Math 1425.P70 Examples for 4.2 & 4.3 Solve the problem. 1) A special-events promoter sells x tickets and Name _____________________________________________ 3) Hatts and Company determines that its marginal cost, in dollars per hat, is given by 3 C'(x) = x + 40, for x ≤ 350. 40 has a marginal-revenue function given by R'(x) = 2x - 1240, where R'(x) is in dollars per ticket. This means that the rate of change of total revenue with respect to the number of tickets sold, x, is R'(x). Find the total revenue from the sale of the first 340 tickets. 2) Rejoyne Inc. has a marginal-profit function Find the total cost of producing the first 260 hats. 4) Accent Woodworkers knows that their given by P'(x) = -3x + 140, where P'(x) is in dollars per unit. This means that the rate of change of total profit with respect to the number of units produced, x, is P'(x). Find the total profit from the production and sale of the first 30 units. Examples for 4.2 & 4.3 marginal cost of producing x feet of custom molding is given by C'(x) = -0.00003x 2 - 0.04x + 90, for x ≤ 1000, where C'(x) is in cents. Approximate their total cost, in dollars, of manufacturing 1000 feet of molding, using 5 subintervals over [0, 1000] and the left endpoint of each subinterval. Write summation notation for the expression. 5) 7 + 14 + 21 + 28 + 35 Find the area under the given curve over the indicated interval. 11) y = 7; [2, 9] y 8 7 6 5 6) 8 + 16 + 24 + 32 + 40 + 48 4 3 2 1 1 2 3 4 5 6 7 7) f(x1 ) + f(x2 ) + f(x3 ) + f(x4 ) + f(x5 ) + f(x6 ) 12) y = 2x + 1; [1, 3] y 9 8 7 8) g(x1 ) + g(x2 ) + . . . + g(x16) 6 5 4 3 2 1 Express the sum without using summation notation. 6 9) ∑ h(xi) i=1 1 2 3 4 5 13) y = x2 + 3; [0, 2] y 9 8 7 6 5 10) 5 ∑ 4i i=1 4 3 2 1 1 Math 1425.P70 Examples for 4.2 & 4.3, page 2 2 3 4 x x 8 9 10 11 x 14) y = 1 ; [0.5, 2] x State what the shaded area represents. 17) y 7 6 5 4 3 2 1 1 2 3 4 x 18) 15) y = ex; [1, 2] y 9 8 7 6 5 4 3 2 1 1 2 19) x 3 16) y = (x - 3)2; [2, 4] y 6 5 4 3 2 1 1 2 3 4 5 6 x Math 1425.P70 Examples for 4.2 & 4.3, page 3 20) 24) y = 2x + 7; [1, 5] 25) y = x2 - 6x + 9; [2, 4] 21) 26) y = 2x - x2 ; [0, 2] 22) 27) y = 3 ; [1, 3] x3 Find the area under the graph of the function over the interval given. 23) y = 2x - x2 ; [0, 2] Math 1425.P70 Examples for 4.2 & 4.3, page 4 28) y = -x2 + 9; [0, 3] 31) A well-drilling company finds that its marginal profit, in dollars, from drilling a well that is x feet deep is given by 3 P'(x) = x. Find the company's profit from drilling a well that is 230 feet deep. 29) y = x2 + 1; [0, 1] 32) A kitchen remodeling company determines that the marginal cost, in dollars per foot, of installing x feet of kitchen countertop is given by C'(x) = 7x-1/3. Solve the problem. 30) A manufacturer determined that its marginal Find the cost of installing an extra 12 feet of countertop after 30 feet have already been ordered. cost per unit produced is given by the function C'(x) = 0.0006x2 - 0.4x + 94. Find the total cost of producing the 401st unit through the 500th unit. 33) A company estimates that its sales will grow continuously at a rate given by the function S'(t) = 15et, where S'(t) is the rate at which sales are increasing, in dollars per day, on day t. Find the sales from the 2nd day through the 10th day. (This is the integral from 1 to 10.) Math 1425.P70 Examples for 4.2 & 4.3, page 5 Answer Key Testname: 1425_SECTION4_2__4_3 1) 2) 3) 4) 5) 6) 7) 8) 9) -$306,000 $2850 $7865.00 $668.00 5 ∑ 7i i=1 6 ∑ 8i i=1 6 ∑ f(xi) i=1 16 ∑ g(xi) i=1 h(x 1) + h(x2 ) + h(x3 ) + h(x4) + h(x 5) + h(x 6) 10) 4 + 8 + 12 + 16 + 20 11) 49 12) 10 26 13) 3 14) 1.39 15) e2 - e 16) 2 3 17) Distance traveled in miles 18) Distance traveled in miles 19) Total mass in grams 20) Total number of births 21) Total cost in dollars 22) Total cost 4 23) 3 24) 52 2 25) 3 26) 4 3 27) 4 3 28) 18 4 29) 3 30) $3569.96 31) $1056.89 32) $25.49 33) $330,356.21 Math 1425.P70 Examples for 4.2 & 4.3, page 6

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