Some suggested repetition for the course MAA508

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1 Some suggested repetition for the course MAA58 Linus Carlsson, Karl Lundengård, Johan Richter July, 14

2 Contents Introduction 1 1 Basic algebra and trigonometry Univariate calculus 5 3 Linear algebra 8 4 Multivariate calculus 11 5 Ordinary differential equations 13

3 Introduction Welcome to Engineering Mathematics at MDH. We have prepared a set of exercises that you can use to self-diagnose if their is any part of mathematics that you need to review. Please have a look at them, and try to solve them. In each section there are some exercises on material that we expect that you have seen before during your education. After each exercise you will find some keywords which you can use when looking for material relevant for solving the exercise. At the end of each section you will also find a list of textbooks which you can use to review the material. Please note that you can also use your old textbooks if you still have them, they almost certainly treat the same material. If you do not manage to solve the problem even after consulting the textbooks, you are of course welcome to contact us: Karl Lundengård (karl.lundengard@mdh.se) Johan Richter (johan.richter@mdh.se) 1

4 1 Basic algebra and trigonometry 1.1. Simplify the following expressions as much as possible. You may assume that a, b are positive numbers. (a) 1/9 4 /3 (b) 6 3 1/4 (c) (d) 9 1/ 6 ( 1 ) ab ( ) b a b 3 a 3/ (e) (f) ( a b ) ( 4 a 3 b 3 ab ) ( ) ab ab 3 (a) 13/9 (b) 3 3 3/4 (c) 1 (d) b 1 (e) a 4 b 1 (f) a 11/1 b /3 Keywords: Power laws, roots 1.. Simplify the following expressions as much as possible. You may assume a, b are positive numbers. (a) ln(e ) log (b) 1 (1) log 1 (1) (c) ln(ab ) ln(a) (d) log 1(9) log 1 (e) ln(3) (e) ln(a+b) ln(a) ln(b) (f) ln(a/e) + ln(1/a) (a) (b) 3 (c) ln(b) (d) ln(3) ( ) a + b (e) ln ab (f) 1 Keywords: Logarithms 1.3. Complete the square of the quadratic polynomials. (a) x + x + 1 (b) x + 4x + 5 (c) x 3x + 1 (a) ( x + 1 ) (b) (x + ) + 1 ( (c) x 3 ) Keywords: Completing the square

5 1.4. Factor the following quadratic polynomials. (a) x + x (b) x + 4x + 3 (c) x 3x + 1 (a) ( x + 1 ) (b) (x + 1)(x + 3) ( (c) x 1 ) (x 1) Keywords: Factorization of polynomials, completing the square 1.5. Expand the following expressions. (a) (x + 3) (b) (x 3) (c) (x + 5) (a) x + 6x + 9 (b) x 6x + 9 (c) 4x + x + 5 Keywords: Binomial theorem 1.6. Simplify the following expressions as much as possible. (a) x 9 x + 3 (b) x + x + 1 x + 1 (c) x3 x + x x 1 (a) x 3 (b) x + 1 (c) x(x 1) Keywords: Rational functions, factorization of polynomials 1.7. Find all solutions to the following equations. (a) x = 9 (b) x + 4x + 3 = (c) x3 x x + 1 = (a) x = 3 and x = 3 (b) x = 1 and x = 3 (c) x = and x = 1 Keywords: Factorization of polynomials, quadratic equations 3

6 1.8. Compute the following square roots, if defined. (a) 9 (b) ( 3) (c) 9 (a) 3 (b) 3 (c) undefined Keywords: Square roots 1.9. Indicate whether the following values are positive, negative or zero. The arguments for the trigonometric functions are expressed in radians. ( ) ( 5π π ) ( ) (a) sin (c) cos 1π 6 7 (e) tan ( (b) tan(3π) (d) cos π ) 3 ( ) ( ) π π 5 (f) sin cos 3 3 (a) Positive (b) Zero (c) Positive (d) Positive (e) Positive (f) Negative Keywords: Trigometric functions, unit circle 1.1. (a) Show that cos(x) + sin (x) = 1 for all x. (b) Show that sin(x + π 3 ) = 1 (sin(x) + 3 cos(x)) for all x. (c) Show that (cos(x) + sin(x)) = 1 + sin(x) for all x. Keywords: Trigonometric identities For more information see: This is basic material that can be found in many calculus or pre-calculus textbooks. You may consult, for example, Calculus by James Stewart (any edition) and the review of algebra you can find on that books homepage. If you want a Swedish book you could use Mot bättre vetande i matematik by Andrejs Dunkels et al. 4

