Practice Problems for Final

Practice Problems for Final (1) (a) Define what it means for a function f(x) to be continuous at x = a. (b) Explain what removable, jump, and infinite discontinuities are. (c) Find and classify (as removable, jump, infinite) all discontinuities of the following function: x 2 1 if x 1 > 1 f(x) = sin(x) 1 if x 1 < 1 3 if x 1 = 1 (2) Explain why the function f(x) = { sin(1/x) if x if x = is discontinuous at x =, and also explain why it is not a removable, jump, or infinity discontinuity. (3) Find and classify the discontinuities of the following function: { 1 x 2 if x 1 f(x) = 1 if x > 1. x 1 (4) Calculate the following limits if they exist. You may not use L Hopital s rule (for those who know what this is). If the limit is infinite, say whether it is ± (a) lim x 2 x 2 1 x 2 4x+3 x (b) lim 2 1 x 1 x 2 4x+3 (c) lim x π sin(x) cos(x π) sec 2 (x) (d) lim x 1+x 1 x (e) lim x 2 + 1 x 2 4 x 2 4 (f) lim x 1 x 1 x 1 (g) lim x x 3 x 3 (h) lim x x 2 +4 2 x+4 2 (i) lim x π 1 cos 2 (x) sin 2 (x) (j) lim x π 3 ζ (k) lim x π/2 sin(x)+sin(x) tan 2 (x) sec 2 (x) (l) lim x (m) lim x x 3 3x+1 3x 3 2x 2 +5 x 2 +3x+1 4x 3 +3x 2 +2x+1 1

(n) lim x x 3 +2x+1 (o) lim x x 2 +5x+6 x x (p) lim x x 2 1 x (q) lim x 6 +1 x x 3 (r) lim x x 3 x 3 +x+1 (5) Use the limit definition of the derivative to calculate the derivatives of the following functions. (a) f(x) = 4x 2 + 1 (b) f(x) = x + 1 (c) f(x) = 1 x 1 (d) f(x) = 1 x 2 +1 (e) f(x) = x 1/4 (6) Evaluate the following limits by first expressing the limit as f (a) for some f(x) and some point x = a, and then finding the limit by calculating the derivative instead. (a) lim x 1 x 9 +1 x+1 (b) lim x 8 3 x 2 x 8 (c) lim x π/2 sin(x) 1 x π/2 (d) lim x 3π sin(x) cos(x) x 3π (7) The tangent line to f(x) at the point x = 1 has equation y = 5x + 3. Find f (1) and f(1). (8) Find all points on the graph of f(x) = x 3 3x 2 where the tangent line has slope 9. (9) Find dy/dx: (a) xy = x + y (b) sin(y) = cos(x) + cos(y) (c) x y = sin(x) csc(y) (d) x 3 + y 3 = 4y (e) 2 y = x y (1) Find the tangent line to the curve at the given point: (a) x 2 + xy y 2 = 1 at (2, 3) (b) x 2 y 2 = 9 at ( 1, 3) (c) 6x 2 + 3xy + 2y 2 + 17y 6 = at ( 1, ) 2

(d) 2xy + π sin(y) = 2π at (1, π/2) (e) y = 2 sin(πx y) at (1, ) (11) If sin(rθ) = 1 dr, find. 2 dθ (12) Consider the curve x 2 + xy + y 2 = 9 (a) Find the two points where the curve crosses the x-axis. (b) Show that the tangent lines to these two points are parallel, and identify the common slope. (13) Consider the curve x 3 + y 3 9xy = (a) Show (4, 2) is on the curve (b) Find the tangent line to the curve at (4, 2) (c) At what point other than the origin does the curve has a horizontal tangent? (14) Suppose the curves y = x 2 + ax + b and y = cx x 2 has a common tangent at (1, ). Find a, b, and c. (15) The curve y = ax 2 + bx + c passes through (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c. (16) Find the points on f(x) = x 3 3x 2 where the slope of the tangent line is minimal. (17) Gas in a cylinder which is maintained at a constant temperature T has volume V and pressure P related by P = nrt V nb an2 V, 2 where a, b, n, and R are constants. Find dp/dv. (18) A rock is thrown vertically upwards from the surface of the moon with initial velocity 24 m/s. The height at time t is s(t) = 24t.8t 2, where s is in meters and t in seconds. (a) Find the velocity and acceleration at time t. (b) How long does it take the rock to reach its highest point? (c) How high does the rock go? (19) The line normal to x 2 + 2xy 3y 2 at (1, 1) intersects the curve at another point. Find this other point. (2) When a metal circular plate is heated in the oven, the radius increases at.1 cm/min. Find the rate at which the area of the plate is changing 5 minutes after it is put into the oven if its initial radius is 2 cm. (21) At noon, a truck is 6 miles north of a car. The truck is traveling due south at a rate of 4 mph and the car is moving east at 6 mph. At what rate is the distance between the car and the truck changing at 12:3 pm? 3

