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MAC 2311 TEST 1 Solutions Spring 2011

# MAC 2311 TEST 1 Solutions Spring 2011 - MAC 2311 CALCULUS...

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Unformatted text preview: MAC 2311 CALCULUS WITH ANALYTIC GEOMETRY I ‘TESTl 0&0? Name ‘ié E i Score 58 Directions: Answer each question showing ALL work for full credit. 1. Use the graph of the function below to determine each statement as TRUE or FALSE. No work is required for this particular problem. (8 points) 1 lim f(x)=1 x—-> —1 + lim f (x) = 2 x-—>2 lim f (x) exists. x—>1 f (x) is deﬁned for all numbers in the interval (~1 ,3). ___F___ ELITIEJFW) =f(1) T T f (x) is continuous at x = ~2— T 1113 f(x) = 111g; f(x) F f (x) is continuous from the right at x = 2 Valencia Community College 2. Sketch the graph of a function that satisﬁes all of the conditions below. Be sure your graph passes the vertical line test. (8 points) 0 f (x) is continuous on (1,3) 0 f (2) = 0 and f (—2) is undeﬁned 0 lingf(x) =1 0 f(x)=—x+3 for 03x31 ° lim f (X) f (0) x—)0 0 lim f(x)=1 x—>—2 0 lim_ f(x)= +00 and lim f(x)=—oo x—>—3 n I_I—IIIIIIII_-_ ﬁIlI-“I III-III- Valencia Community College 3. The deﬁnition for the derivative of a function f at x = a is given below f’(a) : limf(x)—f(a) . X—)Cl x_a a) Use this deﬁnition and algebra to calculate the slope of the tangent line to the graph of the function f (x) = «M + 9 at x = 0. (4 points) 5&0): “M m ‘3 . m‘l‘s ﬂ X—m 7‘ JXdﬁ *3 :: llm X'l'Cl’Cl >920 XUX'H 453\$ : Rm 1 Xﬁo W +3 ”I b) You are given that f (x) = \/x + 9 and f '(——8) = %. Use this information to ﬁnd the equation of the line tangent to the graph of y = f (x) at x = ~8. (3 points) Valencia Community College 4. The table below gives the population of the United States at the beginning of each decade starting in the year 1790. b) d) TABLE 2.12 US population (in millions), I 790—1990 _--l| 3.9 . 205.0 RLS On average, how fast is the US. population changing between 1830 and 1990? Round your answer to the nearest hundredth using the same units of measurement as the table. (2 points) comma N ‘ ' mo— 1230 "‘ i‘ “‘1 WW 6‘1 We W W. Give the IE possible estimate for how fast the US. population is changing during the year 1950 by taking the average of two secant slopes. Round your answer to the nearest hundredth using the same units of measurement as the table. m : 150.?_131~:11 2 ‘ 1v 1 l lq‘ mtom "" mill-ma 2132.3? M\\\lDﬂS 0 Q 31" m: Hui-1503” “22.23 to If P(x) represents the US. population x years after 1790, write a sentence or two explaining what the equation P’(100) = 1.3 means in context of this problem. Be speciﬁc and include units of measurement. (2 points) In {8’30 -PM 0.3» pgeuiaiﬁﬁn was incmmthj 03¢ 0K yam—ER 0% 1:3 mi\\l0h3 <79 Pm?“ Per 3%?- According to the table, during which decade was the US. population growing the fastest? (2 point) 3 l (150 3 Valencia Community College 5. The graphs of f and g are given below. Use them to evaluate each limit, if it exists. If the limit does not exist write DNE in the space provided. (6 points) Iim[f<x>+g<x>l= 3 lim[f(x)—g(x>]= DNE. 11m[f(x).g(x)]= ‘5 x—>1 x——>2 x—>2‘ limg(x)— O~5 11mlx2g(x)J= H9 lim f(x)= \$1 x~—>l f(x) xe4 b93— Valencia Community College 6. Use algebra to compute both of the following limits, if they exist. (8 points) . 2x2—llx+5 11m —2““————— M —x +7x—10 :llV"! W X»5 -(x—iﬁX-§> = \lm —- lax—D **5 x~a W E] . ( 1 1 ] 11m ~ —2——— H—3 x+3 x +7x+12 Valencia Community College 7. Use limits to ﬁnd the constants a and b that makes g continuous everywhere. (4 points) ax—2 if x<1 g(x)= 6*a if x=1 3x2+bx if x>1 U09 Weeci \‘Vm (3(X323U-31G-‘0c X-§l 5.9 “m 566: “M KBXQA‘be D: 3+b X93} X—vjj llm 3(x\=\\m (RX-Q9“? Ora X91" X44: G~Q=Q'Q e~m=3+b 2=Rm e-q:3+b Mb t=~l What does it mean for the function g(x) to be continuous from the right at x = 5 ‘? Use an equation involving limits to answer this question. (2 points) ihst=Mﬂ=7O xast Valencia Community College 8. A particle starts by moving to the right along a horizontal line; the graph of its position function is shown below. The horizontal axis represents time measured in seconds. The vertical axis represents the position of the particle measured in meters. a) Draw a numerically accurate velocity versus time graph for the particle. Be sure to include any “open dots” along with their coordinates in your graph. (4 points) b) During what time interval is the particle moving to the left? (1 point) x> 5 0) During what time intervals is the particle at rest? (1 point) 3 4 ’X< 5 d) What is the total displacement of the particle during the ﬁrst seven seconds? (1 point) 0-wa- Va/encia Community College ...
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MAC 2311 TEST 1 Solutions Spring 2011 - MAC 2311 CALCULUS...

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