7 Univariate calculus.1. Calculate the derivative of the following functions (a) df dx, where f(x) = ax3 + bx c. (b) df, where f(x) = sin(x). dx (a) df dx = 3ax + b. (b) df dx = cos(x). (c) df dx, where f(x) = 5 x. (d) df dx, where f(x) = ex. (c) df dx = 5 x. (d) df dx = ex. Keywords: Derivative of elementary functions, polynomials, trigonometric functions, exponential functions.. Calculate the derivative of the following functions (a) df dx, where f(x) = eax3 +bx c. (b) df dx, where f(x) = sin ( 1 x). (c) df, where f(x) = x sin(x). dx (a) df dx = (3ax + b) e ax3 +bx c. (b) df dx = cos ( 1 x) x. (c) df = sin(x) + x cos(x). dx Keywords: Derivative of a product, derivative of a composite function.3. Calculate the integrals (a) I = (b) I = (c) I = π π b a f(x) dx, where f(x) = 3x + 7. f(x) dx, where f(x) = sin(x). f(x) dx, where f(x) = e x. (a) I =. (b) I =. (c) I = 3(e b e a ). 5

8 .4. Calculate the integral I = x sin(x) + x cos(x) dx. I = 3 sin() + cos() Keywords: Integration by parts (partial integration).5. Let z = 1 + i and w = 3 i where i is the imaginary unit. (a) Find the real and imaginary parts of z and w. (b) Find the conjugate of z and w. (c) Calculate z + w. (d) Calculate z w. (a) Re(z) = 1, Im(z) = 1, Re(w) = 3, Im(w) =. (b) z = 1 i and w = 3 + i. (c) z + w = 1 + s 3 i. (d) z w = ( 3 + 1) + ( 3 1)i. Keywords: Complex numbers, imaginary number, conjugate, absolute value.6. Let z = 1 + i and w = 3 i where i is the imaginary unit. (a) Find the absolute value (modulus) of z and w. (b) Find the argument of z and w. (c) Write z and w in polar form. (d) Write z and w as complex exponentials. (e) Calculate z 3 and w 7. (f) Calculate z w using the complex exponential form. (a) z = and w = 4. (b) arg(z) = π + k π, arg(z) = π 6 + k π where k is any integer. (c) z = ( cos ( ) ( π 4 + i sin π )) ( ( ) ( )) 4, w = 4 cos π 6 + i sin π 6 (d) z = e i π 4, w = 4e i π 6. (e) z 3 = ( cos ( ) ( 3π 4 + i sin 3π 4 (f) z w = 4 e i π 1. )), w 7 = ( cos ( ) ( 5π 6 + i sin 5π )) 6 Keywords: Polar form, Euler s formula, De Moivre s theorem 6

9 .7. Determine the following limits if they exist 1 (a) lim x (b) lim x x x 1 1 x (c) lim arctan(x) sin(x) x (e) lim x (d) lim tan(x) x x π + (f) lim cos(x) x (a) lim diverges towards +, limit does not exist. x 1 (b) lim x x = (c) (d) lim arctan(x) = π x lim tan(x) diverges towards, limit does not exist. x π + sin(x) (e) lim x x (f) lim x = 1 cos(x) is bounded but does not converge, limit does not exist. Keywords: Limits, one-sided limits.8. Show that k=1 1 k(k + 1) = 1 Keywords: Partial fractal decomposition, convergent sum.9. Give explicit expressions for the Taylor expansion of (a) sin(x). (b) e x. (a) sin(x) = k= ( 1) k x 1+k. (b) e x = (1 + k)! k= x k k!. Keywords: Taylor series, Taylor expansion, factorial For more information see: any introductory text on univariate calculus, for example Calculus: Early Transcendentals by James Stewart. 7

10 3 Linear algebra 3.1. We have: A = 1 3, B = 1 3, I = Calculate: (a) transpose: A. (b) addition: A + B. (c) subtraction: A B. (d) multiplication AI. (e) multiplication IA. (f) multiplication: AB. (g) multiplication: BA. (h) What is I called? (i) Is matrix multiplication commutative? (AB = BA for all matrices A, B) 3 1 (a) A = (b) A + B = (c) A B = 6 (d) AI = A (e) IA = A (f) AB = (g) BA = (h) I is called the identity matrix. (i) Matrix multiplication is not commutative. Keywords: Matrix multiplication, identity matrix 3.. Let: A = 1 3 4, B = 1 1, C = Which of the following matrix multiplications can we compute and what is the size of the resulting matrix? (a) AA (b) BB (a) not possible (b) 3 3 (c) AC (d) CA (c) 3 3 (d) 4 4 (e) AB (f) BA (e) not possible (f) 3 4 Keywords: Matrix multiplication 8

