MA 114 Worksheet Calendar – Fall 2016

MA 114 Worksheet Calendar – Fall 2016Thur, Aug 25: Worksheet01 – Review substitution, area between curvesTues, Aug 28: Worksheet02 – Integration by partsThur, Sep 01: Worksheet03 – Partial fractionsTues, Sep 06: Worksheet04 – Special trig integralsThur, Sep 08: Worksheet05 – Numerical integration – trapezoid, midpointTues, Sep 13: Worksheet06 – Simpson’s Rule, Improper integralsThur, Sep 15: Worksheet07 – SequencesTues, Sep 20: Worksheet08 – Review for Exam 01Thur, Sep 22: Worksheet09 – Recursive sequences, seriesTues, Sep 27: Worksheet10 – Series, Integral TestThur, Sep 29: Worksheet11 – Comparison and Limit Comparison TestsTues, Oct 04: Worksheet12 – Alternating series, absolute & conditional convergenceThur, Oct 06: Worksheet13 – Ratio & root testsTues, Oct 11: Worksheet14 – Power seriesThur, Oct 13: Worksheet15 – Taylor seriesTues, Oct 18: Worksheet16 – Review for Exam 02Thur, Oct 20: Worksheet17 – Average value of a functionTues, Oct 25: Worksheet18 – Volumes – known cross-section and washersThur, Oct 27: Worksheet19 – Volumes by washers and shellsTues, Nov 1: Worksheet20 – Arc length & surface areaThur, Nov 3: Worksheet21 – Centers of mass & momentsTues, Nov 8: Worksheet22 – Parametric equationsThur, Nov 10: Worksheet23 – Calculus with parametric equationsTues, Nov 15: Worksheet24 – Review for Exam 03Thur, Nov 17: Worksheet25 – Polar coordinatesTues, Nov 22: Worksheet26 – Calculus with polar coordinatesTues, Nov 29: Worksheet27 – Differential equations, direction fieldsThur, Dec 01: Worksheet28 – Separable equationsTues, Dec 06: Worksheet29 – Conic sectionsThur, Dec 08: Worksheet30 – Review for Final Exam

Fall 2016MA 114 Worksheet 01Thur, Aug 25, 2016MA 114 Worksheet #01: Substitution Review1. Evaluate the following indefinite integrals and indicate the substitutions that you use.ZZ4(a)(d)sec3 x tan x dxdx(1 2x)3Z (b)x2 x3 1(e) ex sin(ex )Z2x 3(c)cos4 x sin x dx(f) 2x 3x2. Evaluate the following definite integrals and indicate the substitutions that you use.Z 7Z π/3 4 3x dx(a)x4 sin x dx(d)0Z π/3π2(b)Zcos(x) cos(sin(x)) dx0Z(c)01ez 1dzez z4x dx1 2x(e)03Zdx6x 13. If f is continuous and(f)0R6f (x) 8, findR2f (3x).R 25R54. If f is continuous and 0 f (x)dx 16, find 0 xf (x2 ) dx.005. Find the area of the region between the graphs of y x2 and y x4 . 6. Find the area of the regions enclosed by the graphs of y x and y 14 x ways.34in two(a) Write this as an integral in x.(b) Solve each equation to express x in terms of y and write an integral with respectto y.7. Find the area of the region enclosed by the graphs of y x 1 and y x3 x2 x 1.18. What is the area of the region bounded by f (x) , x e2 , x e8 and x-axis?xSketching the region might be helpful.9. If f is continuous on [0, 1], show thatZ 1Zf (x)dx 01f (1 x)dx.010. Find the area of the region bounded by the parabola y x2 , the tangent line to theparabola at (1, 1) and the x-axis.

Fall 2016MA 114 Worksheet 02Tues, Aug 30, 2016MA 114 Worksheet #02: Integration by parts1. Which of the following integrals should be solved using substitution and which shouldbe solved using integration by parts?ZZln (arctan(x))2(a)x cos(x ) dx,(c)dx,1 x2ZZ2x(b)e sin(x) dx,(d)xex dx2. Solve the following integrals using integration by parts:ZZ2(a)x sin(x) dx,(f)x4 ln(x) dxZZx(b) (2x 1)e dx,(g)ex sin x dxZZ(c)x sin (3 x) dx,(h)x ln(1 x) dxHint: Make aZsubstitution first, then try integration(d)2x arctan(x) dx,by parts.Z(e)ln(x) dx3. Let f (x) be a twiceZ differentiable function with f (1) 2, f (4) 7, f 0 (1) 5 and4f 0 (4) 3. Evaluatexf 00 (x) dx14. If f (0) g(0) 0 and f 00 and g 00 are continuous, show thatZ aZ a0000f (x)g (x) dx f (a)g (a) f (a)g(a) f 00 (x)g(x) dx.00

