AP Calculus Prep Session I February, 008 Houston ISD Presented by: Advanced Placement Strategies, Inc. Advanced Placement, AP, and Advanced Placement Strategies are trademarks of the College Entrance Examination Board and used under license by AP Strategies, Inc. The College Entrance Examination Board is not involved in the production nor endorses the use of this material.
Calculus Test Format Section I 50% Part A % of Grade Number of Questions 8 multiple choice Time allotted Calculator Use 55 minutes no calculator Part B Section II 50% Part A Part B 7 multiple choice free response questions free response questions 50 minutes graphing calculator required 45 minutes graphing calculator required 45 minutes no calculator
008 Exams Schedule Week Monday, May 5 Tuesday, May 6 Morning Session 8 a.m.* Government and Politics: United States Computer Science A** Computer Science AB** Spanish Language** Wednesday, May 7 Calculus AB** Calculus BC** Thursday, May 8 English Literature** German Language** Afternoon Session noon* Government and Politics: Comparative** French Language** Statistics Chinese Language and Culture Japanese Language and Culture** French Literature** Friday, May 9 United States History European History Studio Art (portfolios due) Week Monday, May Tuesday, May Morning Session 8 a.m.* Biology** Music Theory** Environmental Science** Chemistry** Wednesday, May 4 Italian Language and Culture** English Language** Thursday, May 5 Friday, May 6 Macroeconomics** World History** Human Geography** Spanish Literature** Afternoon Session noon* Physics B** Physics C: Mechanics** Psychology Art History Microeconomics Latin Literature** Latin: Vergil** Afternoon Session p.m. Physics C: Electricity and Magnetism
Limits, Continuity, and the Definition of the Derivative
DEFINITION Derivative of a Function The derivative of the function f with respect to the variable x is the function f whose value at x is f( x) lim provided the limit exists. f ( xh) f( x) h 0 h DEFINITION (ALTERNATE) Derivative at a Point The derivative of the function f at the point x a is the limit f( a) lim x a f ( x) f( a) x a provided the limit exists. DEFINITION Continuity A function f is continuous at a number a if ) f ( a ) is defined (a is in the domain of f ) ) lim f ( x ) x a exists ) lim f ( x ) f ( a ) x a
Limits, Continuity, and the Definition of the Derivative Page of cos h cos. What is lim? h0 h (A) - (B) (C) 0 (D) (E) The limit does not exist. The graph of the function f is shown in the figure below. Which of the following statements about f is true? (A) lim f ( x ) lim f ( x ) x a (B) lim f ( x ) x a (C) lim f ( x ) x b (D) lim f ( x ) (E) x b lim f ( x) x a x b does not exist. The graph of a function f is shown below. Which of the following statements about f is false? (A) f is continuous at x = a (B) f has a relative maximum at x = a (C) x = is in the domain of f (D) lim f ( x) is equal to lim f ( x) (E) x a lim f ( x ) x a exists x a
Limits, Continuity, and the Definition of the Derivative Page of x 4. If a 0, then lim x a x (A) a (B) a (C) 6a (D) 0 (E) nonexistent a a 4 4 is 5. lim x x x (A) 4 (B) (C) 4 (D) 0 (E) does not exist 6. x 9 x lim 9 (A) 6 (B) 6 (C) 0 (D) x (E)
Limits, Continuity, and the Definition of the Derivative Page of 7. If x 6, x f ( x) x, then lim f( x ) x 5, x (A) 5 (B) (C) (D) 0 (E) does not exist 8. If f ( x) and lim x f( x) does not exist, then k (A) (B) (C) (D) (E) x k 9. If the graph of f is as shown below, then lim x f( x) f( x) 5 Y 4 X - - 0 4 5 6 7 8 - - - -4-5 (, -4) (A) 8 (B) 0 (C) (D) 9 (E) 5
0. Limits, Continuity, and the Definition of the Derivative Page 4 of lim x x x (A) 0 (B) (C) (D) (E) does not exist. lim x 0 sin x x (A) (B) 0 (C) (D) (E) does not exist. Let f ( x) x. The lim f( x) x (A) 0 (B) (C) (D) (E) 4x 6. xlim 4 x 5 x (A) 0 (B) (C) (D) (E) none of these
4. Limits, Continuity, and the Definition of the Derivative Page 5 of lim x x x x 6x9 (A) (B) (C) (D) (E) does not exist 5. For x 0,lim h 0 (A) x (B) x (C) x (D) x (E) 0 h x h x 6. lim x x x (A) 8 (B) (C) 0 (D) (E) 8 7. 6n 00 4nkn lim, x k (A) (B) 6 (C) (D) 8 (E)
8. Limits, Continuity, and the Definition of the Derivative Page 6 of 0 x 0 x 0 x lim x 0 x 0 x 0 x 8 5 6 4 4 9 6 7 5 5 (A) 0 (B) (C) (D) 0 (E) 0 9. The lim h 0 x h x tan tan h is (A) 0 sec (B) x (C) sec x (D) cotx (E) nonexistent x a 0. If lim f ( ) L, where L is a real number, which of the following must be true? (A) f ( x) exists (B) f ( x) is continuous at x a (C) f ( x) is defined at x a (D) f a L (E) none of the above. What is lim h 0 h 8 8 80.5 80.5 h (A) 0 (B) 4 (C) (D) (E) the limit does not exist
Limits, Continuity, and the Definition of the Derivative Page 7 of. lim h 0 sin h h sin (A) (B) 0 (C) (D) (E). lim x a x a x a (A) 0 (B) a (C) a a (D) a (E) does not exist 4. lim h 0 x h x tan tan h (A) 0 cot x (B) (C) sec x (D) secx (E) does not exist
Limits, Continuity, and the Definition of the Derivative Page 8 of 5. If lim f( x) 7, which of the following must be true? x I. f is continuous at x II. f is differentiable at x III. f () 7 (A) none (B) II only (C) III only (D) I and III only (E) I, II, and III 6. If x5 x7 f( x) x f() k (A) 0 for x and if f is continuous at x then k = (B) 6 (C) (D) (E) 7 5 7. For what value of c is (A) (B) (C) 6 (D) none (E) 0 x, x f( x) cx 5, x continuous
Limits, Continuity, and the Definition of the Derivative Page 9 of sin x, x 0 8. Consider the function f( x) x k, x 0 x 0, the value of k must be (A) 0 (B) (C) (D) (E) a number > in order for f ( x ) to be continuous at 9. Let x, x f( x) x, x Which of the following statements is correct? (A) f ( x ) is continuous at since f ( x ) is defined at x (B) f ( x ) is continuous at since lim f ( x) exists x (C) f ( x ) is not continuous at since f ( x ) is not defined at x (D) f ( x ) is not continuous at since lim f ( x) does not exist (E) f ( x ) is not continuous at since x lim f ( x) f() x x, x0 0. Use f( x) and find the x, x 0 (A) 0 (B) (C) (D) (E) does not exist lim h0 f xh f( x) h
Limits, Continuity, and the Definition of the Derivative Page 0 of. At x =, the function given by x, x f( x) 6x 9, x is (A) Undefined (B) Continuous but not differentiable (C) Differentiable but not continuous (D) Neither continuous nor differentiable (E) Both continuous and differentiable. If f is a continuous function on [a,b], which of the following is necessarily true? (A) f exist on (a,b) (B) f ( x0 ) is a maximum of f, then f ( x0 ) = 0 (C) lim f ( x) f(lim x), forx ( a, b) xx0 xx0 (D) f ( x) 0 forsomex[ a, b ] (E) the graph of f is a straight line 0. Which of the following is true about the function f if f x I. f is continuous at x = II. The graph of f has a vertical asymptote at x =. III. The graph of f has a horizontal asymptote at y =. x x 5 x? (A) I only (B) II only (C) III only (D) II and III (E) I, II, and III
