Fa 17: MATH 2924 040 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
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
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 ).
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
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?
Q4]... [25 points] Compute the following integrals. 1. e 7 e 3 dx x ln x 2. e x dx 1 + e 2x
Fa 17: MATH 2924 040 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
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
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.
Q2]... [26 points] Evaluate the following two integrals. 1. x ln(x)dx 2. cos 5 (x)dx
Q3]... [22 points] Use a substitution to evaluate the following integral. Show all the steps of your work. dx (x 2 + 9) 2
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
Fa 17: MATH 2924 040 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
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 < 1. 3. 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. 1 4. 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 0. 5. 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 0. 10. 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
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 )
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
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.
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]?
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.
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.
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. 4 1 2 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 1 100. 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
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
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.
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
3. (a) Prove that d dx sin 1 x = 1 1 x 2. (b) Compute 1 1 2 1 1 x 2 dx.
4. Compute x 3 e x2 dx. (Hint: Have you integrated by parts yet?)
5. Compute sec 4 x tan xdx.
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 ).
(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.
(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?
(3) Calculate 3x 2 x (x 1)(x 2 + 1) dx
(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.
(5) Write the equation of the plane through the points a = (1, 0, 1), b = (1, 1, 3), and c = (4, 0, 3).
Name: Section: Math 2924-010, Exam 3 (1) Derive the power series for f(x) = 1 ln(1 x). Hint: Think about the power series for f (x).
(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 + 100 (c) n=1 e n 3 n+1
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.
(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.
(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, ).
Exam 1 Math 2924 Fall 2017 Name: Id: 1 2 3 4 5 6 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
2. (15 pt) Find the inverse function f(x) = (ln x) 2. 3. (15 pt) Find the domain and the derivative of the function f(x) = sin 1 (5x).
4. (15 pt) Find the domain and the derivative of sinh 1 (x). Recall sinh x = ex e x 2. 5. (15 pt) Find the following limit lim x ln ( ) 1 1 x + x
6. (20 pt) Evaluate the integrals. Each is worth 10 points. x sec 2 x dx (ln x) 2 dx
Exam 2 Math 2924 Fall 2017 Section: Name: Id: 1 2 3 4 5 6 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
2. (20 pt) Find the integral 2x + 1 dx using partial fraction decomposition. x 2 (x + 1)
3. (10 pt) Find the exact length of the curve y = x3 3 + 1 4x, 1 x 3. 4. (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.
5. Determine if the following series is convergent; find its sum when convergent. (8 pt) n=1 5 π n (8 pt) n=1 1 1 + (2/3) n
6. Determine if the following series is convergent or divergent. (8 pt) n=1 1 1 + n 2 (8 pt) n=1 n 2 + n + 1 n 4 + n 2 (8 pt) n=1 ( 1) n n
Exam 3 Math 2924 Fall 2017 Section: Name: Id: 1 2 3 4 5 6 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).
2. a). (10 pts) Find a power series representation of 1 1 + x 2. b). (10 pts) Integrate the series and identify the function it represents.
3. (15 pts) Give the first four terms of the binomial series of the function f(x) = 3 2 + x.
4. (15 pts) Use the Cartesian-to-polar method to graph the curve r = 1 + 2 sin θ, 0 θ 2π. r 3 1 0 π 2π θ y 0 x
5. (15 pts) Find the slope of the line tangent to the curve r = 1 + 2 sin θ at θ 0 = π/3.
6. (15 pts) Find the area of region bounded by r = 1 + 2 sin θ, 0 θ π and the x-axis.