Fall 2016MA 114 Worksheet 03Thur, Sept 1, 2016MA 114 Worksheet #03: Integration by Partial Fractions1. Write out the general form for the partial fraction decomposition but do not determinethe numerical value of the coefficients.1(a) 2x 3x 2x 1(b) 2x 4x 4x(c)2(x 1)(x 1)(x 2)2x 5(d)2(x 1)3 (2x 1)2. Compute the following integrals.Zx 9(a)dx(x 5)(x 2)Z1(b)dx2x 3x 2Z 3x 2x2 1(c)dxx3 2x2Z 3x 4(d)dxx2 4Z1(e)dx2x(x 1)3. ComputeZ1 dx x 3x by first making the substitution u 6 x.

Fall 2016MA 114 Worksheet 04Tues, Sept 6, 2016MA 114 Worksheet #04: Special Trig Integrals1. Compute the following integrals:Z 2u3 (a)du16 u20Z1 (b)dxx2 25 x2Z π/2(c)cos2 (x) dxZ0 (d)cos x sin3 x dxZ 2π (e)sin2 31 θ dθ0Zπ/2(2 sin θ)2 dθ(f)0Z (g)Z1 x2dxx43x dx. Hint: Use the sub36 x20stitution x 6u.Z 1/2 (i)x 1 4x2 dx. Hint: Substitute(h)0x u/2.2. Let r 0. Consider the identityZ s 1 1r2 x2 dx r2 arcsin (s/r) s r2 s2220where 0 s r.x 2r x2 .s(b) Using part (a), verify the identity geometrically.(a) Plot the curves y r2 x2 , x s, and y (c) Verify the identity using trigonometric substitution.

Fall 2016MA 114 Worksheet 05Thur, Sept 8, 2016MA 114 Worksheet #05: Numerical Integration1. (a) Write down the Midpoint rule and illustrate how it works with a sketch.(b) Write down the Trapezoid rule and the error bound associated with it.(c) How large should n be in the Midpoint rule so that you can approximateZ 1sin x dx0with an error less than 10 7 ?Z12. Use the Midpoint rule to approximate the value of2e x dx with n 4. Draw a 1sketch to determine if the approximation is an overestimate or an underestimate of theintegral.3. RThe left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate2f (x) dx, where f is the function whose graph is shown. The estimates were 0.7811,00.8675, 0.8632, and 0.9540, and the same number of sub- intervals were used in eachcase.(a) Which rule produced which estimate?(b) Between which two approximations does the true value of1R20f (x) dx lie?0.80.60.40.2004. Draw the graph of f (x) sinR1I 0 f (x) dx.0.51 2x21 1.52in the viewing rectangle [0, 1] by [0, 0.5] and let(a) Use the graph to decide whether L2 , R2 , M2 , and T2 underestimate or overestimateI.(b) For any value of n, list the numbers Ln , Rn , Mn , Tn , and I in increasing order.(c) Compute L5 , R5 , M5 , and T5 . From the graph, which do you think gives the bestestimate of I?

Fall 2016MA 114 Worksheet 05Thur, Sept 8, 20165. The velocity in meters per second for a particle traveling along the axis is given in thetable below. Use the Midpoint rule and Trapezoid rule to approximate the total distancethe particle traveled from t 0 to t 6.t0123456v(t)0.751.341.51.92.53.23.0Page 2 of 2

