Find the derivative and simplify. y = (8x+7)(x² - 3x). y' = Find the derivative and simplify. f(x) = (x12+2x+3)(5x³- 1) f'(x) = Find the indicated derivative. for z = x²+ dx x2 1-x-5x2 dz dx x²+2 For the function y = x+8 at (3, 1), find the following. (Give exact answers. Do not round.) (a) the slope of the tangent line (b) the instantaneous rate of change of the function If the cost C (in dollars) of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by the equation shown below. C(p) = 6100p 100-p For the function y = (x² + 4)(x³- 9x), at (-3, 0) find the following. (a) the slope of the tangent line Find the rate of change of C with respect to p. C'(p) = (b) the instantaneous rate of change of the function Find the indicated derivative. dp - for p = dq 92+2 69-5 dp da Find the indicated derivative. dx dy 1- 2x2 for y= dx x4-2x²+3 Suppose the revenue (in dollars) from the sale of x units of a product is given by R(x) = 36x² + 50x 2x + 2 Find the marginal revenue when 35 units are sold. (Round your answer to the nearest dollar.) $ Interpret your result. When 35 units are sold, the projected revenue from the sale of unit 36 would be $ The demand for q units of a product depends on the price p (in dollars) according to 9 = 512 Vp 1, for p > 0. Suppose that the demand for a product depends on the price p according to 50,000 1 D(p) = --༔ p2 p>0 Find and explain the meaning of the instantaneous rate of change of demand with respect to price when the price is as follows. (a) $16 where p is in dollars. Find and explain the meaning of the instantaneous rate of change of demand with respect to price for the following values of p. Interpret the instantaneous rate of change. ○ If price decreases by $1, the demand will drop by the absolute value of this number of units. (b) $64 If price increases by $1, the demand will drop by the absolute value of this number of units. If price increases by $1, the demand will increase by the absolute value of this number of units. If price increases by the absolute value of this amount, the demand will drop by 1 unit. If price decreases by the absolute value of this amount, the demand will drop by 1 unit. Interpret the instantaneous rate of change. ○ If price decreases by $1, the demand will drop by the absolute value of this number of units. ○ If price increases by $1, the demand will drop by the absolute value of this number of units. If price increases by $1, the demand will increase by the absolute value of this number of units. If price increases by the absolute value of this amount, the demand will drop by 1 unit. If price decreases by the absolute value of this amount, the demand will drop by 1 unit. (a) p= 50 Interpret the instantaneous rate of change. If price increases by $1, the demand will increase by the absolute value of this number of units. If price decreases by this amount, the demand will drop approximately 1 unit. If price increases by this amount, the demand will drop approximately 1 unit. If price decreases by $1, the demand will drop by the absolute value of this number of units. If price increases by $1, the demand will drop by the absolute value of this number of units. (b) p 100 Interpret the instantaneous rate of change. If price increases by this amount, the demand will drop approximately 1 unit. If price increases by $1, the demand will increase by the absolute value of this number of units. If price decreases by this amount, the demand will drop approximately 1 unit. If price increases by $1, the demand will drop by the absolute value of this number of units. If price decreases by $1, the demand will drop by the absolute value of this number of units. Cost is in dollars and x is the number of units. Find the marginal cost function MC for the given cost function. C(x)=60+ 4x MC = Cost is in dollars and x is the number of units. Find the marginal cost function MC for the given cost function. C(x) 900+12x + 0.03x² MC =