Friday 09/15/2017 Midterm I 50 minutes

Size: px
Start display at page:

Download "Friday 09/15/2017 Midterm I 50 minutes"

Transcription

1 Fa 17: MATH Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets. Extra paper is available if you need it. 3. Show all the steps of your work clearly. Question Points Your Score Q1 25 Q2 25 Q3 25 Q4 25 TOTAL 100

2 Miscellaneous expressions and definitions. 1. Inverse functions. g = f 1 means g(f(x)) = x for all x in domain(f) and f(g(x)) = x for all x in domain(g). If g = f 1 and f is differentiable, then so is g and g (x) = 1 f (g(x)). 2. Log and Exp. ln(x) = x 1 dt t e x is the inverse function of ln(x) If dp dt = kp, then P = P 0 e kt. 3. Inverse trig. d dx sin 1 (x) = 1 1 x 2 d dx cos 1 (x) = 1 1 x 2 d dx tan 1 (x) = 1 1+x 2 4. Hyperbolic trig. cosh(x) = ex +e x 2 sinh(x) = ex e x 2 tanh(x) = sinh(x) cosh(x) sech(x) = 1 cosh(x) etc. cosh 2 (x) sinh 2 (x) = 1 d cosh(x) dx = sinh(x) d sinh(x) dx = cosh(x) d tanh(x) dx = sech 2 (x) 5. Inverse hyperbolic trig. sinh 1 (x) = ln(x + x 2 + 1) cosh 1 (x) = ln(x + x 2 1), x 1 tanh 1 (x) = 1 2 ln ( 1+x 1 x), x < 1 d sinh 1 (x) dx = 1 1+x 2 d cosh 1 (x) dx = 1 x 2 1, x 1 d tanh 1 (x) dx = 1 1 x 2, x < 1

3 Q1]... [25 points] 1. Compute the derivative of the function y = ( x) x + x x 2. Compute the derivative of the function y = log 7 (x 2 ).

4 Q2]... [25 points] 1. Using other properties of the hyperbolic trigonometric functions, show that the following is true: d tanh 1 x dx = 1 1 x 2 2. Check that the function y = e t cos(2t) is a solution to the differential equation d 2 y dt 2 + 2dy dt + 5y = 0

5 Q3]... [25 points] The radioactive material, Calctwoium, has a half-life of 100 years. Your answers to the questions below will be numbers; it is OK to describe these numbers as expressions involving other numbers. Since you do not have a calculator, I am not expecting you to give answers as explicit numbers with many decimal places accuracy. 1. What percentage of the original sample of Calctwoium is left after 50 years? 2. How long does it take for 60% of a sample of Calctwoium to decay?

6 Q4]... [25 points] Compute the following integrals. 1. e 7 e 3 dx x ln x 2. e x dx 1 + e 2x

7 Fa 17: MATH Differential and Integral Calculus II Noel Brady Friday 10/20/2017 Midterm II 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets. Extra paper is available if you need it. 3. Show all the steps of your work clearly. Question Points Your Score Q1 26 Q2 26 Q3 22 Q4 26 TOTAL 100

8 1. Trig Addition, Half Angle. Miscellaneous expressions and definitions. cos(a ± B) = cos(a) cos(b) sin(a) sin(b) cos(2a) = cos 2 (A) sin 2 (A) = 2 cos 2 (A) 1 = 1 2 sin 2 (A) sin 2 (x) = (1 cos(2x))/2 cos 2 (x) = (1 + cos(2x))/2 sin(a ± B) = sin(a) cos(b) ± cos(a) sin(b) sin(2x) = 2 sin(x) cos(x). 2. Hyperbolic. sinh(x) = 1 2 (ex e x ) cosh(x) = 1 2 (ex + e x ) 3. Integration by Parts. u dv = uv v du 4. Inverse Trig. d dx sin 1 (x) = 1 1 x 2 d dx tan 1 (x) = 1 1+x 2 dx = 1 x 2 +a 2 a tan 1 ( x) a 5. Trig Substitutions. For a 2 x 2 use x = a sin(θ) For a 2 + x 2 use x = a tan(θ) For x 2 a 2 use x = a sec(θ) 6. Some integrals. tan(x) dx = ln sec(x) + C 7. Jon McCammond Method (for integrals in tan(x) and sec(x)). Let u = sec(x) + tan(x) and v = sec(x) tan(x). Note that and that 8. Arc length. uv = 1 ; u + v 2 du u = sec(x) ; = sec(x)dx = dv v u v 2 = tan(x) 9. Surface area of revolution. A = 2π b L = b x 1 + (y ) 2 dx (about y-axis) a 1 + (y ) 2 dx A = 2π b a a y 1 + (y ) 2 dx (about x-axis) 10. Geometric Series. n i=1 ar i 1 = a(1 rn ) 1 r

9 Q1]... [26 points] 1. Evaluate the following limit. Show the steps of your work. lim x)x x 0 +(tan 2. Compute the arc length of the portion of the graph of y = cosh x which lies over the interval [0, a] in the x-axis. Your answer will be a function of a.

