PASS MOCK EXAM FOR PRACTICE ONLY

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1 PASS MOCK EXAM FOR PRACTICE ONLY Course: MATH 4 ABCDEF Facilitator: Stephen Kimbell December 6 th 25 - (3:-6:) and (8:-2:) in ME 338 (Mackenzie Building 3 rd floor 3 rd block) It is most beneficial to you to write this mock final UNDER EXAM CONDITIONS. This means: Complete the mock final in 3 hour(s). Work on your own. Keep your notes and textbook closed. Attempt every question. After the time limit, go back over your work with a different colour or on a separate piece of paper and try to do the questions you are unsure of. Record your ideas in the margins to remind yourself of what you were thinking when you take it up at PASS. The purpose of this mock exam is to give you practice answering questions in a timed setting and to help you to gauge which aspects of the course content you know well and which are in need of further development and review. Use this mock exam as a learning tool in preparing for the actual exam. Please note: Come to the PASS session with your mock exam complete. There, you can work with other students to review your work. Often, there is not enough time to review the entire exam in the PASS session. Decide which questions you most want to review the Facilitator may ask students to vote on which questions they want to discuss. Facilitators do not bring copies of the mock exam to the session. Please print out and complete the exam before you attend. Facilitators do not produce or distribute an answer key for mock exams. Facilitators help students to work together to compare and assess the answers they have. If you are not able to attend the PASS session, you can work alone or with others in the class. *** I based the mock exam on my exam in 23 with a few minor changes. I think it will be a great representation for what you will see in your exam. I included a few extra longer answer problems if you are curious or interested in how calculus can be applied in engineering! Good Luck writing the Mock Exam!! DISCLAIMER: PASS handouts are designed as a study aid only for use in PASS workshops. Handouts may contain errors, intentional or otherwise. It is up to the student to verify the information contained within.

2 Multiple Choice Calculus Questions f(+h) f(3). Let f(x) = x + x 3. Calculate L = lim h + h a) b) c) - d) DNE 2. Let f(x) = cos (x) x 2 for x and f(x) = A, for x =. What value of A will make f continuous at a) b) /2 c) - d) 3. Evaluate L = lim x2 2x 3 x 3 x 2 9 a) b) 3/2 c) 2/3 d) DNE 4. Evaluate L = lim x sin (tx) sin (5x) a) /5 b) 5 c) t/5 d) DNE 5. Two functions f and g are defined by f(x) = 3x 2 and g(x) = cos(x). What is the value of f(g()) a) -3 b) 3 c) -3.2 d) 6. Find the derivative of the function y = 8x ln (5x+) a) 8(5x+) ln(5x+) 4x (5x+)(ln(5x+)) 2 b) ln (4x) c) 8 ln(5x+) 4x ln(5x+)(5x+) 2 d) 8 5ln(5x+) 7. Find the derivative of the function y = 5x 2 e 3x a) xe 3x (2x + 3) b) 5xe 3x (2x + 3) c) ex 3x (3x + 2) d) 5xe 3x (3x + 2) 8. Find the derivative of the function y = ln(e x2 + ) a) 2xex2 e x2 + b) 2x e x2 c) 2xe x2 ln(e x2 +) d) 2e x2 (e x2 +) 2 9. Find any local maximum or minimum points of the given function y = x 3 3x 2 + a) Minimum at (,), maximum at (2,-3) b) Minimum at (2,-3), maximum at (,) c) Minimum at (2,-3), maximum at (-2,-9) and (,) d) Minimum at (2,-3). Which of the following statements is true? a) f(x) = 2e x is concave down for all x and has no POI b) f(x) = x 5 + is concave up for all x and has no POI c) f(x) = x is concave up for x <,concave down for x > and has a POI at (,5) d) f(x) = (x 5) 3 is concave up for x > 5,concave down for x < 5 and has a POI at (5,)

