(a) Find the area of RR. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
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1 Calculus AB Final Review Name: Revised 07 EXAM Date: Tuesday, May 9 Reminders:. Put new batteries in your calculator. Make sure your calculator is in RADIAN mode.. Get a good night s sleep. Eat breakfast 4. Bring: TI-84, pencils, eraser, watch, and jacket. Work these on notebook paper. Use your calculator only if the problem is labeled calculator. π π cos + cos 4 ) lim = ) ( x )( x) 5x + 8x lim = ) lim h 0 hh x 4 ( x )( x+ ) x 0 x 6 x ax +, x 4) Find aa and bb so that ff is differentiable everywhere, f ( x) = bx, x > x, x 5) f ( x) = Evaluate : f ( x) = x x, x> 6) Let RR be the region bounded by the graphs of yy = xx and yy = xx + 6. (a) Find the area of RR. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line, yy = 9. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line, yy =. (d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the vertical line, xx = 5. x 7) Let R be the region bounded by the graphs of y = x and y =. (a) The region RR is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are rectangles whose heights are three times the lengths of the bases. Write, but do not evaluate, an integral expression for the volume of this solid. (b) The region RR is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Write, but do not evaluate, an integral expression that could be used to find the volume of this solid. f = 5, f 6 =, f = 8, and f 6 =. The function 8) Let ff be a differentiable function such that ( ) ( ) ( ) ( ) gg is differentiable and gg(xx) = ff (xx) for all x. What is the value of g (? ) 9) (Calc.) The first derivative of the function f is defined by f ( x) sin ( x x) = for 0 x. On what intervals is ff increasing? f x = x cos x. How many points of inflection 0) (Calc.) The derivative of the function f is given by ( ) ( ) does the graph of f have on the open interval (, )? (A) One (B) Two (C) Three (D) Four (E) Five
2 ) (Calc.) A particle moves along the xx-axis so that at any time tt > 0, its acceleration is given by ( ) ln ( t ) a t = +. If the velocity of particle is at time tt =, then velocity of the particle at time tt = is: (A) 0.46 (B).609 (C).555 (D).886 (E).46 ) (Calc.) A particle moves along the xx-axis so that at any time tt 0, its velocity is given by v t = ln t+ t+. The total distance traveled by the particle from tt = 0 to tt = is ( ) ( ) (A) (B) (C).540 (D).667 (E).90 ) (004 AB, Calc.) A particle moves along the yy-axis so that its velocity at time tt 0 is given by t ( ) tan ( e ) v t =. At time tt = 0, the particle is at yy =. (Note: tan x= arctan x) (a) Find the acceleration of the particle at time tt =. (b) Is the speed of the particle increasing or decreasing at time tt =? Give a reason for your answer. (c) Find the time tt 0 at which the particle reaches its highest point. Justify your answer. (d) Find the position of the particle at time tt =. Is the particle moving toward the origin or away from the origin at time tt =? Justify your answer. 4) x 5 8 f x 4 6 ( ) Let ff be a function that is twice-differentiable for all real numbers. The table above gives values of ff for 5 f x. selected points in the closed interval xx. Evaluate ( ( )) 5) If θ increases at a constant rate of rad/min, at what rate is x increasing in units/min when x = units? 6) (00-AB 5)