10 Q2]... [26 points] Evaluate the following two integrals. 1. x ln(x)dx 2. cos 5 (x)dx

11 Q3]... [22 points] Use a substitution to evaluate the following integral. Show all the steps of your work. dx (x 2 + 9) 2

12 Q4]... [26 points] 1. Compute the following integral. dx (x 1)(x 2) 2. Say whether the following series is convergent or not. If the series is convergent, find its sum. n=1 4 n+1 5 n

13 Fa 17: MATH Differential and Integral Calculus II Noel Brady Monday 11/20/2017 Midterm III 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets. Extra paper is available if you need it. 3. Show all the steps of your work clearly. Question Points Your Score Q1 30 Q2 20 Q3 25 Q4 25 TOTAL 100

14 Series; Parametric Curves; Polar Coordinates. 1. Geometric Series. n=1 arn 1 converges when r < 1; it converges to the sum 2. Test for Divergence. If n=1 a n converges, then lim n a n = 0. a 1 r when r < Integral Test. For f(x) continuous on [1, ), positive and decreasing to 0, the series n=1 f(n) converges if and only if the improper integral f(x)dx converges Comparison Tests. Direct comparison test: compares series of positive terms, term-by-term. Limit comparison test: compares series of positive terms n=1 a n and n=1 b a n when lim n n bn a finite limit not equal to Root Test. Let lim n a n 1/n = L. If L < 1 then n=1 a n is absolutely convergent, and if L > 1 then it is divergent. 6. Ratio Test. a Let lim n+1 n a n = L. If L < 1 then n=1 a n is absolutely convergent, and if L > 1 then it is divergent. 7. Alternating Series Test. If a n are positive, decreasing to 0, then n=1 ( 1)n 1 a n is convergent. Moreover, the nth partial sum is within a n+1 of the sum of the whole series. 8. Power series. Ratio test is useful for computing the radius of convergence of a power series n=0 c n(x a) n. 9. Taylor and Maclaurin Series. Taylor series for f(x) centered about a is given by f (n) (a) (x a) n n! Maclaurin series for f(x) is the Taylor series for f(x) centered about Remainder Estimate. Taylor s inequality states that if f (n+1) (x) M on the interval [a d, a + d], then n=0 f(x) T n (x) M x a (n+1) (n + 1)! on the interval [a d, a + d]. Here T n (x) is the degree n Taylor polynomial approximation to f(x). = L 11. Parametric curves. Arc length = b a (x (t)) 2 + (y (t)) 2 dt Slope dy = y (t) dx x (t) 12. Polar Coords. x = r cos θ y = r sin θ x 2 + y 2 = r 2 tan θ = y/x Arc length = b r a 2 + ( dr dθ )2 dθ Polar area = b r 2 dθ a 2

15 Q1]... [30 points] Test the following series for convergence or divergence. Show all the steps of your work. ( 1 n + 1 ) 5 5 n n=1 n=1 n! e n2 where e n2 means e (n2 )

16 Q2]... [20 points] Determine whether the following series converges absolutely, converges conditionally, or diverges. Show all the steps of your work. ( ) 1 ( 1) n sin n + 1 n=0

17 Q3]... [25 points] Find the Taylor series for f(x) = sin x centered about π/4. Show all the steps of your work. Does your answer make sense? Hint: angle addition formulae.

18 Q4]... [25 points] Find the formula for the slope dy of the tangent line to the hypocycloid curve as a dx function of the parameter t. The curve is given by the parametric equations x = cos 3 (t) y = sin 3 (t) 1 t 2π Find the formula for the second derivative d2 y dx 2 of the parametric curve above. Your answer will be a function of t. Draw a sketch of the curve, by first answering the following questions. Why does this curve fits inside of the unit circle? What are the points on the curve corresponding to t = 0, π/2, π, 3π/2? What is the slope of the curve at each of these points? What is the concavity of the curve for t in [0, π/2], [π/2, π], [π, 3π/2], and [3π/2, 2π]?

