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1) Peacock Pottery sells large planters

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1) Peacock Pottery sells large planters. Their demand is given by ?(?) = ? = −19? + 1915, where ? is the price per unit and ? is the number of units sold. · a) (5 pts) Find the elasticity of demand, ?(?). · b) (4 pts) Find the elasticity at the price ? = 50 dollar, stating whether the demand is · elastic or inelastic. At a price of $50, will a small increase in price cause total · revenue to increase or decrease? · c) (6 pts) Find the price ? at which Peacock Pottery should sell its planters for · demand to have unit elasticity, and find the maximum revenue. Round your answers to two decimal places. 2. (20pts) The student enrollment at certain college was 2800 in 2015(The base year, t = 0) and was growing continuously at an annual percentage rate of 2.5% per year. · a) Write a function of the form ?(?) = ? ??? that gives the enrollment at this 0 college t years after 2015. · of the enrollment in 2023? Round your answers to a whole number. · c) When will the enrollment reach 4000? · d) When will the enrollment be double what it was in 2015? 3. (10 pts) Find each integral. Consider ? > 0 . a) ∫ (√?5 −9?−3? +5) ?? ? ∫ x5ex6 ?? A kitchen remodeling company determines that the marginal cost, in dollars per foot, of the installing x feet of kitchen countertop is given by −1 ?′(?) = 7? 3 a) (5 pts) Find the cost of installing 30 ft of countertop. b) (5 pts) Find the cost of installing an extra 13 feet of countertop after 30 feet have already been ordered. 5. For the given function, ?(?) = ?3 − 3? + 6 , · a) (6 pts) Find the critical values and determine the coordinates of any extrema using · the second derivative test. · b) (4 pts) Find the points of inflection. · c) (3 pts) Graph the function and label the points of extrema and point of inflection on · your graph · d) (2 pts) State the intervals that the function is increasing or decreasing and · concave up or concave down. ?(?) = −?2 + 8? − 11, 6. (15 pts) For the function ?(?+h) −?(?) a) Find ? (?) by determining lim h . h→0 b) Find the equation of the tangent line at point (3, 4) c) Sketch the graph of the function and the tangent line on the same diagram and label the point of tangency. ′ 7. (15 pts) Let ?(?) = (? − 6)2 be the price, in dollars per unit, that consumers are willing to pay for x units of an item, and ?(?) = ?2 + 12 be the price, in dollars per unit, that producers are willing to accept for x units. · a) Find the (x, y) coordinates of the equilibrium point · Use the Integral formula to compute the following & Label the graphs · b) Find the consumer surplus at the equilibrium point and graph its region. · c) Find the producer surplus at the equilibrium point and 1. Peacock Pottery sells large planters. Their demand is given by ?(?) = ? = −19? + 1915, where ? is the price per unit and ? is the number of units sold. • • a) (5 pts) Find the elasticity of demand, ?(?). b) (4 pts) Find the elasticity at the price ? = 50 dollar, stating whether the demand is • elastic or inelastic. At a price of $50, will a small increase in price cause total • revenue to increase or decrease? • c) (6 pts) Find the price ? at which Peacock Pottery should sell its planters for • demand to have unit elasticity, and find the maximum revenue. Round your answers to two decimal places. 2. (20pts)Thestudentenrollmentatacertaincollegewas2800in2015(Thebaseyear, t = 0) and was growing continuously at an annual percentage rate of 2.5% per year. • a) Write a function of the form ?(?) = ? ??? that gives the enrollment at this 0 • college t years after 2015. • b) Use the model to predict the enrollment in 2023, and find the rate of change • of the enrollment in 2023? Round your answers to a whole number. • • c) When will the enrollment reach 4000? d) When will the enrollment be double what it was in 2015? 3. (10 pts) Find each integral. Consider ? > 0 . a) b) ∫ ? ∫ (√?5 −9?−3? +5) ?? ? ? ?? 5 ?6 A kitchen remodeling company determines that the marginal cost, in dollars per foot, of the installing x feet of kitchen countertop is given by −1 ?′(?) = 7? 3 a) (5 pts) Find the cost of installing 30 ft of countertop. b) (5 pts) Find the cost of installing an extra 13 feet of countertop after 30 feet have already been ordered. 5. For the given function, ?(?) = ?3 − 3? + 6 , • a) (6 pts) Find the critical values and determine the coordinates of any extrema using • the second derivative test. • • b) (4 pts) Find the points of inflection. c) (3 pts) Graph the function and label the points of extrema and point of inflection on • your graph • d) (2 pts) State the intervals that the function is increasing or decreasing and • concave up or concave down. ?(?) = −?2 + 8? − 11 , 6. (15 pts) For the function ?(?+h)−?(?) a) Find ? (?) by determining lim h . h→0 b) Find the equation of the tangent line at point (3, 4) c) Sketch the graph of the function and the tangent line on the same diagram and label the point of tangency. ′ 7. (15 pts) Let ?(?) = (? − 6)2 be the price, in dollars per unit, that consumers are willing to pay for x units of an item, and ?(?) = ?2 + 12 be the price, in dollars per unit, that producers are willing to accept for x units. • a) Find the (x, y) coordinates of the equilibrium point • Use the Integral formula to compute the following & Label the graphs • • b) Find the consumer surplus at the equilibrium point and graph its region. c) Find the producer surplus at the equilibrium point and graph its region