Math 132 Spring 2003 Exam 1 Solutions No calculators with a CAS are allowed Be sure your calculator is set for radians not degrees if you do any calculus computations with trig functions Part I Multiple Choice 5 points problem blacken your answers on the answer card 1 Suppose is the velocity cm sec of a point moving along a straight line At time the point is at position B cm It ends its journey at position B cm At what time did its journey end A 1 sec F 15 sec B 1 5 sec G 18 sec C 2 sec H 16 sec D 5 sec I 2 7 sec E 12 sec J 2 95 sec If the journey ends at time then change of position of the particle displacement l Therefore so Alternate solution Since the position G Since we get G and Then gives when 2 The acceleration due to gravity on the planet Eirene is 4 0 m sec If a projectile is fired upward with an initial velocity of 20 m sec how high will it rise Ignore any forces except gravity A 60 m F 35 m B 55 m G 30 m C 50 m H 25 m D 45 m I 20 m E 40 m J 15 m We take ground level and make up the positive direction We then have acceleration so G Since we get so H Since we get The projectile is highest when that is when and m 3 The temperature outdoors is changing at a rate of G hr during the time interval What is the change in temperature over this time period A F B G C H D I 10 E J 12 The total change theorem says that rate of change of temperature total change in temperature If we substitute so we get l 4 Find B B B A sin B G B 3B G C B B G D B G E sin B G F B G G B B G J ln B G H B B G I 24 12 B Let B so that B B or B B Then B B B B G B G 5 The figure shows the graph of the function C 0 over the interval Define 1 0 List all the values in the interval at which 1 has a local maximum A F I B C F G 2 J D E H 1 has a local maximum at a point if 1 switches from increasing to decreasing at that is 1 w switches from positive near on the left to negative near on the right Since 1 w 0 so we can see that from the figure that this occurs at and 6 A substitution transforms ln B B A F B G B into What is C 2 H E J D 1 I B gives lnBB B Since when B and ln when B we have lnBB B So Substituting ln B Suppose A 4 2 F ln B What is B ln G 5 ln C 4 ln H 2 E ln ln J ln D 4 6 I 1 ln l l l ln l l ln l l ln l l ln so ln 8 Which of the following is equal to B B A lim 8 8 3 8 D lim 8 3 B lim 3 8 8 C lim 3 8 8 8 3 8 8 8 8 3 8 8 3 8 E lim 3 8 8 3 8 8 F lim 3 8 8 8 3 8 3 If we subdivide into 8 equal subintervals then B 8 8 The right endpoints 3 8 of these subintervals are 8 8 8 8 If 0 B B then 0 8 8 0 8 3 8 lim 3 8 8 3 8 8 8 0 3 8 8 0 is the right endpoint Riemann sum for 3 8 8 B B B 8 8 B so 8 9 Suppose 0 B B 0 B B and 0 B B What is 0 B B A F B G C 2 H D I E J 0 B B 0 B B 0 B B 0 B B 0 B B But 0 B B 0 B B 0 B B So 0 B B 10 The graph C 0 B below consists of straight line segments and arcs of circles 0 B B A 0 F 1 1 B 1 G 1 C 1 H 3 1 D 1 I 4 1 E J 3 1 In terms of areas 0 B B E E E E where E E E are areas of triangles E area of 1 square area of quarter circle with 1 So 0 B B 1 1 11 Liquid is draining out of a tank at a rate of L min Find the midpoint approximation using 8 for Include units on your answer and round to 3 decimal places A L min D L min G L J 0 min B L min E L H L C L min F L I 47 min Since is measured in L min and in min has units L it represents the total change in the amonount of liquid in the tank between and If we divide the interval into 8 equal subintervals they are and so their midpoints are and Since B the midpoint approximation is Q L B 12 Let 1 B What is the equation of the tangent line to the graph of C 1 B at the point where the graph crosses the B axis A C B D C B G C B J C B B C B E C B H C B C C B F C B I C B The curve crosses the B axis when 1 B that is B Therefore the tangent line is C 7 B where the slope 7 1 w By the Fundamental Theorem part I 1 w B B B so 1 w So the tangent line is C B 13 For what value of is it true that A F B G 6 1 C 2 D E I 4 J H l Therefore so If this gives 1 If 1 B B cos B A F 5 what is 1 w B 1 G C 2 H D 3 I E 4 J We want 1 w B B where B cos B The chain rule and the Fundamental Theorem part I give B B B sin B cos B B cos B B sin B cos B So 1 w cos sin cos Part II True or False 1 point each Blacken your answers on the answer card B 15 Let E the midpoint approximation Q10 for the integral BB B B Let F the midpoint approximation Q10 for the integral BB B Then E F A True B False B B B Let 0 B BB and 1 B B At each midpoint B 3 0 B 3 1 B 3 so each of the 10 midpoint …
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