19 Math 2924 Test 1 Sept. 22, 2017 Name: Instructions: Justify your work, communicate clearly and mathematically. Calculators are not allowed. Label your problems clearly and mark the work you want graded/not graded. All supporting work should be done on extra pages and stapled to this page - do NOT write on this page. A1. Compute the following: d a. dx arctan(x2 ) 1 b. dx 1 2x 2 A2. Differentiate the function ln(x x + 1). A3. Find the area bounded by the curves y = e x and y = e x. A4. Using techniques from calculus, sketch the graph y = x ln(x). Solve 1 of the following 2 problems - clearly mark which one you want graded. B1. Compute the limit as x goes to infinity of sin( 1 x ) x 2 arctan( 1 x ). B2. Consider the equation d dy f 1 = 1 y=f(a) f (a) Explain when this is valid and give a proof.

20 Math 2924 Test 2 October 20, 2017 Name: Instructions: Justify your work, communicate clearly and mathematically. Calculators are not allowed. Label your problems clearly and mark the work you want graded/not graded. All supporting work should be done on extra pages and stapled to this page - do NOT write answers or work on this page. Some identities: cos(a + b) = cos(a) cos(b) sin(a) sin(b) sin(a + b) = cos(a) sin(b) + sin(a) cos(b). A1. Compute the integral 3/2 0 arcsin(x) dx. A2. Compute cos(4x) cos(3x)dx. 1 x 2 A3. Consider the function over the interval [ 1, 1]. Give a (very rough) sketch of 1 + x the graph of this function and integrate over the given interval. A4. Compute the integral cos(ln x)dx. Solve 1 of the following 2 problems - clearly mark which one you want graded. B1. State and prove the formula for integration by parts. (State the version with definite bounds on the integral, i.e. integrating over an interval [a, b].) B2. Integrate the function 1 x 3/5 1.

21 Math 2924 Test 3 Take-home exam November 20, 2017 Name: Instructions: Justify your work, communicate clearly and mathematically. Calculators are not allowed. Label your problems clearly and mark the work you want graded/not graded. All supporting work should be done on extra pages and stapled to this page - do NOT write answers or work on this page. No book, notes, etc. Turn your exam in on Gradescope by Sunday, 8pm. A1. Determine which of the following converge. If they converge, find the limit. a. a n = e n n+1 b. b n = 1 + ( 1)n 3 4 n 2 c. { 2 3, 4 8, 6 15, 8 24, 10 35, 12 48, 14 63,...} A2. Compute the area which is inside one leaf of r = cos(3θ) and outside r = 1 2 plane. in the polar A3. Compute the surface area of the curve y = x2 y-axis ln x, with x [1, 2], rotated about the A4. Determine which of the following converge. If the series converges, determine how many terms you need to approximate the series with error less than a. n 1 cos(nπ) n 2 6n + 2. b. 0 n 4n 4 2. Solve 1 of the following 2 problems - clearly mark which one you want graded. B1. Let f(x) and g(x) be two real-valued functions with f(x) g(x). Consider the region of the plane bounded by y = f(x), y = g(x), x = a, and x = b. a. Write the formulas for the centroid of the region and dervive. b. Derive these formulas from scratch. B2. Does n ln( ) converge? If so, what is it? n + 1 n=1

22 Name: Math 2924, Exam 1 Formulas: 1. cos 2 x + sin 2 x = 1 2. cos(a + B) = cos A cos B sin A sin B 3. sin(a + B) = sin A cos B + cos A sin B

23 1. Tell me about the function f(x) = x ln x. Do you like it? I want the domain, the limits as x and as x 0 +, the derivative, and the range. Sketch its graph too.

24 2. Compute the following limits (show your work). (a) lim x (ln x x) 1 (b) lim x 0 x x 0 ln(t + π)dt

25 3. (a) Prove that d dx sin 1 x = 1 1 x 2. (b) Compute x 2 dx.

26 4. Compute x 3 e x2 dx. (Hint: Have you integrated by parts yet?)

27 5. Compute sec 4 x tan xdx.

28 Name: Section: Math 2924, Exam 2 1. cos 2 x + sin 2 x = 1 2. cos 2 x sin 2 x = cos(2x) 3. 2 cos x sin x = sin(2x) 4. cos(a + B) = cos A cos B sin A sin B 5. sin(a + B) = sin A cos B + cos A sin B 6. For vectors a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) in R 3, the cross product of a and b is a b = (a 2 b 3 a 3 b 1, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ).

29 (1) Let u = (7, 3), i = (1, 0), and j = (0, 1). (a) Find constants a and b so that u = ai + bj. (b) Compute the projection of u onto i + j. (c) Compute the distance between u and 4i j.

