A marketing specialist determines that when a certain product is released, the number of social media references to it can be modeled by f(t) = 450 + 510g8(t) where t is the number of days since the release. Determine f'(t) = [ r=i i 'b. Use part a. to determine the rate at which the population is increasing in t years. Use exact expressions. P' (t) 2 people per year c. Use part b. to determine the rate at which the population is increasing in the year 2022. Round to the nearest person per year. P'(22) = H 7 7 7 jpeople per year An isotope of the element erbium has a half-life of approximately 10 hours. Initially there are 15 grams of the isotope present. a. Write the exponential function that relates the amount of substance remaining, A(t) measured in grams, as a function of t, measured in hours. A(t) = ' I grams .b. Use part a. to determine the rate at which the substance is decaying after t hours. A'(t) =[ I grams per hour c. Use part b. to determine the rate of decay at 19 hours. Round to four decimal places. A'(19) = grams per hour Let f(x) = 10In(x). Then f'(a) is and f'(3) is f"(ac) is and f" (3 ) isLet f(x) = -3 1n(3x) f' (2 ) = f' (3) =d Find (7*). Type In(x) for the natural logarithm function.If f(a) = 2In(4 + a), f' (20 ) = f' (5) =If f(a) = 5 cos(4 In(a) ), find f' (a). Find f' (5).Let f(x) = 6xIn(x) + 4 A CO f' (2 )Let f(ac) = 5x In x f' (ac) = f' ( e2 ) =\fGiven f(a) = 3ex+3 df dxLet f(a) = 4x + 12 - 19e. Then the equation of the tangent line to the graph of f (a) at the point (0, -7) is given by y = ma + b for m b =\fConsider a closed rectangular box with a square base with side a: and height y. a. Find an equation for the surface area of the rectangular box. 99mg) = b. If the surface area of the rectangular box is 144 square feet, find _y when a: = 4 feet and y = 7 feet. da: _ dw_[J The volume of a right circular cone of radius a: and height y is given by V = Ewmzy. Suppose that the d Dolume of the cone is 1927r cm3. Find dy when m = 6 and y = 16. :1: _[ ] dw A manufacturer can produce 4050 cell phones when :1: dollars is spent on labor and y dollars is spent on g A capital. The equation that relates :n and y is 75:13 4 y4 = 4050. d a. Find a formula in terms of w and y for _y da: ' dy _ dz: [J d b. Find the value of of (1: at the point (81,16). ' Round to 4 decimal places. The number of cars produced when 1: dollars is spent on labor and y dollars is spent on capital invested by l 2 a manufacturer can be modeled by the equation 55:13 3 y3 = 660. d a. Find a formula in terms of an and y for y. ' dz): kl ] da: d b. Find the value of of at the point (27,8). ' Round to 4 decimal places. \fIf va+ vy = 12 and y(4) = 100, find y'(4) by implicit differentiation.Find the slope of the tangent line to the curve 4:82 + 3233/ + 43/3 = 272 at the point (4, 4). Find the slope of the tangent line to the curve at the point (4, 4). Given the equation below, find dy. a: 16:1:3 + 9m64y + y7 = 26 03%: Now, find the equation of the tangent line to the curve at (1, 1). Write your answer in mm + b format y=[ Find the slope of the tangent line to the curve 1x2 + lay + 3y = 5 at the point (1, 1).Find the slope of the tangent line to the curve 2:1:2 my 4y3 2 44 at the point (2, 2). dy given the equation a + y' = -5. Use implicit differentiation to determine dx dy dxGiven the equation below, find dy. 1: Has9 + 69:22:; + y5 = 18 D zi da: Now, find the equation of the tangent line to the curve at (1, 1). Write your answer in ma: + b format In the graph of y = \\7 csc (# - 2x) - e, what is the coordinates of the "SECOND" point? A. (7, -e) B. 7 7 10 , VT - e C. 7 7 20 , V T - e D. 7 7T 20 '- VTT - e) OC OD O B OAIn the graph of y - 2 sin (5x - ) + 1, what is the coordinates of the "FOURTH" point? A. 1071 . -3) 30 ' B. 1071 30 , 2) C. (5, 1) D. (#, -1) OA OD O B OCIn the graph of y = v7 tan(5x - # ) - 2, what is the x - coordinate of the STARTING point? A. - 971 70 B. 14 C. 14 D. 35 OA OD OB OC\fFind: cot (Csc 1 (*)) A. V72 -16 4 16-712 B. 4 C. V2 D. none OA OC OD OBGiven: sin 36 = ", find in exact value, tan(-414) A. 3-e e B. colro C. - V9-e2 3 D. - V9-e2 e OC OA OD OBFind: Sin 1 (sin A. 5 TT 6 B. C. D. None OA OB OC ODFind: sec(123456789987654321TT) A. 0 B. 1 C. -1 D. none OD OC OA O B\fFind: sin ( Cos-1 2 A. N/H B. C. 57T 6 D. none OD OB OC OA