Fall 2016MA 114 Worksheet 06Tues, Sept 13, 2016MA 114 Worksheet #06: Simpson’s Rule & ImproperIntegrals1. (a) Write down Simpson’s rule and illustrate how it works with a sketch.(b) How large should n be in the Simpson’s rule so that you can approximateZ 1sin x dx0 7with an error less than 10 ?Z 212. Approximate the integraldx using Simpson’s rule. Choose n so that your error is1 xcertain to be less than 10 3 . Compute the exact value of the integral and compare toyour approximation.3. Simpson’s Rule turns out to exactly integrateZpolynomials of degree three or less. Showhthat Simpson’s rule gives the exact value ofp(x) dx where h 0 and p(x) ax3 0bx2 cx d. [Hint: First compute the exact value of the integral by direct integration.Then apply Simpson’s rule with n 2 and observe that the approximation and the exactvalue are the same.]4. For each of the following, determine if the integral is proper or improper. If it is improper,explain why. Do not evaluate any of the integrals.Z Z 2sin xx(d)dxdx(a)22 1 x0 x 5x 6Z 2Z π/21(b)dx(e)sec x dx1 2x 10Z 2(c)ln (x 1) dx15. For the integrals below, determine if the integral is convergent or divergent. Evaluatethe convergent integrals.Z 0Z 2x 31(a)dx(c)dx0 2x 3 2x 1Z Z x2(b)xedx(d)sin θ dθ 06. Consider the improper integralZ 1dx.xp1Integrate using the generic parameter p to prove the integral converges for p 1 anddiverges for p 1. You will have to distinguish between the cases when p 1 and p 6 1when you integrate.

MA 114Worksheet 06Fall 20167. Use the Comparison Theorem to determine whether the following integrals are convergent or divergent.Z 2 e x(a)dxx1Z x 1 dx(b)x6 x18. Explain why the following computation is wrong and determine the correct answer. (Trysketching or graphing the integrand to see where the problem lies.)Z 10Z11 12 1dx du2x 82 4 u2121ln x 2 41 (ln 12 ln 4)2 where we used the substitution u(x) 2x 8u(2) 4u(10) 12du 2dx9. A manufacturer of light bulbs wants to produce bulbs that last about 700 hours but, ofcourse, some bulbs burn out faster than others. Let F (t) be the fraction of the companysbulbs that burn out before t hours, so F (t) always lies between 0 and 1.(a) Make a rough sketch of what you think the graph of F might look like.(b) What is the meaning of the derivative r(t) F 0 (t)?R (c) What is the value of 0 r(t) dt? Why?Page 2 of ?

Fall 2016MA 114 Worksheet 07Thur, Sept 15, 2016MA 114 Worksheet #07: Sequences1. (a) Give the precise definition of a sequence.(b) What does it mean to say that lim f (x) L when a ? Does this differ fromx alim f (n) L? Why or why not?n (c) What does it means for a sequence to converge? Explain your idea, not just thedefinition in the book.(d) Sequences can diverge in different ways. Describe two distinct ways that a sequencecan diverge.(e) Give two examples of sequences which converge to 0 and two examples of sequenceswhich converges to a given number L 6 0.2. Write the first four terms of the sequences with the following general terms:n!2nn(b)n 1(c) ( 1)n 1(a)3.nwhere an 2 n 2.(d) {an } n 1 where an (e) {an } n 1(f) {bk } k 1 where bk ( 1)k ).k23. Find a formula for the nth term of each sequence. 11 1 1, , , ,.(a)1 8 27 64 1 1 1 1(b) 1, , , , , . . .2 4 8 16(c) {1, 0, 1, 0, 1, 0, . . .} (d) 21 , 32 , 34 , 45 , 65 , . . . ,4. Suppose that a sequence {an } is bounded above and below. Does it converge? If not,find a counterexample.5. The limit laws for sequences are the same as the limit laws for functions. Supposeyou have sequences {an }, {bn } and {cn } with limn an 15, limn bn 0 andlimn cn 1. Use the limit laws of sequences to answer the following questions. an · c n(a) Does the sequenceconverge? If so, what is its limit?bn 1 n 1 an 3 · c n(b) Does the sequenceconverge? If so, what is its limit?2 · bn 2 n 1

Fall 2016MA 114 Worksheet 09Thur, Sept 22, 2016MA 114 Worksheet #09: Recursive sequences & Series1. Write out the first five terms of Let(a) a0 0, a1 1 and an 1 3an 1 an 2 .an(b) a1 6, an 1 .nan(c) a1 2, an 1 .an 1s 22(d) a1 1, an 1 1.an(e) a1 2, a2 1, and an 1 an an 1 .2. (a) For what values of x does the sequence {xn } n 1 converge?(b) For what values of x does the sequence {nx } n 1 converge? (c) If lim bn 2, find lim bn 3 .n n 3. (a) Determine whether the sequence defined as follows is convergent or divergent:a1 1an 1 4 anfor n 1.(b) What happens if the first term is a1 2?4. A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% permonth and the farmer harvests 300 catfish per month.(a) Show that the catfish population Pn after n months is given recursively byPn 1.08Pn 1 300P0 5000.(b) How many catfish are in the pond after six months?