(22) Sand is being dumped into a conical pile where the height is 1/2 the radius. Suppose the sand is being dumped at a rate of 5 cubic inches per minute. (a) At what rate is the height of the pile increasing at the moment the height is 2 inches? (b) At what rate is the area of the base changing at this instant? (c) At what rate is the circumference of the base changing at this instant? (d) Will the height be increasing more rapidly, more slowly, or at a constant rate as time goes on? (23) A balloon leaves the ground 5 feet away from an observer and rises vertically at a rate of 14 feet per minute. At what rate is the angle of inclination of the observer s line of sight increasing at the moment the balloon is 5 feet above the ground. (24) A street light sits on top of a 15 feet pole. A 6 ft man is walking in a straight line away from the pole at a rate of 6 feet per second. (a) At what rate is the tip of the shadow moving? (b) At what rate is the shadow lengthening? (25) For what a, m, b does the function 3 if x = f(x) = x 2 + 3x + a if < x < 1 mx + b if 1 x 2 satisfy the hypotheses of the Mean Value Theorem on the interval [, 2]? Find the c satisfying the theorem with these choices of a, m, b. (26) Find a so that is continuous at x = 3. f(x) = { ax + a if x < 3 ax 2 1 if x 3 (27) If lim x 2 f(x) 5 x 2 = 3, find lim x 2 f(x). (Hint: Limit laws) (28) If lim x f(x) x 2 (a) lim x f(x) (b) lim x f(x) x = 1, find: (29) Suppose a particle is moving along a straight line with acceleration a(t) = 4 cos(2t), and velocity v(t) and position s(t). At time t =, the velocity v() = 1, and its position at t = is s() = (a) Find v(t). 4

(b) Find s(t). (c) For what t in the interval [, π] is the particle at rest. (3) What are the horizontal asymptotes of f(x) = x x 2 4? (31) Let f(x) be a function with f (x) = 6x + 8. Find f(x) if the graph of f(x) is tangent to the line 3x y = 2 at the point (, 2). (32) Use Rolle s theorem to show that f(x) = x 5 4x + 2 has at most 3 real roots. (33) Use Rolle s theorem to show that f(x) = x 3 + 3x + 1 has at most 1 real root. (34) Determine the area of the largest rectangle that can be inscribed in the triangle formed by the x-axis and the lines y = 3x + 12 and y = 3x + 12. That is, find the area of the largest rectangle that can be drawn with one side on the x-axis and exactly one vertex on each line. (Hint: Draw a picture.) (35) Consider the curve y = 1 x in the first quadrant. (a) Find the point on the curve closest to the origin. (b) Show that the line segment connecting the origin to the point in (a) is perpendicular to the tangent line to y = 1/x at the point from (a). (36) Calculate the absolute maximum and minimum of the following functions on the specified intervals, if they exist. (a) f(x) = x 4 2x 2 + 1 on [ 2, 1] (b) f(x) = 3 sin(x) cos(x) on [ π 3, π 4 ] (c) f(x) = sec(x) on [ π, 5π ] (be careful) 3 6 (d) f(x) = x+3 on [ 3, 1]. x 2 (37) Suppose f(x) is everywhere differentiable and that f(6) = 2. If f (x) 1 for all x, find the largest possible value for f(15). (38) Consider f(x) = x 3 x on [, 2]. Find all c satisfying the conclusions of the Mean Value Theorem. (39) Suppose f(x) = sin(sin(sin(sin(sin(x))))). Find g(x) such that f (x) = g (x) and f() = 1. (4) We want to construct a closed box with a square base and we have only 1 square meters of material. What should the dimensions of the box be if we want to maximize the volume? (41) A window is being built, where the top is a semicircle and the bottom part is a rectangle. If there is 12 meters of framing material, what should the radius of the semicircle be if we want the window to let in a maximal amount of light? 5

