Math 132 Fall 2005 Exam I 1 A Riemann sum for a function Riemann sum if for each the point on an interval in the maximized Calculate the upper Riemann sum for Use a partition of a b into equal length subintervals a 14 f 24 b 16 c 18 g 26 h 28 d 20 e 22 i 30 Solution i The nodes points of the uniform partition are a 1 b 3 N 4 Delta b a N for j from 0 to N do x j a j Delta od j 32 is said to be a n upper subinterval is chosen so that is and f x x 3 3 x 2 Diff f x x factor D f x This calculation tells us that increases for decreases from 1 to 1 so 1 Therefore xi 1 x 0 xi 2 x 1 xi 3 x 3 xi 4 x 4 fnGraph plot f x x 1 3 color PLUM thickness 2 nodes plot seq xi j f xi j j 1 N style POINT symbol CIRCLE color NAVY plots display fnGraph nodes and rect 1 plottools rectangle x 0 f x 0 x 1 0 color wheat rect 2 plottools rectangle x 1 f x 1 x 2 0 color wheat rect 3 plottools rectangle x 2 f x 3 x 3 0 color wheat rect 4 plottools rectangle x 3 f x 4 x 4 0 color wheat plots display fnGraph nodes seq rect j j 1 N ANSWER f xi 1 f xi 2 f xi 3 f xi 4 Delta 2 Calculate a 1 b f c g d 2 h e i j Solution a F unapply int sec theta tan theta theta theta F Pi 3 F 0 3 An antiderivative of If is the function a f b g 1 c h Solution i restart F x x exp x 1 exp x d i 2 e j 2 what is eqn1 Int f x c x 0 1 F 1 F 0 c eqn2 5 2 rhs eqn1 solve eqn2 c 4 Calculate a 1 f b 2 g c d h e i Solution c F x x ln x This is an antiderivative of the integrand F exp 1 F 1 j 5 Suppose that and a 1 b 2 c 3 f 6 g 7 h 8 d 4 e i 9 What is 5 j 10 Solution d restart eqn1 int x 2 f x x 0 3 17 eqn2 int x 2 x 0 3 int f x x 0 3 17 eqn3 9 int f x x 0 2 int f x x 2 3 17 eqn4 9 4 int f x x 2 3 17 solve eqn4 int f x x 2 3 6 Suppose that The Mean Value Theorem for Integrals asserts that there is a point in the interval 1 2 such that where in the interval 1 7 What is a b f g is the average value of c d h e i j Solution f f x x 2 6 x interval 1 2 Ave int f x x interval 3 solve f x Ave x 7 Calculate at a 0 b 1 c 2 d 3 e 4 f 5 g 6 h 7 i 8 j 9 Solution c J Int 9 t tan Pi t 4 t 2 4 t 1 x for Integrand student integrand J simplify subs t 1 Integrand 8 Suppose that What is d 2 e The derivative of F x at a 1 f b c g h Solution b restart with student F x Int sqrt 7 2 t 2 t 0 sin x i j D F x simplify D F Pi 4 Where answer comes from derivative subs t sin x integrand F x D sin x simplify subs x Pi 4 derivative 9 Suppose that What is D F 2 The derivative of F x at x 5 a 13 b 20 c 27 d 33 e 40 f 47 g 53 h 60 i 67 j 73 Solution f F x Int sqrt 144 t 2 t x 3 x 1 simplify D F 5 Where answer comes from h x 3 x 1 g x x derivative subs t h x integrand F x D h x subs t g x integrand F x D g x simplify subs x 5 derivative 10 Calculate a 3 b 4 c 6 d 8 e 9 f 8 g 12 h 15 i 16 j 20 Solution h J1 Int 4 cos x sin x 1 3 x 0 Pi 2 J2 changevar u sin x 1 J1 u value J2 11 Calculate a b c d f g h i Solution i J1 Int x sqrt x 3 x 3 4 J2 changevar u x 3 J1 u J3 Int expand integrand J2 u 0 1 value J3 e j 12 Calculate a b c f g h d e i j Solution e J1 Int 1 x x ln x x 1 exp 1 J2 Int factor integrand J1 x 1 exp 1 J3 changevar u 1 ln x J2 u value J3 13 Find the solutions x a and x b of the equation sin x cos x in the first and third quadrants respectively Calculate the area between and for in a b a 1 f b c 2 g d h e i j 4 Solution e solve sin x cos x plot sin x cos x x Pi 4 5 Pi 4 color NAVY MAROON int sin x cos x x Pi 4 5 Pi 4 14 At irregular intervals during the first 10 seconds of a race a radar gun recorded the following speeds in m s of a runner Given that distance at time is expressed by the formula estimate the distance in m the runner has covered during those 10 seconds Use trapezoids and all the given data a 99 4 b 99 5 f 99 9 g 100 0 c 99 6 d 99 7 h 100 1 i 100 2 e 99 8 j 100 3 Solution d 0 6 8 2 1 6 8 10 2 2 2 10 2 11 0 2 2 11 0 11 6 2 1 11 6 11 8 2 4 15 A swimming pool has the shape of a rectanle with width 12 feet and length 20 feet Measured at 5 foot intervals along its length starting at the shallow end and ending at the deepest end the depths in feet are 1 3 7 9 10 By applying Simpson s Rule with four subintervals what approximation to the volume of the pool in cubic feet is obtained a 1410 b 1420 f 1460 g 1470 c 1430 d 1440 h 1480 e 1 450 i 1490 j 1500 Solution f 12 5 3 1 4 3 2 7 4 9 1 10 16 If is the unique solution of the initial value problem then what for what positive value a b f g is c …
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