3 7) (00 AB 5/BC 5) Match the slope fields with their differential equations. (A) (B) (C) (D) 8) sin x = 9) x y = 0) y = ) x = ) Find the particular solution y = f ( x) to the differential equation = 6x xy where f ( 0) = 7. 4x ) Which of the following is the solution to the differential equation =, where y ( ) =? y (A) y = x for x > 0 (B) y = x 6 for x (C) y = x x> (D) y x x 4) = 4 for > (E) y = x x> 4 6 for.5 4 for Shown above is a slope field for which of the following differential equations? (A) = x (B) y x = (C) y x = (D) y x x = (E) = y y
4 5) (005B AB) (Calc.) A water tank at Camp Newton holds 00 gallons of water at time tt = 0. During the t time interval 0 tt 8 hours, water is pumped into tank at the rate W( t) = 95 tsin gallons per 6 t hour. During same time interval, water is removed from tank at rate R( t) = 75sin gallons per hour. (a) Is the amount of water in the tank increasing at time tt = 5? Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time tt = 8? (c) At what time tt, for 0 tt 8, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion. (d) For tt > 8, no water is pumped into the tank, but water continues to be removed at the rate R( t ) until the tank becomes empty. Let kk be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of kk. 6) 7) Consider the curve given by xx + 4yy = 7 + xxxx (a) Show that dddd = yy xx. dddd 8yy xx (b) Show that there is a point PP with xx-coordinate at which the line tangent to the curve at PP is horizontal. Find the yy-coordinate of PP. d y (c) Find the value of at the point PP found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point PP? Justify your answer. 8) Let R be the region in the first quadrant enclosed by the graphs of ff(xx) = 8xx and gg(xx) = sin(ππππ) as shown in the figure on the right. (a) Write an equation for the line tangent to the graph of ff at xx =. (b) Find the area of RR. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when RR is rotated about the horizontal line, yy =. (d) The region RR is the base of a solid. For this solid, the cross sections perpendicular to the xx-axis are squares. Write, but do not evaluate, an integral expression for the volume of this solid.
5 x 9) Consider the differential equation =. y (a) On the axes provided, sketch a slope field for the differential equation given at the nine points indicated. d y (b) Find in terms of xx and yy. (c) What is the particular solution yy = h(xx) to the differential equation with the initial condition h(0) =? (d) Is the point (0, ) a relative maximum, relative minimum, or neither? Justify your answer. Evaluate. d 5 0) cos ( x ) π x 8 ) e ) sin ( x) 0 ) ( ) ( ) π sin 5x cos 5x 4) (Calc.) A particle moves along the xx-axis so that its velocity at time tt, for 0 t 5, is given by ( ) ln ( ) v t = t t+. The particle is at position, xx = 8 at time tt = 0. 5) If (a) Find all times tt in the open interval 0 < tt < 5 at which the particle changes direction. During which time intervals, for 0 tt 5 does the particle travel to the left? (b) Find the average speed of the particle over the interval 0 tt. x t = 5 +, which of the following is true? y e dt 4 x = e and y = 5 = and = 5 x x 4x (A) = e and y( 0) = 5 (B) = e and y( ) = 5 (C) e y( ) 4x (D) = e and y( 0) = 5 (E) ( ) 6) Consider the differential equation 6x x y =. Let y f ( x) = be a particular solution to this differential equation with the initial condition f ( ) =. (a) Write an equation for the tangent line that passes through the point (, ), and use it to approximate the value of f. d y (b) For < x <, < 0. Is the approximation you found in (a) for an overestimate? Explain. f an underestimate or (c) Find the particular solution y = f ( x) to the differential equation with the initial condition ( ) f =.
