Math 115 Test 1 Sample Problems for Dr. Hukle s Class

Transcription

1 Mat 5 Test Sample Problems for Dr. Hukle s Class. Demand for a Jayawk pen at te Union is known to be D(p) = 26 pens per mont wen te selling p price is p dollars and eac p 3. A supplier for te bookstore is willing to provide S(p) = 48p 2 pens per mont wen te selling price is p dollars eac and p 0.5. Find te equilibrium price and quantity. 2. A diver springs from a board over a swimming pool. Te eigt in feet above te water level of te diver t seconds after leaving te board is given by (t) = 4 + 2t 6t 2. Wit wat velocity does te diver it te water? (A) -4 ft/sec (B) -32 ft/sec (C) -2 ft/sec (D)-20 ft/sec (E) -2.5 ft/sec 3. Market researc as sown tat te price people are willing to pay for gourmet cocolates is given by te demand function p = 20 x. Find te marginal revenue function MR(x). 4. Te matematics office sells sample midterms for 50 cents eac. Te cost of producing x tests (in cents) is given by C(x) = x + x 2. (a) Find te equations for te revenue function R(x) and profit function P (x). (b) For wat production levels is P (x) = 0? Solve tis algebraically. (c) Find te rate of cange of te profit function wit respect to te production level x. (d) Find te rate of cange of te profit function wen x = 50. (e) Find te production level wic yields maximum profit. (Hint: look at te grap.) 5. If te price of a product per item is given by p(x) = x 2 + 2x + 4 and te total cost function is given by C(x) = 8 + x were x is te number of items produced and sold. Find te profit function P (x). How is te profit canging wen te production and sales are x = 6? 6. Given a number c, find te x-coordinate of te point at wic te curve y = x 2 + c/x as a orizontal tangent line. 7. Find te following limits if tey exist. If te limit does not exist write DNE. (a) lim x 4 x 2 6 x 2 2x 8 = 5x 22 3x (d) lim x 0x 22 = (e) lim 3x 2 4 = 7 x 3 (g) 3t t 6x 3 + 4x 2 7x (b) lim = (c) lim t 0 t x x 4 = + 3x (4 + ) 2 6 (f) lim = 0 2x lim x x 2 = () lim = (i) lim = + x 3 x + 3 x x + 3 x + 3 (j) lim x = (k) lim x x 2 = + 3

2 8. Let = 2 x. Find te following using your graping calculator, rounding to 3 decimal places. (a) An equation of te line tangent to f at x =.7 (Use zdecimal window) (b) Find f(.52) and f(3.9) (c) Find f (2.7). 9. A farmer as 20 meters of fencing wit wic to make a rectangular pen. Te pen is to ave one internal fence running parallel to te end fence tat divides te pen into two sections. Te lengt of te larger section is to be twice te lengt of te smaller section. Let x represent te lengt of te smaller section. Find a function in x tat gives te total area of te pens. 0. A tool rental company determines tat it can acieve 500 daily rentals of jackammers per year at a daily rental fee of $30. For eac $ increase in rental price, 0 fewer jackammers will be rented. Let x represent te number of $ increases. Find te function R(x) giving total revenue from rental of te jackammers.. A omeowner wises to enclose 800 square feet of garden space to be laid out in te sape of a rectangle. One side of te space is to ave a stone wall costing $50 per running foot. Te oter tree sides are to ave cedar fencing costing $20 per foot. Let x represent te lengt of te stone wall. Find a function in x giving total cost of te enclosure. 2. A manufacturer as montly fixed costs of $40,000 and a production cost of $8 per unit produced. Te product sells for $2 eac unit. Te profit (or loss) wen te production level is 2,000 units is A) $8000 B) $48,000 C) $40,000 D) -$48,000 E) None of te above. 3. Let = x x 2. (a) Find te slope of te line tangent to te grap of f at te point x = a by using te definition of te derivative. To receive credit you must sow all your work. (b) Using your results from part (a), find an equation of te line tangent to te grap at x = Te function f is continuous on te closed interval [0, 2] and as values tat are given in te table below. Te equation = 3.5 must ave at least two solutions in te interval [0, 2] if k = x 0 2 k 3 (a) 0 (b) 2 (c) (d) 4 (e) 3 5. Carefully grap 2