Limits, Continuity, and the Definition of the Derivative Page of Free Response. Let f be the function defined as follows: x 4, x f( x) ax bx, x, where a and b are constants (a) If a = -/ and b = 4, is f continuous for all x? Justify your answer. (This question implies one should name the intervals of continuity and justify with limits) (b) Describe all values of a and b for which f is a continuous function. (c) For what values of a and b is f both continuous and differentiable.
Limits, Continuity, and the Definition of the Derivative Page of Free Response. Let f be a function defined by x, x f( x) x k, x (a) For what value of k will f be continuous at x =. Justify your answer algebraically. (b) Using the value of k found in part (a), determine whether f is differentiable at x =. Justify your answer. (c) Let k = 4. Determine whether f is differentiable at x =. Justify your answer algebraically.
Relationships Between f, f, f
Calculus Things I Know. If a function is increasing, the first derivative is positive (or occasionally zero, think x.). If the first derivative is positive, the function is increasing.. If a function is decreasing, the first derivative is negative (or occasionally zero, think x.) 4. If the first derivative is negative, the function is decreasing. 5. If a function is concave up, the second derivative is positive. 6. If the second derivative is positive, the function is concave up. 7. If a function is concave down, the second derivative is negative. 8. If the second derivative is negative, the function is concave down. 9. A function has a relative maximum if the first derivative changes sign from positive to negative. 0. A function has a relative minimum if the first derivative changes sign from negative to positive.. A function has a point of inflection if the second derivative is zero or undefined and if the concavity changes sign.. If the first derivative is increasing, the function is concave up.. If the first derivative is decreasing, the function is concave down.
Page of, f Multiple Choice: dy d y. At which of the five points on the graph below are and dx dx both negative? (A) A (B) B (C) C (D) D (E) E. The graph of the derivative of f is shown in the figure below. Which of the following could be the graph of f?
Page of, f. The graph of a twice-differentiable function f is shown in the figure below. Which of the following is true? (A) f () f () f () (B) f () f () f () (C) f () f () f () (D) f () f () f () (E) f () f () f () 4. Let f be a function whose domain is the open interval (,5). The figure above shows the graph of f. Which of the following describes the relative extrema of f and the points of inflection of the graph of f? (A) relative maximum, relative minimum, and no point of inflection (B) relative maximum, relative minimum, and no point of inflection (C) relative maximum, relative minimum, and point of inflection (D) relative maximum and points of inflection (E) relative minimum and points of inflection
5. Page of, f The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maximum on the open interval a < x < b? (A) f only (B) g only (C) h only (D) f and g only (E) f, g, and h 6. Which of the following statements about the function given by true? f ( x) x x 4 is (A) The function has no relative extremum (B) The graph of the function has one point of inflection, and the function has two relative extrema. (C) The graph of the function has two points of inflection, and the function has one relative extremum. (D) The graph of the function has two points of inflection, and the function has two relative extrema. (E) The graph of the function has two points of inflection, and the function has three relative extrema. 4 7. The graph of y x 6x 4x 48 is concave down for (A) x 0 (B) x 0 (C) x or x (D) x or x (E) x
Page 4 of, f 8. Let f be a function defined for all real numbers x. If decreasing on the interval (A) (, ) (B) (, ) (C) (, 4) (D) (, ) (E) (, ) 4 x f ( x ) x, then f is 9. If f ( x ) x ( x )( x ), then the graph of f has inflection points when x = (A) - only (B) only (C) and 0 only (D) and only (E), 0, and only 0. The function f is given by f increasing? (A), (B), (C) 0, (D), 0 (E), 4 f( x) x x. On which of the following intervals is
Page 5 of, f t t. Let g be the function given by gt () 000sin 0cos. For 0 t 8, g is 6 decreasing at a decreasing rate when t = (A) 0.949 (B).07 (C).06 (D) 5.965 (E) 8.000. The derivative of (A) (B) 0 (C) 4 5 x x f( x) 5 attains its maximum value at x = (D) 4 (E) 5. For what value of k will (A) 4 (B) (C) (D) 4 (E) none of these x k x have a relative maximum at x? 4. The maximum value of (A) 0 (B) (C) (D) (E) 4 ( ) 9 f x x x x on, is
5. For what value of k will (A) (B) 6 (C) 0 (D) 6 (E) 8x k x Page 6 of, f have a relative minimum at x 4? 6. The function f given by f( x) x x 4 (A) is increasing for x, decreasing for x, increasing for x (B) is decreasing for x 0, increasing for x 0 (C) is increasing for all x (D) is decreasing for all x (E) is decreasing for x, increasing for x, decreasing for x 7. Let f be a polynomial function with degree greater than. If a b and f( a) f( b), which of the following must be true for at least one value of x between a and b? I. f( x) 0 II. f( x) 0 f x III. 0 (A) None (B) I only (C) II only (D) I and II only (E) I, II and III 8. Given the function defined by graph of f is concave up. 5 f ( x) x 0x, find all values of x for which the (A) x 0 (B) x0 or x (C) x (D) x0 or x (E) x
Page 7 of, f 9. If the graph of value of b? 4 has a point of inflection at (, -6), what is the y x ax bx (A) (B) 0 (C) (D) (E) It cannot be determined from the information given. 0. If the graph of k is (A) (B) (C) (D) (E) 0 f( x) x k x has a point of inflection at x, then the value of 4. The function y x bx 8x has a horizontal tangent and a point of inflection for the same value of x. What must be the value of b? (A) 6 (B) 4 (C) (D) - (E) -6. Let, g be a relative maximum for gx ( ) xhx kx 6. Use the fact that g, (A) -5 (B) -9 (C) 9 (D) 5 (E) 4 is an inflection point to find the value of h k.