(42) Determine the largest rectangle that can be inscribed in a circle of radius 4. (43) Determine the point in the first quadrant on y = x 2 + 1 closest to the point (, 2). (44) Suppose f (x) = sin(x) cos 2 (x). If f(π/2) =, find f(). (45) Sketch the graphs of the following functions: (a) f(x) = x 1/3 (x + 4) (b) f(x) = sin( x ) + x on [ 2π, 2π] (c) f(x) = x 4 2x 2 + 1 (d) f(x) = 1 x 2 +1 (e) f(x) = x 3 x+2 (46) Find the area bounded by the curves y = sin(x), y = cos(x), the y-axis, and the line x = π/2. (47) Find the area bounded by the curves y = sin 2 (x), y = cos 2 (x), the y-axis, and the line x = π/2. (48) (a) Show that f(x) = 1 + x is increasing on [1, 4]. (b) Conclude that (c) Evaluate 3 2 (Hint: It is a u-sub problem). 4 1 4 1 1 + xdx 3 3. 1 + xdx (49) Determine the area of the region bounded by y = 2x 2 + 1 and y = 4x + 16. (5) Find the absolute maximum and minimum of the function f(x) = x 2 2 sin ( x 2) on the interval [, 2π]. (51) Calculate 2 4 x2 dx. (52) Find the area of the region bounded by y = x 2 and y = 2x x 2. (53) Consider the region bounded by the graphs of y = 1 x and y = x 1. (a) Sketch the region. (b) Use integrals to find the area of the region. 2 (c) Find the answer in a much simpler way, using geometry to find the area of that shape. 6

(54) Calculate the area under the following curves finding a general formula for the right endpoint sum R N and taking a limit as N. Then, calculate the area using the Fundamental Theorem of Calculus. (a) f(x) = 2 on [1, 3] (b) f(x) = 2x 1 on [3, 5] (c) f(x) = x 2 + 1 on [1, 3]. (55) State the Fundamental Theorem of Calculus, and use it to do the following: (a) If (b) If find F (π/4). F (x) = tan(x) G(x) = ( t 2 1 + ) t dt, x 3 x sin(t 3 )dt, where x >, find G (x). (56) Consider the function (a) What is F ()? F (x) = x sin(t 2 )dt. (b) Determine where on the interval [, 2π] the function F (x) is increasing/decreasing. (c) Determine the local extrema of F (x) on [, 2π]. (d) Determine where F (x) is concave up/down on the interval [, 2π]. (e) Determine the points of inflection of F (x). (57) Calculate the following integrals. (a) (x 2 + 2x + 3)dx (b) 2 1 x 1 x 2 dx (c) cos(x) + 2 cos 2 (x) dx (d) x 4 +3x 3 dx x 2 (e) π sin(2x)dx (f) 1 (5x + 1)5 dx (g) x 1 xdx (h) 9π 2 sin( x) π 2 x dx (i) 5x+5 (x 2 +2x+3) 3 dx 7

(j) cos(x) sin 3 (x)dx (k) x(1 + x) 1 dx (l) 1 x3 x 2 + 1dx (m) x cos(x 2 + 1)dx (58) A particle moves along a straight line with velocity v(t) = t 2 t. (a) Find the particle s displacement on [, 2]. (b) Find the total distance traveled by the particle on [, 2]. (59) Consider the solid formed by rotating the region in the 1st quadrant bound by the curves y = 3 x and y = x about the y-axis. 4 (a) Find the volume of the solid using the washer method. (You do NOT need to simplify your answer) (b) Find the volume of the solid using the method of cylindrical shells. (You do NOT need to simplify your answer) (6) Determine the volume of the solid formed by rotating the region in the first quadrant bound by the curves x = (y 2) 2 and y = x about the line y = 1. (61) Find the volume of a pyramid with square base whose base is a square of side length L and whose height is h. (62) Find the volume of a solid whose base is a disk of radius r and whose cross-sections are equilateral triangles. (63) Find the volume of the solid generated by rotating the region bounded by y = 2 x 2 and y = 1 around the line y = 1. (64) Consider the region in the first quadrant bounded by the lines y = 8x, y = x, and the curve y = 1 x 2. (a) Find the area by setting up an integral with respect to x and integrating. (b) Find the area by setting up an integral with respect to y and integrating (65) Find the volume of a solid whose base is the unit disc (i.e. the disc bounded by x 2 + y 2 = 1) and whose cross sections are squares. (66) Consider the region bounded by y = and y = x x 2. Find the volume of the solid generated by rotating this region about the line y = 3. (67) Consider the region R in the first quadrant bounded by y = cos(x), y = 1, x = π/2. Set up, but do not evaluate, the integrals to calculate the volume of the region generated by rotating R around: (a) The x-axis. 8

