+ FE Mathematics Review Dr. Omar Meza Assistant Professor Department of Mechanical Engineering

+ Topics covered  Analytic geometry  Equations of lines and curves  Distance, area and volume  Trigonometric identities  Algebra  Complex numbers  Matrix arithmetic and determinants  Vector arithmetic and applications  Progressions and series  Numerical methods for finding solutions of nonlinear equations  Differential calculus  Derivatives and applications  Limits and L’Hopital’s rule  Integral calculus  Integrals and applications  Numerical methods  Differential equations  Solution and applications  Laplace transforms

+ Tips for taking exam  Use the reference handbook  Know what it contains  Know what types of problems you can use it for  Know how to use it to solve problems  Refer to it frequently  Work backwards when possible  FE exam is multiple choice with single correct answer  Plug answers into problem when it is convenient to do so  Try to work backwards to confirm your solution as often as possible  Progress from easiest to hardest problem  Same number of points per problem  Calculator tips  Check the NCEES website to confirm your model is allowed  Avoid using it to save time!  Many answers do not require a calculator (fractions vs. decimals)

+ Equations of lines Handbook page:

+ Equations of lines

+

+  What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?  (A) 2x – 3y – 18 = 0  (B) 2x + 3y + 18 = 0  (C) 3x – 2y – 12 = 0  (D) 3x + 2y + 12 =

+ Equations of lines  What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?  (A) 2x – 3y – 18 = 0  (B) 2x + 3y + 18 = 0  (C) 3x – 2y – 12 = 0  (D) 3x + 2y + 12 = 0  Try using standard form  Handbook pg 3: y = mx + b  Given (x1, y1) = (4, 0)  Given (x2, y2) = (0, -6) Answer is (C)

+ Equations of lines  What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?  (A) 2x – 3y – 18 = 0  (B) 2x + 3y + 18 = 0  (C) 3x – 2y – 12 = 0  (D) 3x + 2y + 12 = 0  Work backwards  Substitute (x1, y1) = (4, 0)  Substitute (x2, y2) = (0, -6)  See what works Alternative Solution Answer is (C)

+ Equations of lines

+

+

+ Quadratic Equation Handbook page:

+ Quadratic Equation Handbook page: A) 1, 2; B) 3, 2; C) 0.5,-3; D) -0.5, -3 Answer is (C)

+ Quadratic Equation

+ Equations of curves

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ Logarithms

+ Logarithms Answer is (D)

+ Logarithms

+ Trigonometry

+ Trigonometry

+ Trigonometry

+ Trigonometry

+ Trigonometry For some angle , csc  = -8/5. What is cos 2  ? For some angle , csc  = -8/5. What is cos 2  ? Use trigonometric identities on handbook. Use trigonometric identities on handbook. Confirm with calculator Confirm with calculator First find  = csc -1 (-8/5) First find  = csc -1 (-8/5) Then find cos 2  Then find cos 2  (A) 7/32 (B) 1/4 (C) 3/8 (D) 5/8 Answer is (A)

+ Trigonometry Answer is (C)

+ Trigonometry

+ Complex Numbers

+

+

+

+ Polar coordinates

+ What is rectangular form of the polar equation r 2 = 1 – tan 2  ? What is rectangular form of the polar equation r 2 = 1 – tan 2  ? (A) –x 2 + x 4 y 2 + y 2 = 0 (A) –x 2 + x 4 y 2 + y 2 = 0 (B) x 2 + x 2 y 2 - y 2 - y 4 = 0 (B) x 2 + x 2 y 2 - y 2 - y 4 = 0 (C) –x 4 + y 2 = 0 (C) –x 4 + y 2 = 0 (D) x 4 – x 2 + x 2 y 2 + y 2 = 0 (D) x 4 – x 2 + x 2 y 2 + y 2 = 0 Polar coordinate identities on handbook Polar coordinate identities on handbook Answer is (D)

+ Polar coordinates

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Matrices

+ Vector

+ Vector

+ Vector

+ Vector

+ Vector calculations For three vectors A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k, what is the product A·(B x C)? For three vectors A = 6i + 8j + 10k B = i + 2j + 3k C = 3i + 4j + 5k, what is the product A·(B x C)? (A) 0 (A) 0 (B) 64 (B) 64 (C) 80 (C) 80 (D) 216 (D) 216 Vector products on handbook Vector products on handbook Answer is (A)

+ Vector calculations Answer is (D)

+ Vector calculations Answer is (A) (-16- 8)i – (-8+16)j + (2+8)k -24i -8j + 10k

+ Geometric Progression The 2 nd and 6 th terms of a geometric progression are 3/10 and 243/160. What is the first term of the sequence? The 2 nd and 6 th terms of a geometric progression are 3/10 and 243/160. What is the first term of the sequence? (A) 1/10 (A) 1/10 (B) 1/5 (B) 1/5 (C) 3/5 (C) 3/5 (D) 3/2 (D) 3/2 Geometric progression on handbook Geometric progression on handbook Answer is (B) Confirm answer by calculating l 2 and l 6 with a = 1/5 and r = 3/2.