Page 8 of, f. Let h be a continuous and differentiable function defined on values of h and h are given in the table below: 0,. Some function x 0 hx 6 hx 4-0 - sin If p x h x, evaluate p. (A) - (B) - (C) 0 (D) (E) Use the following table to help you answer questions 4 and 5: x f x f x 0 7 - - 6 4 4 5-4. What is m if m x f x? (A) -8 (B) - (C) 0 (D) (E) 8 5. What is 0 if x k kx f e? (A) - (B) - (C) (D) (E)
Free Response, No Calculator Page 9 of, f x 0 0 x x x x 4 f ( x ) Negative 0 Positive Positive 0 Negative f ( x) 4 Positive 0 Positive DNE Negative Negative f ( x) Negative 0 Positive DNE Negative 0 Positive Let f be a function that is continuous on the interval [0, 4). The function f is twice differentiable except at x. The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at x. (a) For 0 x 4, find all values of x at which f has a relative extremum. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer. (b) On the axes provided, sketch the graph of the function that has all the characteristics of f. Y O X
Page 0 of x (c) Let g be the function defined by gx ( ) ftdt ( ) on the open interval (0, 4). For 0 x 4, find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer., f (d) For the function g defined in part (c), find all values of x, for 0 x 4, at which the graph of g has a point of inflection. Justify your answer.
Free Response, No Calculator. Page of, f The figure above shows the graph of f, the derivative of the function f, on the closed interval x 5. The graph of f has horizontal tangent lines at x = and x =. The function f is twice differentiable with f() = 6. (a) Find the x-coordinate of each of the points of inflection of the graph of f. Give a reason for your answer. (b) At what value of x does f attain its absolute minimum value on the closed interval x 5? At what value of x does f attain its absolute maximum value on the closed interval x 5? Show the analysis that leads to your answers.
Page of, f (c) Let g be the function defined by gx ( ) xf( x). Find an equation for the line tangent to the graph of g at x =.
Related Rates
Strategy for Solving Related Rate Problems (Calculus, Finney, Demana, Waits, Kennedy) Understand the problem. Identify the variable whose rate of change you seek and the variable (or variables) whose rate of change you know. Develop a mathematical model of the problem. Draw a picture (many of these problems involve geometric figures) and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start. Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know. The formula is often geometric, but it could come from a scientific application. Differentiate both sides of the equation implicitly with respect to time t. Be sure to follow all the differentiation rules. The Chain Rule will be especially critical, as you will be differentiating with respect to the parameter t. Substitute values for any quantities that depend on time. Notice that it is only safe to do this after the differentiation step. Substituting too soon freezes the picture and makes changeable variables behave like constants, with zero derivatives. Interpret the solution. Translate your mathematical result into the problem setting (with appropriate units) and decide whether the result makes sense. Check to make sure you answered the question asked.