30 (2) Consider the polar curve given by r = 6 cos θ. (a) Compute the area inside the curve. (b) Convert the equation into Cartesian coordinates. (c) What shape does this curve describe? (d) Is your answer to part (a) correct?

31 (3) Calculate 3x 2 x (x 1)(x 2 + 1) dx

32 (4) Parameterize a curve that starts at (2, 0) and travels clockwise once around the ellipse whose equation in Cartesian coordinates is given by x2 4 + y2 16 = 1. Write an integral formula for the perimeter (or arclength) of this ellipse. Compute the slope of the tangent line at time t.

33 (5) Write the equation of the plane through the points a = (1, 0, 1), b = (1, 1, 3), and c = (4, 0, 3).

34 Name: Section: Math , Exam 3 (1) Derive the power series for f(x) = 1 ln(1 x). Hint: Think about the power series for f (x).

35 (2) Explain why the each of the following series converges or diverges. (a) n 5 e n6 n=1 (b) n=0 ( 1) n n 1 n (c) n=1 e n 3 n+1

36 n 9 (3) Let f(x) = (x 2) n. I ve done the ratio test for you and determined that the radius n=10 n 2 of convergence is R = 1. Now finish the computation of the interval of convergence.

37 (4) In this problem, you ll compute the sum S = 1 + π π2 2! π3 3! + π4 4! + π5 5!... (a) Using the definition, compute the Taylor series T (x) of f(x) = 2 sin x centered at a = π 4. (Write the first 5-6 terms. Don t try to use summation notation.) (b) Show that T (x) converges at x = 5π. (Hint: Compute the radius of convergence.) 4 (c) Plug x = 5π 4 into both f(x) and the Taylor series to compute S.

38 (5) Answer with True, False, or Can t say not enough information. Justify your answers. (a) If the series a n 3 n absolutely converges, then the series n=0 a n ( 3) n converges? n=0 (b) If the series a n x n diverges for values of x satisfying x > 7, then it converges if n=17 x < 7. (Hint: What can you say about the radius of convergence?) (c) The series n=0 n n 2 n ln(n) sin(n) e n n nn ln(ln n) + 14(n 2 + n) n xn has interval of convergence ( 3, ).

39

40 Exam 1 Math 2924 Fall 2017 Name: Id: Total Show all necessary steps. Answers without supporting work may receive 0 credit. 1. Compute the derivatives f (x). Each is worth 4 points. (6 pt) f(x) = ln(sin 2 x) (6 pt) f(x) = 3 x (8 pt) f(x) = x sin x

41 2. (15 pt) Find the inverse function f(x) = (ln x) (15 pt) Find the domain and the derivative of the function f(x) = sin 1 (5x).

42 4. (15 pt) Find the domain and the derivative of sinh 1 (x). Recall sinh x = ex e x (15 pt) Find the following limit lim x ln ( ) 1 1 x + x

43 6. (20 pt) Evaluate the integrals. Each is worth 10 points. x sec 2 x dx (ln x) 2 dx

44 Exam 2 Math 2924 Fall 2017 Section: Name: Id: Total Show all necessary steps. Answers without supporting work may receive 0 credit. 1. (20 pt) Evaluate the integral 3 0 x dx using trigonometric substitution. 36 x 2

45 2. (20 pt) Find the integral 2x + 1 dx using partial fraction decomposition. x 2 (x + 1)

46 3. (10 pt) Find the exact length of the curve y = x x, 1 x (10 pt) Let a 1 = 1, a 2 = 2, and a n = 2a n 1 a n 2 when n 3. Then a 3 = a 4 = a 5 = Then find a formula for the general term a n.

47 5. Determine if the following series is convergent; find its sum when convergent. (8 pt) n=1 5 π n (8 pt) n= (2/3) n

48 6. Determine if the following series is convergent or divergent. (8 pt) n= n 2 (8 pt) n=1 n 2 + n + 1 n 4 + n 2 (8 pt) n=1 ( 1) n n

49 Exam 3 Math 2924 Fall 2017 Section: Name: Id: Total Show all necessary steps. Answers without supporting work may receive 0 credit. 1. a). (10 pts) Let f(x) = k=1 (x 2) k k 5 k. Find the radius of convergence of f. b). (10 pts) Test the endpoints to determine the interval of convergence of f(x).

50 2. a). (10 pts) Find a power series representation of x 2. b). (10 pts) Integrate the series and identify the function it represents.

51 3. (15 pts) Give the first four terms of the binomial series of the function f(x) = x.

