1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.
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1 . Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent when =4, and a vertical tangent when =5 ) The critical points for f are ) The critical points for f are 3) f has a local maimum when 4) f has its maimum value on 5, 6 when 5) f is decreasing on the interval(s) 6) The graph of f is concave up on the interval(s) 7) The -coordinates of the points of inflection are 8) f has its maimum value when 9) f has its maimum value when ) Does f have a minimum value on 5, 6? Eplain. ) Does f have a minimum value on 5, 6? Eplain.
2 3. Numeric Stem Given that f, f, and f the questions that follow. are all continuous for all, use the information in the table to answer f f f Fill in the blanks with TRUE, FALSE, or NOT DETERMINED ) f has a local minimum at 8 ) f has a local maimum at 4 3) f has a POI when 6 4) f has a POI on the interval 6 5) f is increasing on, 6) f 7 has a solution in, 7) f.5 has a solution in 6, 8 8) f.5 has a solution in 6, 8 9) f.75 has a solution in 6, 8 ) The line y 5 is a horizontal asymptote. ) The line 7 is a vertical asymptote.
3 4. Analytic Stem Find all values of A so that y sina is a solution to d y d 9y, A constant. 5. Numeric Stem The table gives values for the speed and gas mileage of a car at minute intervals. t 3 minutes vt miles per hour mt miles per gallon ) Write a left-hand Riemann sum to approimate the distance traveled in this 3 minute interval. ) Write a left-hand Riemann sum to approimate the total fuel consumption in this 3 minute interval.
4 6. Geometric Stem y The numbers given above in each region give the area of that region. Compute the following numbers. ) The average value of f on, 8. ) fd 4 3) f d 9 4) f d 9 5) f d 4 6) If g f, and g 6, what is g8? 7) If g f, and g 6, what is g4? 8) If g f, and g 6, what are the maimum and minimum values of g on 4, 9?
5 7. Analytic Stem or is it numeric? Calculator needed. If f sin and f, what is f? 8. Geometric Stem. The graph of y f is shown below. This grid is: 6 6 and y 6 ) Estimate f ) Estimate fd 4
6 9. Match the slope fields with their differential equations. Calculus, Hughes-Hallett et.al. (a) y y (b) y (c) y sin (d) y y (e) y y (f) y 4 y Each slope field is sketched for 5 5, 5 y 5
7 . Given the function dt F(). t t (a) F () (b) F() lim (c) Use Trapezoidal Rule with n 4 to approimate F(4). (d) What is the largest meaningful domain for F?. A hemispherical tank of radius 4 feet contains water to a depth of feet. The widest part of the tank is at the top. (a) How much work is done in pumping the water to a point 3 feet above the tank? (b) How much water must be added to the tank to raise the level of water by foot?. lim n n n n n n n n = 3. lim 4. Assume that f and g are differentiable functions defined on all of the real line. Mark each of the following TRUE or FALSE. (a) It is possible that f, f, and f everywhere. (b) f can satisfy: f, f, and f everywhere. (c) f and g can satisfy: f () g () and f () g() for all. (d) If f () g () for all and if f g (e) If f () and f () everywhere then lim f() for some, then f() g() for all. 5. The acceleration due to gravity, g, is given by g GM, where M is the mass of the earth, r is the distance from the center of the earth, and G is the universal gravitational constant. r (a) (b) (c) Show that when r changes by r, the change in the acceleration due to gravity, g is given by g g r r. What is the significance of the negative sign? What is the percent change in g when moving from sea level to the top of Pike s Peak (4.35 km)? Assume the radius of the earth is 64 km. Hughes-Hallett 3rd ed.
8 6. The function V whose graph is sketched below gives the volume of air, V(t), that a man has blown into a balloon after t seconds. Assuming the balloon maintains a spherical shape as it epands, approimately how rapidly is the radius changing after 6 seconds? V 4 3 r 3 7. Let g () 4 sin() e ( ). What value of produces the absolute minimum for interval [,]? g() on the 8. Sketch the graph of a function f () that has all of the given properties. The function f has eactly one discontinuity and one stationary point. lim f(), lim f (), lim f (), lim 3 3 f (), f (), and f (). 9. Let f be differentiable for all real numbers. Which of the following must be in the range of f? a. f(a) f (b) b. f(a) f (b) c. f (a) f (b) d. f (a) f (b) e. None of these. (a) lim n n n n n n n n n (b) lim n n n n 3 n n (c) lim h h h u u du (d) lim f tdt
9 . G() f (t)dt, use the graph of y f() to answer the following questions. y f() (a) (b) (c) (d) ( pts.) Estimate the slope of the curve y G() at ( pts.) Estimate G() ( pts.) Estimate G(3.) G3) (4 pts.) Sketch the graph of y G(). and. y f () B C D A. Let h be the function defined by h() f (t)dt where the graph of f is shown above. The regions labeled A, B, C, and D have the following areas: A B 4, C 3, and D (a) Is h is an even function, an odd function, or neither? Justify your answer. (b) Find the domain of h. (c) Determine all the solutions to h (). (d) Sketch the graph of y h() over its entire domain.
10 3. dy Given the differential equation d y ln y, y. (a) Find the general solution of the differential equation. (b) Find the particular solution for y() e. Epress answer in the form y f(). (c) Eplain why is not in the domain of the solution found in part (b). 4. dy Consider the differential equation d y. (a) Verify that the general solution is y ke. (b) Find the particular solution for y(). y 5. Variables and y are related by the equation 4t dt. d y Show that is proportional to y and determine the constant of proportionality. d y y f(t) t Domain of f = 4 t.5 t Let g() f (t) dt and h() f (t) dt. (a) What is the domain of h? (b) What is the domain of g? (c) (d) For what values of does g ()? Sketch the graph of y g over its entire domain A particle is moving on the -ais, where is in centimeters. Its velocity, v, in cm/sec, when it is at the point with coordinate is given by v 3. Find the acceleration of the particle when it is at the point. Give the units of your answer. Hughes-Hallett et. al. 3rd ed.
11 Answers. No such values of A and B.. ) -3,, 3 ) -3, -,, 4, 5 3) 3 4) -5 5) [-5, ], [3, 6] 6) (-5, -3), (-, ) 7) -3, -, 8) 9) ) Yes, since f is continuous the Etreme Value Theorem applies, f attains both a maimum and a minimum value on [-5, 6]. does not eist at 5 and lim f 5 ) No, f 3. ) False ) True 3) Not Determined 4) True 5) False 6) True 7) Not Determined 8) True 9) True ) Not Determined ) False 4. A 3,,3 5. a) b) miles gallons 6. ) 5/4 5) 3 ) -3 6) 6 3) 9 7) 9 4) 8) Ma = 6, Min = 6
12 7. f sin t dt, f ) f ) fd a) II b) VI c) IV d) I e) III f) V. a) b) c).496 d). a) 4, ft.-lbs. b) 4 cubic feet 3. AB:.88, BC: ln Both a) F b) T c) T d) T e) T 5. b) The acceleration due to gravity decreases as the distance from the center of the earth increases. c).35% in sec 7. y y 3
13 9. c. a..44 b. ln c. d. f. a.,.5 b. anything near.5 c..5 d. below. a. even b. 3 3 c., 6, 3 d. below 3, 7 3. a. y e C b. y e 4 c. would make y and the denominator of dy d would be zero. 4. b. y 3e 5. Constant = 4 6. a..5.5 b. c. 8, 5 8 d. below cm sec/sec
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