(b) The y-axis. (c) The line x = 3 (d) The line y = 5 (e) The line x = 5 (f) The line y = 1 Multiple Choice Questions: Select the correct answer for each question. There is only one correct choice. (68) Suppose f(x) is a differentiable function. Consider the following limits: (I) lim h f(a+h) f(a) h (II) lim x a f(x) f(a) x a (III) lim h f(x+h) f(x) h Which of the above represents f (a) for some number a? (a) (I) only (b) (II) only (c) (I) and (II) only (d) (I), (II), and (III) (69) If b f(x)dx = 5 and b g(x)dx = 1, then which of the following must be true? a a (I) f(x) > g(x) for all x in [a, b] (II) b (f(x) + g(x))dx = 4 a (III) b f(x)g(x)dx = 5 a (a) (II) only (b) (II) and (III) only (c) (I) and (II) only (d) (I) and (III) only (7) If f(x) is continuous on [a, b] and differentiable on (a, b), then which of the following could be FALSE: (a) f (c) = f(b) f(a) b a for some c in (a, b) (b) f (c) = for some c in (a, b) (c) f(x) attains a maximum at some point in [a, b] (d) b f(x)dx exists. a 9

(71) Which of the following represents d b dx a sin(t2 )dt? (a) (b) sin(x 2 ) (c) sin(b 2 ) (d) None of the above. (72) If f(x) is a linear function and < a < b, then which of the following represents b a f (x)dx? (a) (b) 1 (c) b a (d) b2 a 2 2 (73) A particle is moving along a straight line with velocity v(t) = cos(2t). Which of the following represents the total distance traveled on [, π]? (a) (b) 1 (c) 2 (d) 1 (74) If f(x) is a continuous function and F (x) some antiderivative of f(x), then 3 f(2x)dx =? 1 (a) 2F (3) 2F (1) (b) 1 2 F (3) 1 2 F (1) (c) 1 2 F (6) 1 2 F (2) (d) 2F (6) 2F (2) (75) Suppose f(x) is continuous on [a, b] and it has a relative maximum at x = c, where c is in (a, b). Which of the following must be true: (I) f (c) exists (II) If f (c) exists, then f (c) = (III) If f (c) exists, then f (c) < (a) (I) only (b) (II) only (c) (III) only (d) (II) and (III) only 1

π (76) Evaluate lim N N 6N j=1 sin( π + π j) 3 6N (a) (b) 1 (c) 1/2 (d) 1/3 (77) Let f(x) be a twice differentiable function with f(1) = 2 and f(3) = 7. Which of the following must be true? (I) The average rate of change of f(x) on [1, 3] is 5/2 (II) The average value of f(x) on [1, 3] is 9/2 (III) The average value of f (x) on [1, 3] is 5/2 (a) (I) only (b) (I) and (II) only (c) (III) only (d) (I) and (III) only (78) Consider the function F (x) = x 1 + t2 dt. Which of the following is true? (a) F (x) is a strictly increasing function on the set of real numbers. (b) There exists a point c with f (c) =. (c) F (1) < (d) F (x) is not differentiable at x = 1 (79) Which of the following is lim x π/4 tan(x) 1+x 1+ π 4 x π 4 (a) (b) 1 (c) 2 (d) 3? 11

Solutions: (1)(a/b) Look in book (c) Removable discontinuity at x =, jump discontinuity at x = 2. (2) It is not continuous becuase lim x sin(1/x) DNE. It is none of the given discontinuities because for removable and jump at least one sided limits exist (which doesn t happen for this function), and for infinite discontinuities the limit must be infinite from either the right or left, which is also not the case here. (3) Infinite discontinuity at x = 1. (4) (a) 3 (b) 1 (c) (d) 1/2 (e) 1/4 (f) 1 (g) DNE (h) (i) 1 (j) π 3 ζ (k) 1 (l) 1/3 (m) (n) (o) 1 (p) 1 (q) 1 (r) 1 (5) (a) f (x) = 8x (b) f (x) = 1 2 x (c) f (x) = 1 (x 1) 2 (d) f (x) = (6) (a) 9 2x (x 2 +1) 2 (e) f (x) = 1 4x 5/4 (b) 1/12 12