52 4. (15 pts) Use the Cartesian-to-polar method to graph the curve r = sin θ, 0 θ 2π. r π 2π θ y 0 x

53 5. (15 pts) Find the slope of the line tangent to the curve r = sin θ at θ 0 = π/3.

54 6. (15 pts) Find the area of region bounded by r = sin θ, 0 θ π and the x-axis.

Without fully opening the exam, check that you have pages 1 through 13.

Without fully opening the exam, check that you have pages 1 through 13. MTH 33 Solutions to Exam November th, 08 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. MTH 33 Exam 2 November 4th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. MTH 33 Exam 2 April th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all

More information

MTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13.

MTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13. MTH 33 Solutions to Exam 2 November 5, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through

More information

cosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L =

cosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L = Integrals Volume: Suppose A(x) is the cross-sectional area of the solid S perpendicular to the x-axis, then the volume of S is given by V = b a A(x) dx Work: Suppose f(x) is a force function. The work

More information

MTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12.

MTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12. MTH 33 Solutions to Exam 2 April 9, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through

More information

MTH 133 Exam 2 November 16th, Without fully opening the exam, check that you have pages 1 through 12.

MTH 133 Exam 2 November 16th, Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response

More information

There are some trigonometric identities given on the last page.

There are some trigonometric identities given on the last page. MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during

More information

Math 230 Mock Final Exam Detailed Solution

Math 230 Mock Final Exam Detailed Solution Name: Math 30 Mock Final Exam Detailed Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and

More information

Math 113 Final Exam Practice

Math 113 Final Exam Practice Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three

More information

Practice problems from old exams for math 132 William H. Meeks III

Practice problems from old exams for math 132 William H. Meeks III Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are

More information

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and

More information

Math Final Exam Review

Math Final Exam Review Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot

More information

Things you should have learned in Calculus II

Things you should have learned in Calculus II Things you should have learned in Calculus II 1 Vectors Given vectors v = v 1, v 2, v 3, u = u 1, u 2, u 3 1.1 Common Operations Operations Notation How is it calculated Other Notation Dot Product v u

More information

Math 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2

Math 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2 Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos

More information

Multiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question

Multiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question MA 114 Calculus II Spring 2013 Final Exam 1 May 2013 Name: Section: Last 4 digits of student ID #: This exam has six multiple choice questions (six points each) and five free response questions with points

More information

Completion Date: Monday February 11, 2008

Completion Date: Monday February 11, 2008 MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,

More information

Math 1310 Final Exam

Math 1310 Final Exam Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space

More information

Math 226 Calculus Spring 2016 Exam 2V1

Math 226 Calculus Spring 2016 Exam 2V1 Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate

More information

You can learn more about the services offered by the teaching center by visiting

You can learn more about the services offered by the teaching center by visiting MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources

More information

Math 122 Test 3. April 15, 2014

Math 122 Test 3. April 15, 2014 SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not

More information

Math 113 Winter 2005 Key

Math 113 Winter 2005 Key Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple

More information

Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).

Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics). Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental

More information

Math 113/113H Winter 2006 Departmental Final Exam

Math 113/113H Winter 2006 Departmental Final Exam Name KEY Instructor Section No. Student Number Math 3/3H Winter 26 Departmental Final Exam Instructions: The time limit is 3 hours. Problems -6 short-answer questions, each worth 2 points. Problems 7 through

More information

f(f 1 (x)) = x HOMEWORK DAY 2 Due Thursday, August 23rd Online: 6.2a: 1,2,5,7,9,13,15,16,17,20, , # 8,10,12 (graph exponentials) 2.

f(f 1 (x)) = x HOMEWORK DAY 2 Due Thursday, August 23rd Online: 6.2a: 1,2,5,7,9,13,15,16,17,20, , # 8,10,12 (graph exponentials) 2. Math 63: FALL 202 HOMEWORK Below is a list of online problems (go through webassign), and a second set that you need to write up and turn in on the given due date, in class. Each day, you need to work

More information

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions Spring 5, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card,

More information

MATH 101: PRACTICE MIDTERM 2

MATH 101: PRACTICE MIDTERM 2 MATH : PRACTICE MIDTERM INSTRUCTOR: PROF. DRAGOS GHIOCA March 7, Duration of examination: 7 minutes This examination includes pages and 6 questions. You are responsible for ensuring that your copy of the

More information

ECM Calculus and Geometry. Revision Notes

ECM Calculus and Geometry. Revision Notes ECM1702 - Calculus and Geometry Revision Notes Joshua Byrne Autumn 2011 Contents 1 The Real Numbers 1 1.1 Notation.................................................. 1 1.2 Set Notation...............................................

