Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

MATH 115 Test 1 (Sec: R - 1.4) Year: 20161 Name: KEY Sec Number: Answer the following questions to the best of your ability showing all appropriate work to obtain full credit. Questions that involve word problems should be answered using a sentence, and appropriate units provided, in order to receive full credit. 1. (4 pts) Write the following expression as an equivalent expression in the form xn . q √ 8 x8 8 1 8 2 =⇒ x 8 = x 16 = x0.5 2. (4 pts) Use factoring to solve the polynomial equation: 2x3 + 6x2 − 20x = 0 =⇒ 2x(x2 + 3x − 10) = 0 =⇒ 2x(x + 5)(x − 2) = 0 =⇒ roots x = 0, x = −5, and x = 2 3. (4 pts) Solve the following equation: 17 2x2/5 = 6 x1/2 9 10 =⇒ 102 = 2x1/2 x2/5 =⇒ 102 = 2x 10 =⇒ x = (51) 9 ≈ 78.94056 1 4032 4. (6 pts) Use the following figure to evaluate the given limits. Write D.N.E. for a limit that does not exist. 6 4 2 -10 -5 5 10 -2 -4 -6 lim f (x) = 4 lim 0.0 x→−8− f (x) = x→−4.0− lim f (x) = lim f (x) = lim x→−4.0+ 2 x→0− x→4− lim f (x) = x→−8+ 3 −2 f (x) = 0.0 lim f (x) = 2 x→0+ lim f (x) = x→4+ lim f (x) = x→−8 −1 5. (4 pts) Simplify the following expression: (x8 )3 x3 x5 =⇒ x24 = x24−8 = x16 x8 D.N.E. lim f (x) = 0.0 x→−4.0 lim f (x) = x→0 lim f (x) = x→4 2 D.N.E. 6. (5 pts) Calculate the following limits: (a) lim x→6+ 13 = x−6 ∞ x−7 = x→7 x2 − 7x 1 ≈ 0.143 7 (b) lim (c) lim y→9 9−y √ = 3− y 6 (a + 3)5 = a→−5 a + 1 8 ≈ 8.0 80x lim = − 6 − 4x x→( 64 ) ∞ (d) lim (e) 7. (10 pts) Find the instantaneous rate of change function f 0 (x) for: f (x) = 2x2 + 7x + 4 Using the definition f 0 (x) = = = = = f (x + h) − f (x) h→0 h 2x2 + 22xh + 2h2 + 7x + 7h + 4 − 2x2 − 7x − 4 lim h→0 h 4xh + 2h2 + 7h lim h→0 h h lim (4x + 2h + 7) h→0 h 4x + 7 lim Thus, f 0 (x) = 4x + 7 8. (8 pts) Bobby decides to mow lawns to earn money. The initial cost of his lawn mower is $300. Gasoline and maintenance costs are $5 per lawn. • Find a cost function C(x) for the total cost of mowing x lawns. The cost function for mowing x lawns is: C(x) = 5 x + 300 • Bobby determines that the total-profit for the lawn-mowing business is given by P (x) = 11x − 300. Find a function for the total revenue for mowing x lawns. How much does Bobby charge per lawn? As profit is revenue minus cost the revenue function can be found by adding P (x) and C(x): R(x) = 16 x Bobby is charging $16 per lawn. • How many lawns must Bobby mow before he begins to make a profit? 11x − 300 = 0 −→ x = 300 = 27.27 11 Bobby will make a profit once he has mowed 28 lawns. • What is the average rate of change in Bobby’s revenue per lawn? Bobby’s revenue is increasing at a rate of $16 per lawn. 9. (4 pts) Let g(x) = 1x − 6, 3x + 2, x < −9 x ≥ −9 Use the definition of continuity to decide if g(x) is continuous at x = −9. Note you must use the limit definition in order to receive full credit. (a) g(−9) = −25. Thus, we can evaluate g at x = −9. (b) limx→−9− g(x) = −15 and limx→−9+ g(x) = −25. Thus, the limit does not exist. (c) There is no need to check further. Not Continuou