Related Rates P. of 7 Circles:. (calculator) The radius of a circle is increasing at a constant rate of 0. meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 0 meters? (A) 0.04 m / sec (B) 0.4 m / sec (C) 4 m /se c (D) 0 m / sec (E) 00 m / sec. (calculator) The radius of a circle is decreasing at a constant rate of 0. centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? (A) 0.C (B) 0. C (C) 0. C (D) 0. C (E) 0. C
Related Rates P. of 7. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is (A) (B) (C) (D) (E) 4. When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is (A) 4 (B) 4 (C) (D) (E)
Related Rates P. of 7 Triangle Area: 5. (calculator) If the base b of a triangle is increasing at a rate of inches per minute while its height h is decreasing at a rate of inches per minute, which of the following must be true about the area A of the triangle? (A) A is always increasing. (B) A is always decreasing. (C) A is decreasing only when b h. (D) A is decreasing only when b h. (E) A remains constant. 6. The rate of change of the area of an equilateral triangle with respect to its side s at s is approximately: (A) 4 (B) (C) (D) 7 (E)
Related Rates P. 4 of 7 7. Let BAC in triangle ABC with AB c and AC b, where b and c are constants and c b. Side BC changes length as the measure of changes. Find the instantaneous rate of change of the area of triangle ABC when. Assume the instantaneous rate of change of is. (A) cb (B) cb (C) cd (D) cd (E) cb Triangles Pythagorean Theorem 8. (calculator) A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? (A) 57.60 (B) 57.88 (C) 59.0 (D) 60.00 (E) 67.40
Related Rates P. 5 of 7 9. The top of a 5-foot ladder is sliding down a vertical wall at a constant rate of feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall? (A) 7 feet per minute (B) 8 (D) 7 8 feet per minute (E) 5 7 feet per minute (C) 4 feet per minute 7 4 feet per minute z y x dz dx dy 0. The sides of the rectangle above increase in such a way that and. At dt dt dt the instant when x 4 and y, what is the value of dx dt? (A) (B) (C) (D) 5 (E) 5
Related Rates P. 6 of 7. A missile rises vertically from a point on the ground 75000 feet from a radar station. If the missile is rising at the rate of 6500 feet per minute at the instant when it is 8000 feet high, what is the rate of change, in radians per minute, of the missile s angle of elevation from the radar station at this instant? (A) 0.75 (B) 0.9 (C) 0.7 (D) 0.469 (E) 0.507 Volume--Cones. The volume of a cone of radius r and height h is given by V r h. If the radius and the height both increase at a constant rate of centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? (A) (B) 0 (C) 4 (D) 54 (E) 08
Related Rates P. 7 of 7 Volume Cubes. The volume V measured in cubic inches of un-melted ice remaining from a melting ice cube after t seconds is V 000 40t0.t. How fast is the volume changing when t 40seconds? (A) 6 in in (B) 4 sec sec (C) 0 in sec (D) 0 in sec in (E) 4 sec 4. If the volume of a cube is increasing at a rate of 00 in min 0 inches, then the rate at which the edge is changing is at the instant when the edge is (A) 4 in min (B) in min (C) in min (D) in min (E) 4 in min
Related Rates P. 8 of 7 5. The volume of a cube in increasing at the rate of 0 cubic centimeters per second. How fast, in square centimeters per second, is the surface area of the cube increasing at the instant when each edge of the cube is 0 centimeters long? (A) 4 (B) (C) 4 (D) 6 (E) 8 Volume/Surface Area Spheres 6. If the radius of a sphere is increasing at the rate of inches per second, how fast, in cubic inches per second, is the volume increasing when the radius is 0 inches? (A) 800 (B) 800 (C) 00 (D) 40 (E) 80
Related Rates P. 9 of 7 7. The radius r of a sphere is increasing at the uniform rate of 0. inches per second. At the instant when the surface area S becomes 00 square inches, what is the rate of increase, in cubic inches per second, in the volume V? 4 S 4 r and V r (A) 0 (B) (C).5 (D) 5 (E) 0 8. The volume of an expanding sphere is increasing at a rate of cubic feet per second. When the volume of the sphere is 6 cubic feet, how fast, in square feet per second, is the surface area increasing? (A) 8 (B) 6 (C) 8 (D) 8 (E) 0
Related Rates P. 0 of 7 00 AB 6 Form B Ship A is traveling due west toward Lighthouse Rock at a speed of 5 kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 0 km/hr. Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in the figure above. (a) Find the distance, in kilometers, between Ship A and Ship B when x 4 km and y km. (b) Find the rate of change, in km/hr, of the distance between the two ships when x 4 km and km. y
Related Rates P. of 7 (c) Let be the angle shown in the figure. Find the rate of change of, in radians per hour, when x 4 km and y km.
Related Rates P. of 7 999 AB 6 In the figure above, line is tangent to the graph of y at point P, with coordinates x w, w, where w > 0. Point Q has coordinates ( w, 0). Line crosses the x-axis at point R, with coordinates ( k,0). (a) Find the value of k when w =. (b) For all w > 0, find k in terms of w.
Related Rates P. of 7 (c) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of k with respect to time? (d) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of the area of PQR with respect to time? Determine whether the area is increasing or decreasing at this instant.
Related Rates P. 4 of 7 995 AB5 and BC As shown in the figure above, water is draining from a conical tank with height feet and diameter 8 feet into a cylindrical tank that has a base with area 400 square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of ( h ) feet per minute. (The volume V of a cone with radius r and height h is V r h. ) (a) Write an expression for the volume of water in the conical tank as a function of h. (b) At what rate is the volume of water in the conical tank changing when h? Indicate units of measure.
Related Rates P. 5 of 7 (c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h? Indicate the units of measure.
Related Rates P. 6 of 7 00 AB 5 and BC 5 A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of 5 h cubic inches per second. (The volume V of a cylinder with radius r and height h is V r h.) (a) Show that dh h dt 5.
Related Rates P. 7 of 7 (b) Given that h = 7 at time t = 0, solve the differential equation function of t. dh dt h for h as a 5 (c) At what time t is the coffeepot empty?
Derivatives and Rate of Change
Theorems THEOREM: Intermediate Value Theorem for Continuous Functions (IVT) A function y f ( x) that is continuous on a closed interval a, b takes on every value between f ( a ) and f ( b ). In other words, if y 0 is between f ( a ) and f ( b ), then y0 f( c) for some c in a, b. Example: The function above is continuous over the interval 0 x. Therefore, every y-value between and 4 is guaranteed somewhere on the interval, at least one time.
THEOREM: Mean Value Theorem for Derivatives (MVT) If y f ( x) is continuous at every point of the closed interval a, b and differentiable at every point of its interior ( a, b ), then there is at least one point c in ( a, b) at which f( c) Example: f ( b) f( a) b a For the given function, the slope of the segment connecting (0,) and (,4) is. If the function is continuous on the closed interval and differentiable on the open interval, there is at least one place where the slope of the tangent line will be. This occurs when x.