More information

Homework Problem Answers

Homework Problem Answers Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln

More information

Math 113 Fall 2005 key Departmental Final Exam

Math 113 Fall 2005 key Departmental Final Exam Math 3 Fall 5 key Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.. Fill in the blanks with the correct answer. (a) The integral

More information

Final Exam Review Quesitons

Final Exam Review Quesitons Final Exam Review Quesitons. Compute the following integrals. (a) x x 4 (x ) (x + 4) dx. The appropriate partial fraction form is which simplifies to x x 4 (x ) (x + 4) = A x + B (x ) + C x + 4 + Dx x

More information

FINAL EXAM CALCULUS 2. Name PRACTICE EXAM SOLUTIONS

FINAL EXAM CALCULUS 2. Name PRACTICE EXAM SOLUTIONS FINAL EXAM CALCULUS MATH 00 FALL 08 Name PRACTICE EXAM SOLUTIONS Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of

More information

Math 113 (Calculus 2) Exam 4

Math 113 (Calculus 2) Exam 4 Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems

More information

More Final Practice Problems

More Final Practice Problems 8.0 Calculus Jason Starr Final Exam at 9:00am sharp Fall 005 Tuesday, December 0, 005 More 8.0 Final Practice Problems Here are some further practice problems with solutions for the 8.0 Final Exam. Many

More information

a k 0, then k + 1 = 2 lim 1 + 1

a k 0, then k + 1 = 2 lim 1 + 1 Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if

More information

MATH141: Calculus II Exam #4 7/21/2017 Page 1

MATH141: Calculus II Exam #4 7/21/2017 Page 1 MATH141: Calculus II Exam #4 7/21/2017 Page 1 Write legibly and show all work. No partial credit can be given for an unjustified, incorrect answer. Put your name in the top right corner and sign the honor

More information

Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018

Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018 Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement

More information

Without fully opening the exam, check that you have pages 1 through 11.

Without fully opening the exam, check that you have pages 1 through 11. MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show

More information

Math 162: Calculus IIA

Math 162: Calculus IIA Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ

More information

Math 113 Winter 2005 Departmental Final Exam

Math 113 Winter 2005 Departmental Final Exam Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple

More information

MATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.

MATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval. MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =

More information

Math 152 Take Home Test 1

Math 152 Take Home Test 1 Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I

More information

Part I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3

Part I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3 Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (

More information

Mathematics 111 (Calculus II) Laboratory Manual

Mathematics 111 (Calculus II) Laboratory Manual Mathematics (Calculus II) Laboratory Manual Department of Mathematics & Statistics University of Regina nd edition prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry University of Regina Department

More information

Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.

Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n. .8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x

More information

2 Recollection of elementary functions. II

2 Recollection of elementary functions. II Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic

More information

y = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx

y = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,

More information

RED. Math 113 (Calculus II) Final Exam Form A Fall Name: Student ID: Section: Instructor: Instructions:

RED. Math 113 (Calculus II) Final Exam Form A Fall Name: Student ID: Section: Instructor: Instructions: Name: Student ID: Section: Instructor: Math 3 (Calculus II) Final Exam Form A Fall 22 RED Instructions: For questions which require a written answer, show all your work. Full credit will be given only

More information

Exam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:

Exam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA: Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may

More information

Last/Family Name First/Given Name Seat #

Last/Family Name First/Given Name Seat # Math 2, Fall 27 Schaeffer/Kemeny Final Exam (December th, 27) Last/Family Name First/Given Name Seat # Failure to follow the instructions below will constitute a breach of the Stanford Honor Code: You

More information

The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin

The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin Math : Practice Final Answer Key Name: The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. Problem : Consider the definite integral I = 5 sin ( ) d.

More information

APPM 1360 Final Exam Spring 2016

APPM 1360 Final Exam Spring 2016 APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan

More information

10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.

10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1. 10.1 Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1 Examples: EX1: Find a formula for the general term a n of the sequence,

More information

Inverse Trig Functions

Inverse Trig Functions 6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)

More information

Math 122 Test 3. April 17, 2018

Math 122 Test 3. April 17, 2018 SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator

More information

MA CALCULUS II Friday, December 09, 2011 FINAL EXAM. Closed Book - No calculators! PART I Each question is worth 4 points.

MA CALCULUS II Friday, December 09, 2011 FINAL EXAM. Closed Book - No calculators! PART I Each question is worth 4 points. CALCULUS II, FINAL EXAM 1 MA 126 - CALCULUS II Friday, December 09, 2011 Name (Print last name first):...................................................... Signature:........................................................................

