce lost gallon We know at a production level 100 gallons , cast increases by $ 3998.75 , Co... $200/15 + $3998.76 = $204173. 765. ($3.2) Find the equation y = ma + b of the tangent line of (I, f(x)) for each of the following functions. Round your answers for m and b to three decimal places if necessary. a. (10pts) f(x) = 213 - ed with (0, f(0)) Finding the Slope on f'(x) = = [2x1 -(* ] = 2.2x' -0* = 4x -* f ' co ) = 4 ( 0 ) - e" = 0- 1 = 1 - slope m Finding the value b f(0] = 260 )" - 2" = 0 - 1 =-1 + (0, 1 ) = (*,y) -1 = - 1(0)+b -1 = b Y = - 1x - 1 + equation of tangent live b. (15pts) f(x) = In (- ) with (2e, f(2e)). Hint: use the properties of logs to simplify this function. f (x) = In ( *") = In(x3) - 19(8) = 3 ln(x)-In(8) finding the slope m fix) = $ [3 In(x)-In (8] ] = 3.4 -0 = 2 (' cze ) = 30562 - slope m finding the value be f(zel = 3 1 ( 20)-In (8) = mm((ze) )-luke) = In( Be? ) - In(?) = In(set " ) = me ' ) = 3 ( x , y ) = (Ze, 3 ) 3 = = (20)+ b ze 3 : 3+b Ze T y= 0. 562x