Mostly free response questions for calc 2 in college.
PROBLEM LIST IV Math 1226, Summer II - 2021 Fazle Rabby This list covers everything from this course. 1. L’Hôpital’s Rule (1) For each of the following limits, find its value or explain why it does not exist. (a) lim x→0 | sin(x)|x (b) lim x→∞ x− sin(x) x+ sin(x) (c) lim x→1+ ( 1 x− 1 − 1 lnx ) (d) lim x→0 (cosx)cot 2(x) (e) lim x→∞ x√ x2 + 1 (f) lim x→0 ln | sinx| ln |x| (2) Let f and g be differentiable functions such that g′(x) 6= 0 for all x in some interval I containing 0, except possibly at 0. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. 1) If f(0) = g(0) = 0 and g′(0) 6= 0 then lim x→0 f(x) g(x) = f ′(0) g′(0) , given that f ′(0) and g′(0) exist. 2) If lim x→0 f(x) = lim x→0 g(x) = 0 and lim x→0 f ′(x) g′(x) = 1, then lim x→0 f(x) g(x) = 1. 3) If lim x→0 f(x) = lim x→0 g(x) = 1 and lim x→0 f ′(x) g′(x) = −1, then lim x→0 f(x) g(x) = −1. 4) If lim x→0 f(x) = lim x→0 g(x) = 0 and lim x→0 f ′(x) g′(x) =∞, then lim x→0 f(x) g(x) =∞. 5) If lim x→0 f(x) = lim x→0 g(x) = 0 and lim x→0 f ′(x) g′(x) does not exist, then lim x→0 f(x) g(x) also does not exist. 2. Area (1) Sketch the region in the xy-plane enclosed by the curves y = 2x3 − x2 − 5x and y = −x2 + 3x, and then find its area. (2) Sketch the region in the xy-plane defined by the inequalities x − 2y2 ≥ 0 and 1− x− |y| ≥ 0, and then find its area. 3. Volume (1) A solid lies between the planes perpendicular to the x-axis at x = −1 and x = 1. The cross-sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x2 to the parabola y = 2− x2. Find its volume. (2) The base of a solid is the region bounded by the curve y = 2 √ sinx and the x-axis on the interval [0, π]. Find its volume given that the cross-sections perpendicular to the x-axis are (a) equilateral triangles and (b) squares with bases running from the x-axis to the curve. (3) Find the volume of a tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3, 4 and 5 meters. (4) The disk x2 + y2 ≤ r2 is rotated about the line x = R, where R > r, to generate a solid shaped like a doughnut, called a torus. Find its volume. 2 PROBLEM LIST IV (5) Find the volume of the solid obtained by rotating the region bounded by the curves y = x sinx and y = 0 on the interval [0, π], about the line x = π. (6) For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. 1) Of two solids of revolution, the one with the greater volume is obtained by revolving the region in the plane with the greater area. 2) A cube is a solid of revolution. 4. Work (1) A 5 lb bucket is attached to a rope, which weighs 1.6 lb and is 20 ft long, hanging from the top of a tall building. (a) How much work is required to pull up the bucket and the rope to the top of the building, at a constant speed? (b) How much work is required to pull up the bucket and the rope only 10 ft, at a constant speed? (2) A spring at rest is 6 ft long. Suppose the force required to stretch the spring to 9 ft in length is 4 lb. How much work, measured in ft-lb, is needed to stretch the spring from a length of 9 ft to a length of 13 ft? (3) Find the work required to pump the water out of a tank of height 4 m having the shape of a paraboloid of revolution obtained by revolving the parabola 4y = x2 about the y-axis. Use the fact that the density of water is 1000 kg/m3. (4) For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. 1) It takes more work to lift a 50 lb weight 10 ft slowly than to lift it the same distance quickly. 2) A cable hangs vertically from a winch at the top of a tall building. The work required for the winch to pull up the top half of the cable is half of the work required to pull up the entire cable. 