More information

MTH 133 Final Exam Dec 8, 2014

MTH 133 Final Exam Dec 8, 2014 Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Problem Score Max Score 1 5 3 2 5 3a 5 3b 5 4 4 5 5a 5 5b 5 6 5 5 7a 5 7b 5 6 8 18 7 8 9 10 11 12 9a

More information

Have a Safe Winter Break

Have a Safe Winter Break SI: Math 122 Final December 8, 2015 EF: Name 1-2 /20 3-4 /20 5-6 /20 7-8 /20 9-10 /20 11-12 /20 13-14 /20 15-16 /20 17-18 /20 19-20 /20 Directions: Total / 200 1. No books, notes or Keshara in any word

More information

C3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009)

C3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009) C3 Revision Questions (using questions from January 2006, January 2007, January 2008 and January 2009) 1 2 1. f(x) = 1 3 x 2 + 3, x 2. 2 ( x 2) (a) 2 x x 1 Show that f(x) =, x 2. 2 ( x 2) (4) (b) Show

More information

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx. Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the

More information

M152: Calculus II Midterm Exam Review

M152: Calculus II Midterm Exam Review M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance

More information

Final Exam 2011 Winter Term 2 Solutions

Final Exam 2011 Winter Term 2 Solutions . (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L

More information

SECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.

SECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),

More information

Math 190 (Calculus II) Final Review

Math 190 (Calculus II) Final Review Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the

More information

FINAL EXAM CALCULUS 2. Name PRACTICE EXAM

FINAL EXAM CALCULUS 2. Name PRACTICE EXAM FINAL EXAM CALCULUS 2 MATH 2300 FALL 208 Name PRACTICE EXAM Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of your

More information

- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of

More information

MAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists

MAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists Data provided: Formula sheet MAS53/MAS59 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics (Materials Mathematics For Chemists Spring Semester 203 204 3 hours All questions are compulsory. The marks awarded

More information

Calculus II (Math 122) Final Exam, 19 May 2012

Calculus II (Math 122) Final Exam, 19 May 2012 Name ID number Sections C and D Calculus II (Math 122) Final Exam, 19 May 2012 This is a closed book exam. No notes or calculators are allowed. A table of trigonometric identities is attached. To receive

More information

Final Exam. Math 3 December 7, 2010

Final Exam. Math 3 December 7, 2010 Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.

More information

SOLUTIONS TO EXAM II, MATH f(x)dx where a table of values for the function f(x) is given below.

SOLUTIONS TO EXAM II, MATH f(x)dx where a table of values for the function f(x) is given below. SOLUTIONS TO EXAM II, MATH 56 Use Simpson s rule with n = 6 to approximate the integral f(x)dx where a table of values for the function f(x) is given below x 5 5 75 5 5 75 5 5 f(x) - - x 75 5 5 75 5 5

More information

Calculus II. George Voutsadakis 1. LSSU Math 152. Lake Superior State University. 1 Mathematics and Computer Science

Calculus II. George Voutsadakis 1. LSSU Math 152. Lake Superior State University. 1 Mathematics and Computer Science Calculus II George Voutsadakis Mathematics and Computer Science Lake Superior State University LSSU Math 52 George Voutsadakis (LSSU) Calculus II February 205 / 88 Outline Techniques of Integration Integration

More information

MA 114 Worksheet Calendar Fall 2017

MA 114 Worksheet Calendar Fall 2017 MA 4 Worksheet Calendar Fall 7 Thur, Aug 4: Worksheet Integration by parts Tues, Aug 9: Worksheet Partial fractions Thur, Aug 3: Worksheet3 Special trig integrals Tues, Sep 5: Worksheet4 Special trig integrals

More information

Integration Techniques

Integration Techniques Review for the Final Exam - Part - Solution Math Name Quiz Section The following problems should help you review for the final exam. Don t hesitate to ask for hints if you get stuck. Integration Techniques.