3 { 3 x + x < = 2x x Te grap of a function f is sown above. Wic of te following statements about f is false?. f is not continuous at x = x 0 2. x = x 0 is in te domain of f. 3. lim = lim x x 0 x x lim x x 0 f(x 0 ). 5. f is not differentiable at x = x Sow tere is a real number tat is equal to its cube plus one. 8. Let = /x. Wen simplified, te difference quotient (A) x(x+) (B) x( x) (C) x 2 (D) x(x ) (E) x(x+) f(x + ) becomes 9. Find te slope of te line tangent to te grap of = by using te limit definition of te x + 7 slope. Using tis answer, find te equation of te line tangent to te grap of = at x=7. x Consider te grap of sown to te rigt. 3

4 (a) lim = x ( ) (b) lim = x (c) lim x 2 = (d) f(2) = (e) f() = (f) is NOT continuous at x = (g) is NOT differentiable at x = (f) lim x 2 + = y 3 2!4!3!2! 2 3! x 2. Let f be te function defined by x +. Te domain of f is x 2 A) (, 2) (2, ) B) (, 2] [2, ) C) [, 2) (2, ) D) [, ) E) None of te above 22. Te following data is known about te function f and g. Find te value of: t f(t) f (t) g(t) g (t) (a) d [f(t)g(t)] at t = 3. dt (b) d dt (g(t) t 2 ) at t = 5. (c) d ((f(t) + g(t)) at t =. dt 23. If we correctly calculate te derivative of = x directly from te definition of derivative, wic of te following will appear in our calculations? (A) lim 0 x 2 + (B) lim x 2 + (C) lim 0 2x + (D) lim 2x + (E) lim 0 x + (F) lim x Let y = x 7. Wat is te definition of dy dx? x 7 (A) lim 0 x 7 (F) lim (B) lim 0 (x + ) 7 (C) lim (x + ) Find a function and a real number a suc tat f (a) = lim 0 (D) lim 0 (x + ) 7 x (E) lim (x + ) 7 x 7 4

5 26.Wat is te profit in terms of te cost C(x) to produce x units and te unit price p(x) at wic x units will sell? (A) x(p(x) C(x)) (B) p(x) xc(x) (C) p(x) + xc(x) (D) xp(x) C(x) (E) None of te above 27. A diswaser company can produce up to 00 diswasers per week. Sales experience indicates tat te manufacturer can sell x diswasers per week at price p were p and x are related by te demand function p = 600 3x. Te company s weekly fixed costs are 4000 dollars and production records sow tat it costs 50x + 0.5x 2 dollars to make x diswasers.. Find te Cost function C(x) and compute te total cost of manufacturing 75 diswasers in a week. 2. Find te Average Cost function C(x) and compute te average cost at a weekly production level of 75 diswasers. 3. Find te Marginal Revenue function MR(x) and use it to estimate te revenue received wit te sale of te 75t diswaser in a week. 4. Find te Profit function P (x) and compute te profit earned from te sale of 75 diswasers. True or False Questions:. If is not defined at x = a, ten lim does not exist. T F 2. If bot lim and lim exist, ten lim exists. T F + 3. If lim and lim g(x) exist, ten lim g(x) exists. T F 4. If lim = 0 and lim g(x) = 0, ten lim does not exist. T F g(x) 5. If lim = L 0 and lim g(x) = 0, ten lim does not exist. T F g(x) 6. If lim exists, ten f is continuous at x = a. T F 7. If is continuous at x = a, ten lim exists. T F 8. If is continuous at x = x 0, ten as a derivative at x = x 0. T F 9. If is differentiable at x = x 0, ten is continuous at x = x 0. T F 0. If and g(x) are continuous, ten is continuous. T F g(x). If f is differentiable at x = a, ten lim exists. T F 2. If f and g are differentiable, ten d dx (g(x)) = d dx d dx g(x). T F 5