5. Average Vaule of a function (1) For each of the following functions, find its average value on the interval [ 0, π 4 ] . (a) f(x) = 1− 2 cos2(x) (b) g(x) = x sec2(x) (2) For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. 1) If f is an integrable function defined on [0, π] such that ∫ π 0 f(t)dt = π, then there exists a number 0 < c="">< π such that f(c) = 1. 2) if a continuous variable force of average value 10 n on the interval 1 ≤ x ≤ 3 meters is applied to move an object along the x-axis from x = 1 to x = 3 meters, then the work done by the force in moving the object is 20 j. 6. techniques of integration (1) evaluate the following integrals. (a) ∫ 1 0 √ x− x2 dx (b) ∫ √ x2 + 2x dx (c) ∫ x √ 1− x4 dx (d) ∫ √ 3− 2x− x2 dx problem list iv 3 (2) if f is a quadratic function such that f(0) = 1 and ∫ f(x) x2(x+ 1)3 dx is a rational function, then what is the value of f ′(0)? (3) for each of the following statements, determine whether it is true or false. if it is true, prove it. if it is false, give a counter example to disprove it. 1) if two elementary functions don’t have elementary antiderivatives, then so does their sum. 2) if two elementary functions don’t have elementary antiderivatives, then so does their product. 3) if two elementary functions have elementary antiderivatives, then so does their quotient. 7. approximate integration (1) approximate ln 2 using the trapezoidal rule with n = 3. (2) suppose we want to approximate the integral i = ∫ b a f(x)dx, where f is a positive function defined on [a, b], by partitioning [a, b] into n subintervals of equal width ∆x. for each of the following statements, determine whether it is true or false. if it is true, prove it. if it is false, give a counter example to disprove it. 1) if f is concave down on [a, b] then the trapezoidal rule provides an underes- timate of i. 2) if f is concave down on [a, b] then the midpoint rule provides an underestimate of i. 3) if f is concave up on [a, b] then the trapezoidal rule provides an overestimate of i. 4) if f is concave up on [a, b] then the midpoint rule provides an overestimate of i. 5) in order to apply the simpson’s rule n must be an even number. 6) if f is a polynomial of degree 3 or lower, then simpson’s rule provides the exact value of i. 8. improper integrals (1) for each of the following integrals, find its value or show that it is divergent. (a) ∫ ∞ 2 dx x2 + 2x− 3 (b) ∫ ∞ 1 dx√ x+ x √ x (c) ∫ ∞ 1 tan−1(x) x2 dx (d) ∫ ∞ −∞ dx 4x2 + 4x+ 5 (e) ∫ 4 0 dx x2 − x− 2 (f) ∫ 4 0 ln(x)√ x dx (g) ∫ 3 0 x (x2 − 1)2/3 dx (h) ∫ 1 0 x lnxdx (2) for each of the following integrals, determine whether it converges or diverges. (a) ∫ ∞ −∞ dx (x2 + 1)2 (b) ∫ ∞ 1 xe−xdx (c) ∫ 2 0 dx (2− x)2 (d) ∫ ∞ 1 1 + sin2 x√ x dx (e) ∫ ∞ 0 sin2 x x2 dx (f) ∫ ∞ −∞ sin2 x 1 + x2 dx 4 problem list iv (3) for each of the following statements, determine whether it is true or false. if it is true, prove it. if it is false, give a counter example to disprove it. 1) if f is a continuous, decreasing function on [1,∞) such that lim x→∞ f(x) = 0, then the improper integral ∫ ∞ 1 f(x) dx converges. 2) if f is a continuous function on [1,∞) such that the improper integral ∫ ∞ 1 f(x) dx diverges, then lim x→∞ f(x) 6= 0. 3) if f is a continuous function on [1,∞) such that lim x→∞ f(x) = 1, then the improper integral ∫ ∞ 1 f(x) dx diverges. 4) if f is a continuous function on [0,∞) such that the improper integral ∫ ∞ 1 f(x) dx converges, then the improper integral ∫ ∞ 0 f(x) dx converges as well. 5) if f is a continuous function on (0,∞) such that the improper integral ∫ ∞ 1 f(x) dx converges, then the improper integral ∫ ∞ 0 f(x) dx converges as well. 6) if f is a continuous function such that lim t→∞ ∫ t −t f(x) dx exists, then the im- proper integral ∫ ∞ −∞ f(x) dx converges. 