student: Sterling White Instructor: Davrd boot! Assignment: 105 Related Rates Date: 11/10/22 __ Course: Math1325 Calculus for Business 3 dX Assume that x = x(t) and y: y(t). Let y =x + 3 and E=3 when x =1. . dy Find when x z 1. dt {1 t: = I: (Simplify your answer.) 5. dx Assume that x =x(t) and y = y(t). Find 3;. using the following information. dy x2+y2=4.52: E: 2whenx= -1.4and y=1.6 '- dy dx Assume x and y are functions of t. Evaluate ET for 4xy - 5x + 2y3 ' - 88, with the conditions E = - 6, x = 2, y = - 3. dy tit-[:1 (Type an exact answer in simplified form.) A point is moving on the graph of xy= 30. When the point is at (6,5), its x-coordinate is increasing by 6 units per second. How fast is the y~coordinate changing at that moment? The y-coordinate is (1) at |:] units per second. (Simplifyr your answer.) (1) C) decreasing 1:} increasing i. The radius of a spherical balloon is increasing at the rate of 0.4 cm/minute. How fast is the volume changing when the radius is 7.4 cm? The volume is changing at a rate of [: cm3/minute. (Type an integer or a decimal. Round to one decimal place as needed.) i. Boyle's law for enclosed gases states that if the volume is kept constant, the pressure P and temperature T are related by the equation P T where k is a constant. if the temperature is increasing at 4 kelvins per hour, what is the rate of change of pressure when the temperature i: 250 kelvins and the pressure is 500 pounds per square inch? The rate of change of pressure is [:l pounds per square inch per hour. (Simplify your answer.) A point is moving along the graph of xy4 = 648. When the point is at (2.9), its x-coordinate is changing at the rate of - 5 units per minute. How fast is the ycoordinate changing at that moment? The y-coordinate is changing by :l (1) -- (Type an integer or a fraction. Simplify your answer.) (1) C3 units. C} units per minute. C) minutes. Suppose that for a company manufacturing calculators, the cost, revenue, and profit equations are given by 2 X c=70,000+40x, swam-E. P=Rc where the production output in 1 week is x calculators. lf production is increasing at a rate of 600 calculators per week when production output is 4,000 calculators. Find the rate of increase (decrease) in cost, revenue, and profit. A) Costs are (1) _____ at the rate of 33:: per week at this production level. (Simplify your answer.) B) Revenue is (2) _ at the rate of $:: per week at this production level. (Simplify your answer.) C) Prots are (3) __._ at the rate of $|:j per week at this production level. (Simplify your answer.) (1) (:3 increasing (2) C: increasing (3) G increasing C) decreasing C) decreasing (:2! decreasing Student: Sterling White Instructor: David Scott Date: 11/10/22 Course: Math 1325 Calculus for Business Assignment: 10.5 Implicit Differentiation Use implicit differentiation to find y' and then evaluate y' at the point (1,7). y - 6x +5=0 y' = y' la, 7) = L (Simplify your answer.) !. Use implicit differentiation to find y' and then evaluate y' at (2,7). 7x5 - y- - 7=0 y'=] y'| (2,7) = (Simplify your answer.) 1. Use implicit differentiation to find y' for the equation below and then evaluate y' at the indicated point. y' + 4y +6x = 0; (-2,2) y'= y'l(-2,2) = L (Simplify your answer.) Use implicit differentiation to find z' and then evaluate z' at ( - 4, 2). xz + 8= 0 Z'= z'|( -4,2) = (Simplify your answer.) i. Use implicit differentiation to find y' and then evaluate y' at (5, - 3). 6xy+ y + 93 = 0 y' = y'|(5, -3) =L (Simplify your answer.) Use implicit differentiation to find y', and then evaluate y' for x y - 4x-=0 at the point (2,4). y' = y'|(2,4) = (Simplify your answer.) " Use implicit differentiation to find y' and then evaluate y' at ( - 2,0). 16ed = xi + y3 y'= y'l(-2,0) =1 (Simplify your answer.)i. Use implicit amTerentiation to Tina y and then evaluate y at ( - 1, 1). x - y = Iny y' =_ y'l(- 1, 1) = (Simplify your answer.) . Use implicit differentiation to find y' and then evaluate y' at ( - 1,1). x Iny + 4y = 3x y' = y' l( - 1, 1) = L (Simplify your answer.) ". Find x' for x(t) defined implicitly by x+ tox + t* -7 =0 and then evaluate x' at the point ( - 1,2). x' = *'|(-1.2) = L (Simplify your answer.) The demand x is the number of items that can be sold at a price of $p. For x= p* - 2p+ 1100, find the rate of change of p with respect to x by differentiating implicitly. The rate of change of the price p with respect to the demand x is