3 . Evaluate sec2 (lnx) dx x a) tan(lnx) + C b) ln(secx) + C c) 2sec(lnx) + C d) ln(tanx) + C π 2. Evaluate the definite integral sin 2 x 2 cos2 x dx 2 a) π 6 b) π 2 c) π 4 d) π 8 3. Evaluate I = e 4x sin ( x 2 ) dx a) 2 e4x (3 sin ( x ) + 4 cos 2 (x)) + C b) 2 23 e4x (2 sin ( x ) + 3 cos 2 (x )) + C 2 c) 65 e4x (2 sin ( x ) + 6 cos 2 (x)) + C d) 2 5 e4x (sin ( x ) cos 2 (x )) + C 2 4. Evaluate the definite integral e x 3 (x 2 + 2x)dx 3 a) b) 27e-36 c) e- d) 2e+4 e 5. Evaluate the definite integral (xlnx) 2 dx a) e 4 b) e3 2 c) 5e d) e Evaluate I = 4 x 4 dx a) ln x ln x + 2 tan x + C b) ln x tan x + C c) ln x 4ln x + 2 tan x + C d) 2ln x + ln x + + tan x + C 7. Let f(x) = sin(sin3x). Evaluate f (π/2) a) b) c) 2 d) 3 arcsin (5x) 8. Evaluate the following limit: L = lim x x 2 a) 5 b) /5 c) d) DNE 9. Given that f is such that its inverse F exists, f ( 5) = 4, F(2) = 5, find the value of the derivative of F at x=2. a) 4 b) /4 c) 5 d) /5 2. Let y be given implicitly as a differentiable function of x by 2x = xy + y 2. Then the slope of the tangent line to the curve y = y(x) at the point (x, y) where x =, y = is equal to: a) 2 b) /2 c) 3 d) /3

4 f(5+h) f(5) 2. Let f(x) = 2 x 5. Calculate L = lim h h a) L= b) L=5 c) L=-5 d) DNE 22. Let f(x) = x Evaluate the second derivate of f at x= a) 4 b) c) /2 d) DNE 23. Find an expression for the volume V of the solid of revolution obtained by rotating the region bounded by the graph of y = x 3, y = x 2 +, x = and x = about the y axis. a) πx(x 2 x 3 )dx d) 2πx 2 ( x 2 x 3 )dx b) 2πx( + x 2 x 3 )dx c) 2πx( x 2 x 3 )dx 24. Evaluate the improper integral 3 2 x x e dx a) 2 b) 2 c) 6 d) 25. Find the area of the region enclosed by the curves y x, y 2x x 2 2 a) /3 b) 9 c) 2/3 d) 6 END OF MATH 4 CORE CONTENT

5 26. Number of Cycles to Failure due to Fatigue BONUS ENGINEERING QUESTIONS Parts in machines and structures have natural imperfections and may have small cracks created during production. These small cracks can propagate under cyclical stresses the structure or machine may be exposed. For example an engine turbine blade is under cyclical stresses during takeoff and landing when differing thrust levels are required. Due to changes in rotational speed, temperature, and pressure, resulting cyclical stresses are put on the blade. This is known as fatigue. The differential equation that describes the growth of the crack in terms of the number of cycles is: da dn = A(K)m Let s say we want to find how many cycles before the part will fail. How can we solve this? Hint: what are you solving for, also this involves INTEGRATION. dn da = A(K) m = A(Yσ πa) m Givens: Y =.45, parameters: A =2.*, m = 3., initial crack size: a o =.3* 3 m, stress: σ= MPa K = Yσ πa, stiffness of material = 55 MPa m, largest allowable crack size before fast fracture: a f =? N f =?

6 27. Differential Equation solution using Partial Fractions: With differential equations, sometimes the equations are too complicated to simply deal with it normally using conventional integration and differentiation in the time domain. Instead, you will learn in later years that you can transform the equation into a simpler algebraic form using the Laplace transform and convert the problem into the s-domain. This is what we will do here to solve the following differential equation. x + 2x + 5x = 3, x() =, x () = If we take the Laplace transform, we can convert the equation into the form: s 2 X(s) + 2sX(s) + 5X(s) = 3 s And then we can solve for the solution in the Laplace domain: X(s): X(s) = 3 s(s 2 +2s+5) However, at this point we want to convert back to the time domain using the inverse Laplace function; however because of the nature of the solution it is too difficult to determine the inverse directly. First, we have to use PARTIAL FRACTIONS to split up the fraction into an easier form to take the inverse. Your job is to find the partial fraction expansion of the solution X(s):

7 28. Secular Variation in Orbital Elements due to J2 First of all, Orbital mechanics is based upon inertial 2 body Keplerian conditions. Put more simply, basic orbital mechanics considers only the effect of one larger body (earth/sun/moon etc) and the smaller satellite body. Additionally, the kinematics of the motion of orbits is defined by the simple, perfect condition known as a Keplerian orbit. However, in real situations, there are perturbing forces that change the orbital elements. For example atmospheric drag, solar radiation, and most importantly J2 or the effect of the oblateness of the earth. Therefore, calculations must be made to accommodate for these perturbing accelerations. Secular Variation is the term that describes the change in an orbital element over one orbit. In other words it T can be described as a = da 2π da dt, it can also be written as a = dθ. dt θ dt From analysing orbital conditions & parameters of the perturbing acceleration, θ = μa a 2, da dt = 2a2 μa f θ Given: T= hrs convert to s f θ =.5sinθ a = km (semi major axis) μ = 3986km (gravitational parameter) Find the secular change in semi major axis a:

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