6 7) (Calc.) Oil is leaking from a tanker at the rate of ( ) 0. hours. How much oil leaks out of the tanker from time tt = 0 to tt = 0? R t t = 000 gallons per hour, where tt is measured in e 8) (Calc.) A pizza, heated to a temperature of 50 degrees Fahrenheit ( FF), is taken out of an oven and placed in 0.4t a 75 FF room at time tt = 0 minutes. The temperature of the pizza is changing at a rate of 0e degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes? 9) (Calc.) The density function in people per mile for the population of the small coastal town of Westport, WA, is given by p( x) = 000( x) where xx is the distance along a straight highway from the ocean and pp(xx) is measured in people per mile. The town extends for two miles from the ocean so that 0 xx. Find the population of Westport. π f x = x, then f = 9 40) If ( ) cos( ) 4) ax + cx lim x bx + = t t 4) (Calc.) The velocity, in ft/sec, of a particle moving along the xx-axis is given by the function v( t) = e + te. What is the average velocity of the particle from time tt = 0 to tt =? (A) ft/sec (B) ft/sec (C).809 ft/sec (D) ft/sec (E) 79.4 ft/sec 4) If yy = xxxx + xx +, then when xx =, =? 44) Given below, the graphs of the functions ff and gg are shown. The value of lim xx ff gg(xx) =. 45) 7x sin x lim x 0 x + sin ( x) = ( ) 46) If f ( x) = sin ln ( x), then '( ) f x =
7 47) Three graphs labeled I, II, and III are shown above. One is the graph of ff, one is the graph of ff and one is the graph of ff. Which of the following correctly identifies each of the three graphs? 48) The local linear approximation to the function gg at xx = is yy = 4xx +. What is the value of gg + gg? 49) Write the following integral expressions is equal to lim n k n + k n n with a range of [0,]. = 50) The right triangle shown in the figure above represents the boundary of a town that is bordered by a highway. The population density of the town at a distance of x miles from the highway is modeled by D( x) x = + where DD(xx) is measured in thousands of people per square mile. According to the model, which of the following expressions gives the total population, in thousands, of the town? (A) 0 4 x + (B) 4 8 x + (C) 0 4 x x 4 + (D) ( ) x x + 5) xx 0 ff(xx) 5 7 ff (xx) 5 4 The table above gives selected values of a differentiable and decreasing function ff and its derivative. If gg is the inverse function of ff, what is the value of gg ()? 5) (calc) The second derivative of a function gg is given by gg (xx) = xx + cos xx + xx. For 5 < xx < 5, on what open intervals is the graph of gg concave up?
8 5) (calc) Let h be the function defined by h(xx) = the value of gg(4)? xx 5 +. If gg is an antiderivative of h and gg() =, what is 54) (Calculator) Let RR be the region in the first quadrant bounded by the graph of gg, and let SS be the region in the first quadrant between the graphs for ff and gg, as shown in the figure above. The region in the first quadrant bounded by the graph of ff and the coordinate axis has an area of.4. The function gg is given by π x g( x) = ( x+ 6 ) cos, and the function ff is not explicitly given. The graphs intersect at the point, (4,0). 8 (a) Find the area of SS. (b) A solid is generated when SS is revolved about the horizontal line, yy = 5. Write, but do not evaluate, an expression involving one or more integrals that gives the volume. (c) Region RR is the base of the art sculpture. All of the points of RR at a distance of xx from the yy-axis, the height of the sculpture is given by h(xx) = 4 xx. Find the volume of the sculpture. 55) tt (minutes) rr(tt) Rotations per minute Rochelle rode a stationary bicycle. The number of rotations per minute of the wheel of the stationary bicycle at time tt minutes during Rochelle s ride is modeled by a differentiable function rr for 0 tt 9 minutes. Values of rr(tt) for the selected values of tt are shown on the table above. a) Estimate rr (4). Show the computations that lead to your answer. b) Is there a time tt, for tt 5, at which rr(tt) is 06 rotations per minute? Justify your answer? c) Use a left Riemann s Sum with four subintervals indicated by the data in the table to approximate 9 9 ( ). Use the correct units, explain the meaning of ( ) r t dt r t dt in the context of the problem. 0 0 d) Sarah also rode a stationary bicycle. The number of rotations per minute of the wheel of the stationary bicycle at time tt minutes during Sarah s ride is modeled by the function s, defined by πt ( ) 40 0π sin 8 s t = + for 0 t 9 minutes. Find the average number of rotations per minute of the wheel of the stationary bicycle for 0 t 9 minutes. 56) Let ff be a continuous function defined on the closed interval [,4]. The graph of ff, consisting of three x line segments, is shown. Let gg be the function defined by g ( x) 5 ( ) (a) Find gg(4). = + f t dt. (b) On what intervals is gg increasing? Justify your answer. (c) On the closed interval find the absolute minimum value of gg and find the absolute maximum value of gg. Justify your answers. (d) Let h(xx) = xx gg(xx). Find h ().
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