7) if f and g are continuous functions on [1,∞) such that both ∫ ∞ 1 f(x) dx and∫ ∞ 1 g(x) dx diverge, then so does ∫ ∞ 1 [f(x) + g(x)] dx. 8) if f and g are positive and continuous functions on [1,∞) such that both∫ ∞ 1 f(x) dx and ∫ ∞ 1 g(x) dx diverge, then so does ∫ ∞ 1 f(x)g(x) dx. 9) if f and g are positive and continuous functions on [1,∞) such that both∫ ∞ 1 f(x) dx and ∫ ∞ 1 g(x) dx converge, then so does ∫ ∞ 1 f(x) g(x) dx. 10) if ∫ ∞ 1 f(x) dx converges then so does ∫ ∞ 1 g(x) dx, where f and g are contin- uous functions such that g(x) ≤ f(x) for all x in the interval [1 π="" such="" that="" f(c)="1." 2)="" if="" a="" continuous="" variable="" force="" of="" average="" value="" 10="" n="" on="" the="" interval="" 1="" ≤="" x="" ≤="" 3="" meters="" is="" applied="" to="" move="" an="" object="" along="" the="" x-axis="" from="" x="1" to="" x="3" meters,="" then="" the="" work="" done="" by="" the="" force="" in="" moving="" the="" object="" is="" 20="" j.="" 6.="" techniques="" of="" integration="" (1)="" evaluate="" the="" following="" integrals.="" (a)="" ∫="" 1="" 0="" √="" x−="" x2="" dx="" (b)="" ∫="" √="" x2="" +="" 2x="" dx="" (c)="" ∫="" x="" √="" 1−="" x4="" dx="" (d)="" ∫="" √="" 3−="" 2x−="" x2="" dx="" problem="" list="" iv="" 3="" (2)="" if="" f="" is="" a="" quadratic="" function="" such="" that="" f(0)="1" and="" ∫="" f(x)="" x2(x+="" 1)3="" dx="" is="" a="" rational="" function,="" then="" what="" is="" the="" value="" of="" f="" ′(0)?="" (3)="" for="" each="" of="" the="" following="" statements,="" determine="" whether="" it="" is="" true="" or="" false.="" if="" it="" is="" true,="" prove="" it.="" if="" it="" is="" false,="" give="" a="" counter="" example="" to="" disprove="" it.="" 1)="" if="" two="" elementary="" functions="" don’t="" have="" elementary="" antiderivatives,="" then="" so="" does="" their="" sum.="" 2)="" if="" two="" elementary="" functions="" don’t="" have="" elementary="" antiderivatives,="" then="" so="" does="" their="" product.="" 3)="" if="" two="" elementary="" functions="" have="" elementary="" antiderivatives,="" then="" so="" does="" their="" quotient.="" 7.="" approximate="" integration="" (1)="" approximate="" ln="" 2="" using="" the="" trapezoidal="" rule="" with="" n="3." (2)="" suppose="" we="" want="" to="" approximate="" the="" integral="" i="∫" b="" a="" f(x)dx,="" where="" f="" is="" a="" positive="" function="" defined="" on="" [a,="" b],="" by="" partitioning="" [a,="" b]="" into="" n="" subintervals="" of="" equal="" width="" ∆x.="" for="" each="" of="" the="" following="" statements,="" determine="" whether="" it="" is="" true="" or="" false.="" if="" it="" is="" true,="" prove="" it.="" if="" it="" is="" false,="" give="" a="" counter="" example="" to="" disprove="" it.="" 1)="" if="" f="" is="" concave="" down="" on="" [a,="" b]="" then="" the="" trapezoidal="" rule="" provides="" an="" underes-="" timate="" of="" i.="" 2)="" if="" f="" is="" concave="" down="" on="" [a,="" b]="" then="" the="" midpoint="" rule="" provides="" an="" underestimate="" of="" i.="" 3)="" if="" f="" is="" concave="" up="" on="" [a,="" b]="" then="" the="" trapezoidal="" rule="" provides="" an="" overestimate="" of="" i.="" 4)="" if="" f="" is="" concave="" up="" on="" [a,="" b]="" then="" the="" midpoint="" rule="" provides="" an="" overestimate="" of="" i.="" 5)="" in="" order="" to="" apply="" the="" simpson’s="" rule="" n="" must="" be="" an="" even="" number.