Derivatives and Rate of Change Page of Graphing Calculator allowed.. Let f be the function given by ( ) sin A) 0 B) f x x. The first positive root for f is C).77 D).507 E). For 0 x 0, how many points of intersection are there for the graphs of x y e and y sin x? A) 0 B) C) D) E) 4. The root for f x x is 0 ( ) ( x 9) 0 A) 4.7 B).8 C).07 D).8 E).4 4. For x > 0, find the root for f( x) x 89x40. A) 79 B) 84 C) 89 D) 40 E) none of these e 5. If f( x) e x x e e x x, how many horizontal asymptotes does f have? A) 0 B) C) D) E) 4 4 6. Let f ( x) x x 8x. The equation of the line tangent to f(x) at x = is y = 4x 5. Calculate the difference between the zero of the function f that occurs in the interval [0, ] and the zero of the line tangent to f at x =. A) 0. B) 0.4 C) 0.57 D) 0.679 E).5
7. How many relative extrema are there for the function g if 4 g( x) x x 60x 0 Page of A) B) C) D) 4 E) 5, then f () x 8. If f ( x) xcosxlnxe A) 0 B).000 C). D).406 E).469 9. The slope of the line tangent to the curve y x ln x at x = is closest to A) 0.76 B) 0.80 C) 0.84 D) 0.90 E) 0.94 0. The graph of the function f given f ( x) e x x changes concavity at x = A).58 B).5 C).46 D).40 E).4 9x. The function f is defined on the closed interval [, 4] by f( x) x. For what values of x in the interval [, 4] is the instantaneous rate of change equal to the average rate of change of f over the interval [, 4]. A) x =.00 B) x =. C) x =. D) x =. E) x =.44. If c satisfies the conclusion of the Mean Value Theorem for f ( x) secx on the interval x, then c is A) 0.94 B) 0.75 C) 0.787 D) 0.799 E).574
Page of. A particle moves along the x-axis so that at time t its position is given by xt () ( t)( t). For what values of t is the velocity of the particle increasing? A) t B) 0 t C) t D) t or t E) all t 4. For f ( x) sin x and g( x) 0.5x on the interval,, the instantaneous rate of change of f is greater than the instantaneous rate of change of g for which values of x? A) 0.8 B) 0 C) 0.9 D). E).5 5. sec x lim x x A).00 B).990 C) 0 D).04 E) Does not exit 6. Let f be the function given by x g x f x e x and let g be the function given by 6. At what value of x do the graphs of f and g have parallel tangent lines? (A) 0.70 (B) 0.567 (C) 0.9 (D) 0.0 (E) 0.58 7. The first derivative of the function f is given by f x cos x. How many x 5 critical values does f have on the open interval (0, 0)? (A) One (B) Three (C) Four (D) Five (E) Seven
Page 4 of 8.The function f given by f( x) x x 4 is (A) increasing for x, decreasing for x, increasing for x (B) decreasing for x 0, increasing for x 0 (C) increasing for all x (D) decreasing for all x (E) decreasing for x, increasing for x, decreasing for x 9. Which of the following is an equation of the line tangent to the graph 4 f ( x) x x at the point where f( x) (A) (D) y8x5 (B) y x 7 (C) y x0.76 y x0. (E) y x.46 0. The graph of the function x x x 6 7 cosx changes concavity at x = (A).58 (B).6 (C).675 (D).894 (E).7. If y x8, what is the minimum value of the product xy? (A) 6 (B) 8 (C) 4 (D) 0 (E)
Page 5 of. Let f ( x) x. If the rate of change of f at x = c is twice its rate of change at x =, then c = (A) (D) 4 (B) (C) 4 (E). An equation of the line tangent to the graph of x y x at the point (, 5) is (A) x y8 (B) x y 8 (C) xy 64 (D) xy 66 (E) xy 4. The slope of the line normal to the graph of y ln(sec x) at x is 4 (A) (B) (C) (D) (E) nonexistent x 5. If f( x) sin, then there exists a number c in the interval x that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c? (A) (B) 4 (C) 5 6 (D) (E)
Page 6 of 6. A particle moves along a line so that at time t, where 0 t, its position is given t by st () 4cost 0. What is the velocity of the particle when its acceleration is zero? (A) 5.9 (B) 0.74 (C). (D).55 (E) 8. No calculator. 7. If f ( x) x, then f (4) (A) - 6 (B) - (C) (D) 6 (E) 8 8. If x xyy 7, then in terms of x and y, dy dx (A) (B) (C) (D) (E) x y x y x y x y x y x y x y y x y
9. If y tan xcot x, then dy dx Page 7 of (A) sec xcsc x (B) sec x csc x (C) secx cscx (D) sec x csc x (E) sec x csc x 0. If f ( x) xcosx, which of the following could be f ( x )? (A) (B) (C) (D) (E) x cos x x x cos x x x cos xx x sin x x sin x, then f (0). If f( x) x x (A) 4 (B) 0 (C) (D) 4 (E). If f ( x) sine x (A) cose x x x (B) cose e x x (C) cose e (D) x x e cose (E) x x e cose, then f ( x)
Page 8 of. The table below gives values for f, f, g,and g at selected values of x. If hx ( ) f( gx ( )), then h() (A) (B) 6 (C) 8 (D) 9 (E) x f ( x ) f ( x) gx ( ) g( x) 6 4 4-4 4 4. The function g is continuous for x 5 and differentiable for x 5. If g( ) and g(5), which of the following statements are true? There exists c, where I. gc () 0 II. g() c 0 III. gc () c 5, such that: (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 5. The function g is continuous for x 4 and differentiable for x 4. If g( ) 5 and g(4), which of the following statements are true? There exists c, where I. gc () 0 II. g() c III. g() c c 4, such that: (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III
Page 9 of 6. If the line tangent to the graph of the function f at the point (,5) passes through the point (, ), then f () (A) (B) (C) (D) (E) 7. Let f be a twice-differentiable function with a positive first derivative and a negative second derivative. If g() and g() 0, which of the following could be g()? (A) 6 (B) 6 (C) 8 (D) 0 (E) 8. Which of the following functions could be increasing at an increasing rate? (A) x f ( x ) (D) 0 4 4 6 x f ( x) 9 5 4 0 (B) x f ( x ) 0 0 8 4 4 (E) x f ( x) 6 4 4 0 (C) x ( ) f x 0 5 4 9
Page 0 of Free Response, No Calculator. The twice-differentiable function f is defined for all real numbers and satisfies the following conditions: f (0), f (0) 4, and f (0). (a) The function g is given by gx ( ) e ax f( x) for all real numbers, where a is a constant. Find g (0) and g (0) in terms of a. Show the work that leads to your answers. (b) The function h is given by constant. Find x 0. h( x) hx ( ) cos kx f( x) for all real numbers, where k is a and write an equation for the line tangent to the graph of h at
Page of Free Response, No Calculator Consider the curve given by x 4y 7 x y. (a) Show that dy y x. dx 8y x (b) Show that there is a point P with x-coordinate at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P. (c) d y Find the value of at the point P found in part (b). Does the curve have a local dx maximum, a local minimum, or neither at the point P? Justify your answer.