+ Roots of nonlinear equations Newton’s method is being used to find the roots of the equation f(x) = (x – 2) 2 – 1. Find the 3 rd approximation if the 1 st approximation of the root is 9.33 Newton’s method is being used to find the roots of the equation f(x) = (x – 2) 2 – 1. Find the 3 rd approximation if the 1 st approximation of the root is 9.33 (A) 1.0 (A) 1.0 (B) 2.0 (B) 2.0 (C) 3.0 (C) 3.0 (D) 4.0 (D) 4.0 Newton’s method on handbook Newton’s method on handbook Answer is (D)

+ Application of derivatives

+

+

+

+

+

+

+

+

+

+ Limits What is the limit of (1 – e 3x ) / 4x as x  0? What is the limit of (1 – e 3x ) / 4x as x  0? (A) -∞ (A) -∞ (B) -3/4 (B) -3/4 (C) 0 (C) 0 (D) 1/4 (D) 1/4 L’Hopital’s rule on handbook L’Hopital’s rule on handbook Answer is (B) You should apply L’Hopital’s rule iteratively until you find limit of f(x) / g(x) that does not equal 0 / 0. You can also use your calculator to confirm the answer, substitute a small value of x = 0.01 or

+ Application of derivatives The radius of a snowball rolling down a hill is increasing at a rate of 20 cm / min. How fast is its volume increasing when the diameter is 1 m? The radius of a snowball rolling down a hill is increasing at a rate of 20 cm / min. How fast is its volume increasing when the diameter is 1 m? (A) m 3 / min (A) m 3 / min (B) 0.52 m 3 / min (B) 0.52 m 3 / min (C) 0.63 m 3 / min (C) 0.63 m 3 / min (D) 0.84 m 3 / min (D) 0.84 m 3 / min Derivatives on handbook; volume of sphere on handbook page 10 Derivatives on handbook; volume of sphere on handbook page 10 Convert cm to m, convert diameter to radius, and confirm final units are correct. Answer is (C)

+ Evaluating integrals

+

+

+

+

+

+

+

+

+

+ Evaluate the indefinite integral of f(x) = cos 2 x sin x Evaluate the indefinite integral of f(x) = cos 2 x sin x (A) -2/3 sin 3 x + C (A) -2/3 sin 3 x + C (B) -1/3 cos 3 x + C (B) -1/3 cos 3 x + C (C) 1/3 sin 3 x + C (C) 1/3 sin 3 x + C (D) 1/2 sin 2 x cos 2 x + C (D) 1/2 sin 2 x cos 2 x + C Apply integration by parts on handbook Apply integration by parts on handbook Answer is (B)

+ Evaluating integrals Evaluate the indefinite integral of f(x) = cos 2 x sin x Evaluate the indefinite integral of f(x) = cos 2 x sin x (A) -2/3 sin 3 x + C (A) -2/3 sin 3 x + C (B) -1/3 cos 3 x + C (B) -1/3 cos 3 x + C (C) 1/3 sin 3 x + C (C) 1/3 sin 3 x + C (D) 1/2 sin 2 x cos 2 x + C (D) 1/2 sin 2 x cos 2 x + C Alternative method is to differentiate answers Alternative method is to differentiate answers Answer is (B)

+ Applications of integrals What is the area of the curve bounded by the curve f(x) = sin x and the x-axis on the interval [  /2, 2  ]? What is the area of the curve bounded by the curve f(x) = sin x and the x-axis on the interval [  /2, 2  ]? (A) 1 (A) 1 (B) 2 (B) 2 (C) 3 (C) 3 (D) 4 (D) 4 Need absolute value because sin x is negative over interval [ , 2  ] Need absolute value because sin x is negative over interval [ , 2  ] Answer is (C)

+ Differential Equations

+

+

+

+ Differential equations What is the general solution to the differential equation y’’ – 8y’ + 16y = 0? What is the general solution to the differential equation y’’ – 8y’ + 16y = 0? (A) y = C 1 e 4x (A) y = C 1 e 4x (B) y = (C 1 + C 2 x)e 4x (B) y = (C 1 + C 2 x)e 4x (C) y = C 1 e -4x + C 1 e 4x (C) y = C 1 e -4x + C 1 e 4x (D) y = C 1 e 2x + C 2 e 4x (D) y = C 1 e 2x + C 2 e 4x Solving 2nd order differential eqns on handbook Solving 2nd order differential eqns on handbook Answer is (B)

+ Laplace transforms Find the Laplace transform of the equation f”(t) + f(t) = sin  t where f(0) and f’(0) = 0 Find the Laplace transform of the equation f”(t) + f(t) = sin  t where f(0) and f’(0) = 0 (A) F(s) =  / [(1 + s 2 )(s 2 +  2 )] (A) F(s) =  / [(1 + s 2 )(s 2 +  2 )] (B) F(s) =  / [(1 + s 2 )(s 2 -  2 )] (B) F(s) =  / [(1 + s 2 )(s 2 -  2 )] (C) F(s) =  / [(1 - s 2 )(s 2 +  2 )] (C) F(s) =  / [(1 - s 2 )(s 2 +  2 )] (D) F(s) = s / [(1 - s 2 )(s 2 +  2 )] (D) F(s) = s / [(1 - s 2 )(s 2 +  2 )] Laplace transforms on handbook Laplace transforms on handbook Answer is (A)

+ Preguntas? Comentarios?

+ Muchas Gracias !