More information

10. e tan 1 (y) 11. sin 3 x

10. e tan 1 (y) 11. sin 3 x MATH B FINAL REVIEW DISCLAIMER: WHAT FOLLOWS IS A LIST OF PROBLEMS, CONCEPTUAL QUESTIONS, TOPICS, AND SAMPLE PROBLEMS FROM THE TEXTBOOK WHICH COMPRISE A HEFTY BUT BY NO MEANS EXHAUSTIVE LIST OF MATERIAL

More information

Notes of Calculus II (MTH 133) 2013 Summer. Hongli Gao

Notes of Calculus II (MTH 133) 2013 Summer. Hongli Gao Notes of Calculus II (MTH 133) 2013 Summer Hongli Gao June 16, 2013 Chapter 6 Some Applications of the Integral 6.2 Volume by parallel cross-sections; Disks And Washers Definition 1. A cross-section of

More information

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209 PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:

More information

Brief answers to assigned even numbered problems that were not to be turned in

Brief answers to assigned even numbered problems that were not to be turned in Brief answers to assigned even numbered problems that were not to be turned in Section 2.2 2. At point (x 0, x 2 0) on the curve the slope is 2x 0. The point-slope equation of the tangent line to the curve

More information

Fall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes

Fall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Math 00 Calculus Lectures Chapter 8 Series Numeration of sections corresponds to the text James Stewart, Essential Calculus, Early Transcendentals, Second edition. Section 8. Sequences A sequence is a

More information

n=1 ( 2 3 )n (a n ) converges by direct comparison to

n=1 ( 2 3 )n (a n ) converges by direct comparison to . (a) n = a n converges, so we know that a n =. Therefore, for n large enough we know that a n

More information

Test 3 - Answer Key Version B

Test 3 - Answer Key Version B Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop,

More information

Virginia Tech Math 1226 : Past CTE problems

Virginia Tech Math 1226 : Past CTE problems Virginia Tech Math 16 : Past CTE problems 1. It requires 1 in-pounds of work to stretch a spring from its natural length of 1 in to a length of 1 in. How much additional work (in inch-pounds) is done in

More information

Name: Answer Key David Arnold. Math 50B Integral Calculus May 13, Final Exam

Name: Answer Key David Arnold. Math 50B Integral Calculus May 13, Final Exam Math 5B Integral Calculus May 3, 7 Final Exam Name: Answer Key David Arnold Instructions. (9 points) Follow the directions exactly! Whatever you are asked to do, you must do to receive full credit for

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.

More information

Turn off all cell phones, pagers, radios, mp3 players, and other similar devices.

Turn off all cell phones, pagers, radios, mp3 players, and other similar devices. Math 25 B and C Midterm 2 Palmieri, Autumn 26 Your Name Your Signature Student ID # TA s Name and quiz section (circle): Cady Cruz Jacobs BA CB BB BC CA CC Turn off all cell phones, pagers, radios, mp3

More information

MATH 1207 R02 MIDTERM EXAM 2 SOLUTION

MATH 1207 R02 MIDTERM EXAM 2 SOLUTION MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find

More information

MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section:

MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section: MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, 2016 Name: Section: Last 4 digits of student ID #: This exam has five true/false questions (two points each), ten multiple choice questions

More information

Math 116 Second Midterm November 14, 2012

Math 116 Second Midterm November 14, 2012 Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

More information

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009.

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009. OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK Summer Examination 2009 First Engineering MA008 Calculus and Linear Algebra

More information

Chapter 7: Techniques of Integration

Chapter 7: Techniques of Integration Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration

More information

Math Review for Exam Answer each of the following questions as either True or False. Circle the correct answer.

Math Review for Exam Answer each of the following questions as either True or False. Circle the correct answer. Math 22 - Review for Exam 3. Answer each of the following questions as either True or False. Circle the correct answer. (a) True/False: If a n > 0 and a n 0, the series a n converges. Soln: False: Let

More information

Practice Differentiation Math 120 Calculus I Fall 2015

Practice Differentiation Math 120 Calculus I Fall 2015 . x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although

More information

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12 First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please

More information

1 + x 2 d dx (sec 1 x) =

1 + x 2 d dx (sec 1 x) = Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating

More information

n=0 ( 1)n /(n + 1) converges, but not

n=0 ( 1)n /(n + 1) converges, but not Math 07H Topics for the third exam (and beyond) (Technically, everything covered on the first two exams plus...) Absolute convergence and alternating series A series a n converges absolutely if a n converges.

More information

Formulas to remember

Formulas to remember Complex numbers Let z = x + iy be a complex number The conjugate z = x iy Formulas to remember The real part Re(z) = x = z+z The imaginary part Im(z) = y = z z i The norm z = zz = x + y The reciprocal

More information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

5 Integrals reviewed Basic facts U-substitution... 4

5 Integrals reviewed Basic facts U-substitution... 4 Contents 5 Integrals reviewed 5. Basic facts............................... 5.5 U-substitution............................. 4 6 Integral Applications 0 6. Area between two curves.......................

More information

Integrated Calculus II Exam 1 Solutions 2/6/4

Integrated Calculus II Exam 1 Solutions 2/6/4 Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (

More information