Position, Velocity, and Acceleration (Motion)
AP Calculus AB Position, Velocity, and Acceleration (Motion along a line) Suppose an object is moving along a straight line (say x-axis) so that it s position x, as a function of time t, on that line is given by x f() t. Average velocity of the object over the time interval ttot dt is given by dx f ( t dt) f ( t) change in position, or. dt dt change in time Instantaneous velocity of the object is the derivative of the position function x f() t with respect to time. vt () x() t dx Speed is the absolute value of the velocity. Speed v() t. dt Acceleration is the derivative of velocity with respect to time. at () v() t x() t vt () dt xt () c, a() t dt v() t c Total distance traveled from time t t to t t is given by t TDT v() t dt. t
Speeding Up or Slowing Down Sign Convention: When the object is moving in the right direction or moving upward then the velocity is positive (Graph of velocity vs. time is above the t axis). When the object is moving in the left direction or moving downward then the velocity is negative. (Graph of velocity vs. time is below the t axis V t Time 0 t v, a v, a v, a v, a t 4 4 t 6 6 t 8 Object is slowing down Object is speeding up Object is slowing down Object is speeding up
What you need to know about motion along the x-axis: There are four closely related concepts that you need to keep straight: position, velocity, speed and acceleration. You need to understand how these relate to one another. If x(t) represents the position of a particle along the x-axis at any time t, then the following statements are true.. Initially means when time t = 0.. At the origin means x(t) = 0.. At rest means velocity v(t) = 0. 4. If the velocity of the particle is positive, then the particle is moving to the right. 5. If the velocity of the particle is negative, then the particle is moving to the left. 6. To find average velocity, divide the change in position by the change in time. 7. Instantaneous velocity is the velocity at a single moment (instant!) in time. 8. If the acceleration of the particle is positive, then the velocity is increasing. 9. If the acceleration of the particle is negative, then the velocity is decreasing. 0. Speed is the absolute value of velocity.. If the velocity and acceleration have the same sign (both positive or both negative), then speed is increasing.. If the velocity and acceleration are opposite in sign (one is positive and the other is negative), then speed is decreasing.. To determine total distance traveled over a time interval, you must find the sum of the absolute values of the differences in position between all resting points. 4. There are ways to use an anti-derivative that are easily confused. Watch out!
Position, Velocity, and Acceleration (Motion) Page of 7. The velocity of a particle moving on a line at time t is v t 5t meters per second. How many meters did the particle travel from t 0 to t 4? (A) (B) 40 (C) 64 (D) 80 (E) 84. If the position of a particle on the x-axis at time t is particle for 0t is (A) 45 (B 0 (C) 5 (D) 0 (E) 5 5t, then the average velocity of the. A particle moves along the x-axis so that at any time t 0 its position is given by x t t t 9t. For what values of t is the particle at rest? (A) No values (B) only (C) only (D) 5 only (E) and 4. The position of a particle moving along the x-axis is x t sin t t 0. When t, the acceleration of the particle is cos t for time (A) 9 (B) 9 (C) 0 (D) (E) 9 9 5. A particle moves along a line so that at time t, where 0 t, its position is given by t s t 4cost 0. What is the velocity of the particle when its acceleration is zero? (A) 5.9 (B) 0.74 (C). (D).55 (E) 8.
Position, Velocity, and Acceleration (Motion) Page of 7 6. A particle moves along the x-axis so that its acceleration at any time t is a t t7. If the initial velocity of the particle is 6, at what time t during the interval 0 t 4 is the particle farthest to the right? (A) 0 (B) (C) (D) (E) 4 7. A particle starts from rest at the point,0 and moves along the x-axis with a constant positive acceleration for time t 0. Which of the following could be the graph of the distance s t of the particle from the origin as a function of time? 8. Two particles start at the origin and move along the x-axis. For 0 t 0, their t respective position functions are given by x sin t and x e. For how many values of t do the particles have the same velocity? (A) None (B) One (C) Two (D) Three (E) Four
Position, Velocity, and Acceleration (Motion) Page of 7 Questions 9 0 refer to the following situation. 4 v 0 4 5 6 7 8 t - - A bug begins to crawl up a vertical wire at time t 0. The velocity v of the bug at time t, 0 t 8, is given by the function whose graph is shown above. 9. At what value of t does the bug change direction? (A) (B) 4 (C) 6 (D) 7 (E) 8 0. What is the total distance the bug traveled from t 0 to t 8? (A) 4 (B) (C) (D) 8 (E) 6. A particle moves along the x-axis so that at any time t 0 its velocity is given by vt () ln( t) t. The total distance traveled by the particle from t = 0 to t = is (A) 0.667 (B) 0.704 (C).540 (D).667 (E).90 t (sec) 0 4 6 a(t) (ft/sec ) 5 8. The data for the acceleration at () of a car from 0 to 6 seconds are given in the table above. If the velocity at t = 0 is feet per second, the approximate value of the velocity at t = 6, computed using a left-hand Riemann sum with three subintervals of equal length, is (A) 6 ft/sec (B) 0 ft/sec (C) 7 ft/sec (D) 9 ft/sec (E) 4 ft/sec
Free Response Questions 999 AB, Calculator allowed. Position, Velocity, and Acceleration (Motion) Page 4 of 7 A particle moves along the y-axis with velocity given by vt () tsin( t ) for t 0. (a) In which direction (up or down) is the particle moving at time t.5? Why? (b) Find the acceleration of the particle at time t.5. Is the velocity of the particle increasing at t.5? Why or why not? (c) Given that yt () is the position of the particle at time t and that y(0), find y(). (d) Find the total distance traveled by the particle from t 0 to t.