(c) (d) 1 (7) f(1) = 8, f (1) = 5. (8) (3, ) and ( 1, 4) (9) (a) y = 1 y x 1 (b) y = (c) y = (d) y = (e) y = sin(x) cos(y)+sin(y) y2 cos(x) csc(y) y y 2 sin(x) csc(y) cot(y) x 3x2 4 3y 2 y y+1 (1) (a) Tangent y 3 = 7 (x 2) 4 (b) Tangent y 3 = 3(x + 1) (11) r/θ (c) Tangent y = 6 (x + 1) 7 (d) Tangent y π = π (x 1) 2 2 (e) Tangent y = 2π(x 1) (12) (a) (±3, ) (b) The common slope is 2 (13) (a) Plug it in and show the equation holds. (b) y 2 = 5 (x 4) 4 ( ) (c) 3 54, 54 2/3 3 (14) a = 3, b = 2, c = 1 (15) a = 1, b = 1, and c = (16) (, 2) (17) dp dv = nrt + 2n2 (V nb) 2 V 3 (18) (a) v(t) = 24 1.6t and a(t) = 1.6 (b) 15 s (19) (3, 1) (c) s(15) meters (2) 2π(2.5)(.1) cm 3 /min 13

(21) 4 mph. 5 (22) (a) inches per minute. 16π (b) 5 sq. in. per minute. (c) 5/4 in/min (d) more slowly (23) 7/5 radians per second (24) (a) 1 ft/s (b) 4 ft/s (25) a = 3, m = 1, b = 4, and with these, c = 3/4 (26) a = 1/3 (27) 5 (28) (a) (b) (29) (a) v(t) = 2 sin(2t) + 1 (b) s(t) = cos(2t) + t + 1 (3) y = ±1 (c) t = 7π, 11π 12 12 (31) f(x) = x 3 + 4x 2 + 3x 2 (32) f (x) = 5x 4 4, which has 2 roots. Therefore f(x) can have at most 3 real roots by Rolle. (33) f (x) > for all x, and so f(x) can have at most one root by Rolle. (34) 24 (35) The point is (1, 1) (36) (a) Max of 9 at x = 2 and min of at x = ±1 (37) 88 (b) Max of 2 at x = 2π/3 and min of 2 at x = π/3 (c) No global max or min exist. (d) Max of at x = 3 and min of 4 at x = 1. (38) c = 2/ 3 14

(39) g(x) = sin(sin(sin(sin(sin(x))))) + 1 (4) It is a cube with side length 5/3 meters. (41) 12/(4 + π) meters. (42) It should be a square, with side length 4 2. (43) (1/ 2, 3/2) (44) 1/3 (45) Use a graphing calculator or WolframAlpha to check. (46) 2 2 2 (47) 1 (48) Its derivative is always positive for x >, so it is increasing on [1, 4]. The integral is 16 5 3 8 15 2. (49) 64/3 (5) Absolute maximum of π at x = 2π and absolute minimum of π 3 3 at x = 2π/3. (51) 2π (52) 1/3 (53) The area is 2. (54) (a) 4 (b) 14 (c) 32/3 (55) (a) 2 2 2 (56) (a) (b) 3x 2 sin(x 9 ) 1 2 x sin(x3/2 ) (b) Increasing on (, π) ( π, 2π) (c) Local max at x = π (d) Concave up on (, π/2) ( 3π/2, 2π) (e) Inflection points at x = π/2 and x = 3π/2 (57) (a) x3 3 + x2 + 3x + C (b) 1 (c) sin(x) + 2 tan(x) + C 15

(d) x3 3 + 3 2 x2 + c (e) (f) 66 3 (g) 2 3 (1 x)3/2 + 2 5 (1 x)5/2 + C (h) (i) 5 4(x 2 +2x+3) 2 + C (j) sin4 (x) 4 + C (k) 1 12 (1 + x)12 1 11 (1 + x)11 + C (l) 2 15 (1 + 2) (m) 1 2 sin(x2 + 1) + C (58) (a) 2/3 (b) 1 (59) Both methods should give you 512π 21. (6) 63π/2 (61) L 2 h/3. (62) 4 3 r 3 (63) 16π/5 (64) 3/2 (65) 16/3 (66) 31π/3 (67) (a) π π/2 1 cos 2 (x)dx (68) c (69) a (7) b (b) 2π π/2 x(1 cos(x))dx (c) 2π π/2 (x + 3)(1 cos(x))dx (d) π π/2 (5 cos(x)) 2 4 2 dx (e) 2π π/2 (5 x)(1 cos(x))dx (f) π π/2 2 2 (cos(x) + 1) 2 dx 16

(71) a (72) a (73) c (74) c (75) b (76) c (77) d (78) a (79) b 17