Math 132 Fall 2006 Exam I 1 A Riemann sum for a function Riemann sum if for each the point in the minimized Calculate the lower Riemann sum for Use a partition of a b into equal length subintervals a 7 42 b 2 c 9 4 f 3 g 13 4 h 7 2 Solution e f x x 2 plot f x x 3 2 5 2 d 5 2 i 15 4 on an interval e 11 4 j 4 is said to be a lower subinterval is chosen so that and is a 3 2 b 5 2 N 4 Delta b a N with plottools with plots Warning the names arrow and changecoords have been redefined r1 rectangle 3 2 f 1 2 1 2 0 color COLOR RGB 0 94 0 94 0 65 r2 rectangle 1 2 f 1 2 3 2 0 color COLOR RGB 0 94 0 94 0 65 r3 rectangle 3 2 f 3 2 5 2 0 color COLOR RGB 0 94 0 94 0 65 fnPlot plot f x x 3 2 5 2 thickness 2 color MAROON display r1 r2 r3 fnPlot f 1 2 f 0 f 1 2 f 3 2 Delta 2 Calculate a 1 f b c g d 2 h e i j Solution c int sec theta 2 theta 0 Pi 3 J Int sec theta 2 theta antiderivative value J definiteIntegral subs theta Pi 3 antiderivative subs theta 0 antiderivative simplify definiteIntegral 3 An antiderivative of is the function If then what is a 1 b 4 3 c 5 4 d 5 3 e 3 2 f 7 4 g 7 3 h 9 4 i 8 3 j 5 2 Solution a restart F x 3 x 2 4 x 2 x 2 x 1 eqn F b F 0 1 solve eqn b 4 Calculate a 1 b 2 c 3 f 6 g 7 h 8 d 4 i 9 e 5 j 10 Solution c F x 7 x 2 9 x 2 3 This is an antiderivative of the integrand F 3 F 0 5 Suppose that and a 1 b 2 c 3 f 6 g 7 h 8 d 4 i 9 e 5 j 10 Solution d restart eqn int 2 f x x 1 4 22 eqn2 10 int f x x 1 4 22 eqn3 10 int f x x 1 1 int f x x 1 4 22 eqn3 10 int f x x 1 1 8 22 What is solve eqn3 int f x x 1 1 6 Suppose that The Mean Value Theorem for Integrals asserts that there is a point in the interval 1 7 such that b f g is the average value of the interval 1 7 What is a where c 3 h 4 d e i j Solution i f x x 2 1 m int f x x 1 7 7 1 solve f c m c 7 Calculate at a 0 b 1 c 2 d 3 e 4 f 5 g 6 h 7 i 8 j 9 Solution e subs t 1 7 t 2 9 t 2 3 for in 8 Suppose that What is The derivative of F x at a 1 b f c g d 2 h e i j Solution g F x int sqrt 1 t 2 t 0 tan x D F Pi 4 9 Suppose that What is D F 1 The derivative of F x at x 1 a 3 b 4 c 5 d 6 e 7 f 8 g 9 h 10 i 11 j 12 Solution j F x int sqrt 8 t 2 t x x 5 D F 1 10 Calculate a 3 b 4 c 5 d 6 e 7 f 8 g 9 h 10 i 11 j 12 Solution c int 4 x 2 x 3 1 3 x 0 1 student changevar u x 3 1 Int 4 x 2 x 3 1 3 x 0 1 u This is the required change of variable value 11 Calculate a b c d e f g h i j Solution f int x sqrt x 1 x 1 2 J student changevar u x 1 Int x sqrt x 1 x 1 2 u This is the required change of variable J1 Int expand student integrand J u 0 1 value J1 12 Calculate a b c d e f g h i Solution b int exp x 1 exp x x 0 1 J student changevar u 1 exp x Int exp x 1 exp x x 0 1 u This is the required change of variable value J 13 Calculate the area between a 8 3 b 3 f 21 4 g 14 3 and c 10 3 d 7 2 h 9 2 i 5 e 4 j 16 3 Solution h solve x 2 1 x 1 x plot x 1 x 2 1 x 1 2 thickness 2 2 color NAVY PLUM j int x 1 x 2 1 x 1 2 14 Tle Lorenz function of a certain country has the following values Using trapezoids and all the given data obtain an estimate for the area under a 2910 b 2920 f 2960 g 2970 c 2930 h 2980 d 2940 i 2990 e 2950 j 3000 Solution e 0 5 2 20 5 15 2 20 15 30 2 20 30 50 2 20 50 70 2 10 70 100 2 10 15 By applying Simpson s Rule with four subintervals what approximation of the area under the graph of and over the x axis is obtained a 301 6 b 307 6 f 252 5 g 152 3 c 158 3 d 209 4 h 201 4 e 107 2 i 103 2 j 256 5 Solution g f x 16 x 4 f 2 4 f 1 2 f 0 4 f 1 f 2 3 16 If is the unique solution of the initial value problem then what is a b c f g h Solution h eqn int 1 y y int 2 x x C eqn2 y solve eqn y eqn3 subs y exp 1 x 0 eqn2 d i e j eqn4 C solve eqn3 C eqn5 subs eqn4 eqn2 eqn6 simplify subs x 2 eqn5 17 The height where of water in a leaking tank is given by the differential equation is the constant radius of the hole through which the water leaks If the initial height i e at of the water was 144 and if the tank becomes empty at a 0 f b 1 d c g h Solution i eqn dsolve diff y t t r 2 sqrt y t y t eqn1 subs t 0 y t 144 eqn eqn2 C1 solve eqn1 C1 eqn3 subs eqn2 eqn then what is the value of e i j eqn4 y t solve eqn3 y t subs y t 0 t 8 eqn4 solve r 18 Calculate a b c f g …
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