00 AB 4 Form B Position, Velocity, and Acceleration (Motion) Page 5 of 7 A particle moves along the x-axis with velocity at time t 0 given by vt () e t. (a) Find the acceleration of the particle at time t =. (b) Is the speed of the particle increasing at time t =? Give a reason for your answer. (c) Find all values of t at which the particle changes direction. Justify your answer. (d) Find the total distance traveled by the particle over the time interval 0t.
00 AB and BC Position, Velocity, and Acceleration (Motion) Page 6 of 7 A car is traveling on a straight road with velocity 55 ft/sec at time t 0. For 0 t 8 seconds, the car s acceleration a(t), in ft/sec, is the piecewise linear function defined by the graph above. (a) Is the velocity of the car increasing at t = seconds? Why or why not? (b) At what time in the interval 0 t 8, other than t = 0, is the velocity of the car 55 ft/sec? Why? (c) On the time interval 0 t 8, what is the car s absolute maximum velocity, in ft/sec, and at what time does it occur? Justify your answer. (d) At what times in the interval 0 t 8, if any, is the car s velocity equal to zero? Justify your answer.
00 AB Form B Position, Velocity, and Acceleration (Motion) Page 7 of 7 A particle moves along the x-axis so that its velocity v at any time t, for 0 t 6, is given by v(t ) e sint. At time t 0, the particle is at the origin. (a) Sketch the graph of v(t) for 0 t 6. (b) During what intervals of time is the particle moving to the left? Give a reason for your answer. (c) Find the total distance traveled by the particle from t 0 to t 4. (d) Is there any time t, 0 t 6, at which the particle returns to the origin? Justify your answer.
Advanced Integration Topics
Advanced Integration Topics Integration by Parts udv uv vdu Integration by Partial Fractions with Linear Factors. Divide if Improper If the integrand is a fraction with the degree of the numerator greater than the degree of the denominator, divide the numerator by the denominator and then apply steps and.. Factor the Denominator Completely factor the denominator into m linear factors and create a sum of m fractions, each having one of the factors as a denominator.. Solve for the numerator values and integrate the decomposition A A A Am dx ax b cx d ex f rx t Improper Integrals with Infinite Integration Limits. If f is continuous on the interval a, then b, f x dx lim f x dx. a b a b b. If f is continuous on the interval,b f x dx lim f x. If f is continuous on the interval, c c, then dx., then a a f x dx f x dx f x dx, where c is any real number. In the first two cases, the improper integral converges if the limit exists --- otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.
Advanced Integration Topics Page of 8 What are advanced techniques of integration? There are two techniques that you are responsible for knowing on the AP exam. The first is called integration by parts and is based on your knowledge of the product rule. udv uv vdu. xsin xdx. ln xdx. Let s review the tabular method: x xe dx
Advanced Integration Topics Page of 8 The second technique is called partial fractions. 4. x 4 dx 5. x x dx 5 x 6. dx x x
Advanced Integration Topics Page of 8 Now, we should be able to use our knowledge of integration to deal with improper integrals. There are two types of improper integrals: i) integrals that are improper because one or both of the limits of integration are infinite 7. dx x 8. dx x ii) integrals that are improper because there is an infinite discontinuity either between or at the limits of integration 9. 4 x dx
0. 0 x dx Advanced Integration Topics Page 4 of 8 What might this look like on the AP exam? 985 BC 5 Let f be the function defined by f (x ) ln x for 0 x and let R be the region between the graph of f and the x-axis. (a) Determine whether region R has finite area. Justify your answer.
980 BC 6 Advanced Integration Topics Page 5 of 8 Let R be the region enclosed by the graphs of y e x, x k (k 0), and the coordinate axes. (a) Write an improper integral that represents the limit of the area of the region R as k increases without bound and find the value of the integral if it exists. 97 BC 5 Determine whether or not your reasoning. 0 xe x dx converges. If it converges, give the value. Show
995 AB Advanced Integration Topics Page 6 of 8 A particle moves along the y-axis so that its velocity at any time vt () tcost. At time t = 0, the position of the particle is y =. (c) Write an expression for the position yt () of the particle. t 0 is given by 996 BC Consider the graph of the function h given by hx ( ) e x for 0 x. (a) Let R be the unbounded region in the first quadrant below the graph of h. Find the volume of the solid generated when R is revolved about the y-axis.
00 BC 5 Advanced Integration Topics Page 7 of 8 Let f be the function satisfying f ( x) x f( x), for all real numbers x, with and lim f( x) 0. x f () 4 (a) Evaluate x f( x) dx. Show the work that leads to your answer. Multiple Choice:. x sec xdx x (a) x tan x C (b) tan x C (c) sec x sec xtan x C (d) x tan xln cos x C (e) x tan x ln cos x C. (a) xx (b) 0 dx 9 (c) 0 5 ln (d) 8 ln 5 (e) ln 5
Advanced Integration Topics Page 8 of 8. 4 x 9 x dx (a) 7 (b) 7 (c) 9 7 (d) 9 7 (e) DNE 4. x f( x) dx = (a) (c) (e) (b) f ( x) fx x f( x) x f x dx x x f( x) f x C x f xdx x x (d) ( ) x f x f x dx dx 5. xcos xdx 0 (a) (b) - (c) (d) (e)
Area and Volume
DEFINITION Area Between Curves If f and g are continuous with ( ) ( ) f x g x throughout, ( ) the curves y f( x) and y g x from a to b is the integral of b a A f( x) g( x) dx. a b, then the area between f g from a to b, DEFINITION Volume of a Solid The volume of a solid of known integrable cross section area A( x ) from is the integral of A from a to b, x a to x b V b A( x) dx. a
006 AB and BC Area and Volume P. of 6 Let R be the shaded region bounded by the graph of shown above. y ln x and the line y x, as (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y. (c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis.