11 3.3. Given the following linear equation system: 4a + b + 4c + d = 13 4a + 4b + 3c + 3d = 4 4a + 4b + c + d = 17 3a + 3b + 3c + d = 18 (a) Write the system using matrices and column vectors. (b) Solve the system. (b) a = 1, b =, c = 1, d = 3. Keywords: Linear equation system, Gaussian elimination 3.4. (a) Show that this linear equation system has a unique solution. x + y z = x + y = 3 x y + z = 1 (b) Show that this linear equation system has infinitely many solutions. x + y z = 1 x + y + z = 3x + y z = 3 (c) Show that this linear equation system has no solution. x y + z = 3 x + 3y z = x + y z = 1 Keywords: Gaussian elimination, linear dependence 3.5. Based on what you already know, for which linear equation systems in problem 3.4. will (a) The determinant of the corresponding matrix be equal to zero? (b) The rows (or columns) of the corresponding matrix be linearly independent? (c) The rank of the corresponding matrix be smaller than 3? (a) This is true for the systems in (b), (c) (b) This is true for the system in (a) (c) (b), (c) Keywords: Determinant, linear dependence, rank 9

12 3.6. Calculate this determinant 3 1 det(a) = det(a) = 84 Keywords: Determinant, rule of Sarrus 3.7. Let 1 3 A = 3 1, B = (a) Show that B is the inverse of the matrix A. (b) If the linear equation systems in 3.4. Ax = y, for which systems would the inverse A 1 exist? Keywords: Matrix inverse 3.8. We have: Calculate: (a) a b (b) C b (c) C b a b (a) a b = 4 6 (b) C b = a =, b = 1, C = (d) Is a an eigenvalue belonging to C? (e) If a is an eigenvalue, is b an eigenvector belonging to a? (c) C b a b = (d) a is an eigenvalue of C. (e) b is an eigenvector of C which corresponds to the eigenvalue a. Keywords: Eigenvector, eigenvalue For more information see: any introductory text on linear algebra (vector algebra), for example Elementary linear algebra by Howard Anton and Chris Rorres or Grundläggande linjär algebra by Hillevi Gavel. 1

13 4 Multivariate calculus 4.1. Calculate the following partial derivatives of the given functions. (a) df f(x, y), where f(x, y) = y sin(x/y). dx x (b) df f(x, y), where f(x, y) = y sin(x/y). dx y (a) f x (x, y) = cos(x/y). (b) f y (x, y) = sin(x/y) x y cos(x/y). 4.. Describe the domain of definition of the following functions, also sketch the domain of definition. (a) f(x, y) = y sin(x/y). (b) f(x, y) = ln ( x y ). (c) f(x, y) = 4 x y. (a) y. (b) x < y < x, if x > together with x < y < x, if x <. (c) The disk x + y Use the chain rule to calculate the following derivatives: (a) df df(x, y), where f(x, y) = x y and x = h(t) and y = g(t). dx dt (b) df f(x, y), where f(x, y) = x y and x(t, s) = ts and y(t, s) = e t/s. dx t (a) df df(x, y) = h(t) h (t) g (t). dx dt (b) df f(x, y) = s ( ts + t ) e t/s. dx t 4.4. Calculate the following integrals. Remember, it is sometimes useful to change the order of integration. (a) I = (b) I = (c) I = (a) I = 1. (b) I =. 1 dxdy. sin(x )(y 1/) dxdy. x 3 sin(yx ) dxdy. 11

14 (c) I = 1 sin(1). Keywords: Partial derivatives, domain of definition, chain rule, multiple integral For more information see: any introductory text on multivariate calculus, for example Calculus: Early Transcendentals by James Stewart. 1

15 5 Ordinary differential equations 5.1. Solve the following first order ordinary differential equations, either by using ingegrating factor (IF) or by separation of varables (SoV). (a) y + y + 1 =. (b) y xy = x. (c) y = y cos(x). (d) y + y = x. (a) y = C e x 1. (IF) or (SoV). (b) y = C e x / 1. (IF) or (SoV). (c) y = ( C sin(x) ) 1. (SoV). (d) y = x x + + C e x. (IF). 5.. Solve the following homogeneous second order ordinary differential equations by help of the Characteristic equation. (a) y + y 3y =. (b) y 4y + 4y =. (c) y + 8y + 5y =. (a) y = C 1 e x + C e 3x. (b) y = (C 1 x + C )e x. (c) y = e 4x( C 1 sin(3x) + C cos(3x) ) One can always check if a function is a solution to a differential equation by showing that the left hand side is equal to the right hand side. Show that: y (a) y = x sin(x) is a solution to = tan(x) + x. cos(x) (b) f(x, t) = (4πt) 1/ e x 4t is a solution to the heat equation f(x, t) f(x, t) x =. t Keywords: Separable ordinary differential equations, linear ordinary differential equations of first and second order, solutions to differential equations For more information see: any introductory text on differential equations, for example Calculus: Early Transcendentals by James Stewart. 13

Further Mathematics SAMPLE. Marking Scheme

Further Mathematics SAMPLE. Marking Scheme Further Mathematics SAMPLE Marking Scheme This marking scheme has been prepared as a guide only to markers. This is not a set of model answers, or the exclusive answers to the questions, and there will