="" 6)="" if="" f="" is="" a="" polynomial="" of="" degree="" 3="" or="" lower,="" then="" simpson’s="" rule="" provides="" the="" exact="" value="" of="" i.="" 8.="" improper="" integrals="" (1)="" for="" each="" of="" the="" following="" integrals,="" find="" its="" value="" or="" show="" that="" it="" is="" divergent.="" (a)="" ∫="" ∞="" 2="" dx="" x2="" +="" 2x−="" 3="" (b)="" ∫="" ∞="" 1="" dx√="" x+="" x="" √="" x="" (c)="" ∫="" ∞="" 1="" tan−1(x)="" x2="" dx="" (d)="" ∫="" ∞="" −∞="" dx="" 4x2="" +="" 4x+="" 5="" (e)="" ∫="" 4="" 0="" dx="" x2="" −="" x−="" 2="" (f)="" ∫="" 4="" 0="" ln(x)√="" x="" dx="" (g)="" ∫="" 3="" 0="" x="" (x2="" −="" 1)2/3="" dx="" (h)="" ∫="" 1="" 0="" x="" lnxdx="" (2)="" for="" each="" of="" the="" following="" integrals,="" determine="" whether="" it="" converges="" or="" diverges.="" (a)="" ∫="" ∞="" −∞="" dx="" (x2="" +="" 1)2="" (b)="" ∫="" ∞="" 1="" xe−xdx="" (c)="" ∫="" 2="" 0="" dx="" (2−="" x)2="" (d)="" ∫="" ∞="" 1="" 1="" +="" sin2="" x√="" x="" dx="" (e)="" ∫="" ∞="" 0="" sin2="" x="" x2="" dx="" (f)="" ∫="" ∞="" −∞="" sin2="" x="" 1="" +="" x2="" dx="" 4="" problem="" list="" iv="" (3)="" for="" each="" of="" the="" following="" statements,="" determine="" whether="" it="" is="" true="" or="" false.="" if="" it="" is="" true,="" prove="" it.="" if="" it="" is="" false,="" give="" a="" counter="" example="" to="" disprove="" it.="" 1)="" if="" f="" is="" a="" continuous,="" decreasing="" function="" on="" [1,∞)="" such="" that="" lim="" x→∞="" f(x)="0," then="" the="" improper="" integral="" ∫="" ∞="" 1="" f(x)="" dx="" converges.="" 2)="" if="" f="" is="" a="" continuous="" function="" on="" [1,∞)="" such="" that="" the="" improper="" integral="" ∫="" ∞="" 1="" f(x)="" dx="" diverges,="" then="" lim="" x→∞="" f(x)="" 6="0." 3)="" if="" f="" is="" a="" continuous="" function="" on="" [1,∞)="" such="" that="" lim="" x→∞="" f(x)="1," then="" the="" improper="" integral="" ∫="" ∞="" 1="" f(x)="" dx="" diverges.="" 4)="" if="" f="" is="" a="" continuous="" function="" on="" [0,∞)="" such="" that="" the="" improper="" integral="" ∫="" ∞="" 1="" f(x)="" dx="" converges,="" then="" the="" improper="" integral="" ∫="" ∞="" 0="" f(x)="" dx="" converges="" as="" well.="" 5)="" if="" f="" is="" a="" continuous="" function="" on="" (0,∞)="" such="" that="" the="" improper="" integral="" ∫="" ∞="" 1="" f(x)="" dx="" converges,="" then="" the="" improper="" integral="" ∫="" ∞="" 0="" f(x)="" dx="" converges="" as="" well.="" 6)="" if="" f="" is="" a="" continuous="" function="" such="" that="" lim="" t→∞="" ∫="" t="" −t="" f(x)="" dx="" exists,="" then="" the="" im-="" proper="" integral="" ∫="" ∞="" −∞="" f(x)="" dx="" converges.="" 7)="" if="" f="" and="" g="" are="" continuous="" functions="" on="" [1,∞)="" such="" that="" both="" ∫="" ∞="" 1="" f(x)="" dx="" and∫="" ∞="" 1="" g(x)="" dx="" diverge,="" then="" so="" does="" ∫="" ∞="" 1="" [f(x)="" +="" g(x)]="" dx.="" 8)="" if="" f="" and="" g="" are="" positive="" and="" continuous="" functions="" on="" [1,∞)="" such="" that="" both∫="" ∞="" 1="" f(x)="" dx="" and="" ∫="" ∞="" 1="" g(x)="" dx="" diverge,="" then="" so="" does="" ∫="" ∞="" 1="" f(x)g(x)="" dx.="" 9)="" if="" f="" and="" g="" are="" positive="" and="" continuous="" functions="" on="" [1,∞)="" such="" that="" both∫="" ∞="" 1="" f(x)="" dx="" and="" ∫="" ∞="" 1="" g(x)="" dx="" converge,="" then="" so="" does="" ∫="" ∞="" 1="" f(x)="" g(x)="" dx.="" 10)="" if="" ∫="" ∞="" 1="" f(x)="" dx="" converges="" then="" so="" does="" ∫="" ∞="" 1="" g(x)="" dx,="" where="" f="" and="" g="" are="" contin-="" uous="" functions="" such="" that="" g(x)="" ≤="" f(x)="" for="" all="" x="" in="" the="" interval="">