006 AB and BC Form B Area and Volume P. of 6 x x x Let f be the function given by f ( x) cosx. Let R be the shaded region in 4 the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and the line, the line tangent to the graph of f at x = 0, as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y. (c) Write, but do not evaluate, an integral expression that can be used to find the area of S.
Area and Volume Free Response 004 AB Form B Area and Volume P. of 6 Let R be the region enclosed by the graph of y x, the vertical line x = 0, and the x-axis. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y =. (c) Find the volume of the solid generated when R is revolved about the vertical line x = 0.
00 AB and BC Form B Area and Volume P. 4 of 6 Let f be the function given by f ( x) 4x x, and let be the line y 8 x, where is tangent to the graph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f, the line, and the x-axis, as shown above. (a) Show that is tangent to the graph of y f( x) at the point x =. (b) Find the area of S. (c) Find the volume of the solid generated when R is revolved about the x-axis.
Area and Volume P. 5 of 6 00 AB and BC Let f and g be the functions given by f (x ) e x and g( x) ln x. (a) Find the area of the region enclosed by the graphs of f and g between x and x. (b) Find the volume of the solid generated when the region enclosed by the graphs of f and g between x and x is revolved about the line y 4. (c) Let h be the function given by h(x ) f (x ) g( x). Find the absolute minimum value of h(x ) on the closed interval value of h(x ) on the closed interval answers. x, and find the absolute maximum x. Show the analysis that leads to your
Area and Volume P. 6 of 6 000 AB and BC Let R be the shaded region in the first quadrant enclosed by the graphs of y cosx, and the y-axis, as shown in the figure above. (a) Find the area of the region R. y e x, (b) Find the volume of the solid generated when the region R is revolved about the x-axis. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.
Area and Volume P. 7 of 6 Area and Volume Multiple Choice Area. What is the area of the region between the graphs of x =? y x and y x from x = 0 to (A) (B) 8 (C) 4 (D) 4 (E) 6. The area of the region enclosed by the graph of y x and the line y = 5 is (A) (B) (C) (D) (E) 4 6 8 8. The area of the region in the first quadrant between the graph of x - axis is (A) 4 and the y x x (B) 8 (C) (D) (E) 6
Area and Volume P. 8 of 6 4. Which of the following represents the area of the shaded region in the figure? (A) (D) d f y dy (B) d f x dx (C) c b a f b f a (E) dc f b f a b a f b fa 5. The area of the region enclosed by the curve x and x 4is y x, the x-axis, and the lines (A) 5 6 (B) n (C) 4 n (D) n (E) n6 6. The area of the region enclosed by the graphs of y x and yx x is (A) (B) (C) 4 (D) (E) 4 7. The area of the region in the first quadrant that is enclosed by the graphs of y x 8and yx 8 (A) 4 (B) (C) 4 (D) (E) 65 4
Area and Volume P. 9 of 6 8. The area of the region bounded by the lines x 0, x, and y0 and the curve y e x (A) e (B) e (C) e (D) e (E) e 9. What is the area of the region completely bounded by the curve the line y 4 6 and y x x (A) (B) 7 (C) 9 (D) 6 (E) 0. The region bounded by the x-axis and the part of the graph of ycos x between x x k is three times the area of the region for k x,then k (A) arcsin (B) arcsin (C) (D) (E) 4 6 4. The area of the region bounded by the curve x y e, the x -axis, the y -axis, and the line x is (A) 4 e e (B) 4 e (C) 4 e (D) 4 e e (E) 4 e
Area and Volume P. 0 of 6. A region in the plane is bounded by the graph of y, the x-axis, the line x m, x and the line x m, m0. The area of this region (A) is independent of m (B) increases as m increases (C) decreases as m increases (D) decreases as m increases when m ; increases as m increases when m (E) increases as m increases when m ; decreases as m increases when m. The area in the first quadrant that is enclosed by the graphs of x y and x 4 y is (A) 4 (B) 8 (C) 4 (D) 0
Area (calculator active) Area and Volume P. of 6 4. If 0 k and the area under the curve y cos x from x = k to then k = x is 0., (A).47 (B).44 (C).77 (D).0 (E) 0.46 5. What is the area of the region in the first quadrant enclosed by the graphs of y cos x, y = x, and the y-axis? (A) 0.7 (B) 0.85 (C) 0.400 (D) 0.600 (E) 0.947
Area and Volume P. of 6 Volume 6. If the region enclosed by the y-axis, the line y =, and the curve y x is revolved about the y-axis, the volume of the solid generated is (A) (B) (C) (D) (E) 5 6 6 5 8 7. Let R be the region in the first quadrant enclosed by the graph of y x, the line x = 7, the x-axis, and the y-axis. The volume of the solid generated when R is revolved about the y-axis is given by 7 (A) 7 x dx (B) xx 0 (C) x dx (D) xx (E) y 0 7 0 dy 0 0 dx dx 8. The region enclosed by the x-axis, the line x =, and the curve y x is rotated about the x-axis. What is the volume of the solid generated? (A) (B) (C) 9 6 (D) 9 (E) 5
Area and Volume P. of 6 x 9. A region in the first quadrant is enclosed by the graphs of y e, x =, and the coordinate axes. If the region is rotated about the y-axis, the volume of the solid that is generated is represented by which of the following integrals? (A) x xe dx (B) 0 0 e x dx (C) 4 x 0 e dx (D) e y nydy (E) 0 4 e nydy 0 0. The volume of the solid obtained by revolving the region enclosed by the ellipse x 9y 9 about the x-axis is (A) (B) 4 (C) 6 (D) 9 (E). The region enclosed by the graph of y x, the line x =, and the x-axis is revolved about the y-axis. The volume of the solid generated is (A) 8 (B) 5 (C) 6 (D) 4 (E) 8. The region in the first quadrant bounded by the graph of y = sec x, x, and the 4 axes is rotated about the x-axis. What is the volume of the solid generated? (A) 4 (B) (C) (D) (E) 8. The first region in the first quadrant bounded by y = cos x, y = sin x, and the y-axis is rotated about the x-axis. The volume of the resulting solid is (A) (B) (C) (D) (E) 4