MCV4U - Practice Mastery Test #9 & #10
|
|
- Jacob Lester
- 5 years ago
- Views:
Transcription
1 Name: Class: Date: ID: A MCV4U - Practice Mastery Test #9 & #10 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. x + 2 is a factor of i) x 3 3x 2 + 6x 8 ii) x a. i only b. ii only c. i and ii d. neither 2. If f(x) = 2-5x and f(a) = -13 then a = a. -3 b. 3 c d Which of the following limits do not exist? i) lim 3 x ii) lim 3 x iii) lim 3 x x 3 + x 3 x 2 a. i) only b. ii) only c. iii) only d. i) and iii) x + h x 4. The expression lim h 0 h is most likely to be... a. the derivative of a function c. slope of a secant b. the value of a derivative d. none of the above 5. If s(t) = t 3 + 2t then v(t) = a. 3t b. 3t 2 c. 6t + 2 d t 6. The number lines for y,, and d 2 y are shown below. All zeros are shown. Assume the function is 2 continuous for all x ò. U means undefined y U d 2 y U The x-coordinates of all points of inflection are a. -3, -1 and 1 b. -3, -1 and 3 c. -3 and 1 d. -3 and 3 e. none 1
2 Name: ID: A 7. If y = 3x 2 ˆ ( x + 3), then = a. ( 6x) ( x + 3) + ( 1) 3x 2 ˆ c. ( 6x) ( 1) + ( x + 3) 3x 2 ˆ b. ( 6x)(1) 3x 2 ˆ ( x + 3) d. ( 6x ) 3x 2 ˆ + ( x + 3) 8. If y = ( 3x + 2) x 2 ˆ, then = a. ( 3x + 2) x 2 ˆ + ( 3) ( 2x + 3) c. ( 3) ( 2x + 3) + ( 3x + 2) x 2 ˆ b. ( 3) ( 2x + 3) ( 3x + 2) x 2 ˆ d. ( 3) x 2 ˆ + ( 2x + 3) ( 3x + 2) 9. Given f(x) = 4 x and g(x) = x + 2. If k(x) = f û g(x), then the domain of y=k(x) is... Ï a. Ô Ì x x 2 or x 2, x ò Ô Ï ÓÔ Ô c. Ô Ì x 2 x 2, x ò Ô ÓÔ Ô Ï b. Ô Ì x x 2, x ò Ô Ï ÓÔ Ô d. Ô Ì x x 2, x ò Ô ÓÔ Ô 10. If y = 3x 2 5 ˆ, then = a. 5 3x 2 4 ˆ 4 ( 6x) c. 5( 6x) 3x 2 ˆ b. 3x 2 4 ˆ 4 ( 6x) d. 5( 6x ) 11. Complete the identity tanx = a. b. 1 c. sin 2 x d. sinx cos 2 x 12. If y = cos 4x 2 ˆ then a. = 8x cos 4x 2 ˆ b. = 8x sin 4x 2 ˆ c. = 4x 2 ˆ cos( 8x) + 8x sin 4x 2 ˆ d. = 4x 2 ˆ sin( 8x) 8x cos 4x 2 ˆ 13. What is the next number in the sequence 10,30,90,270,...? a. 810 b. 450 c. 360 d
3 Name: ID: A 14. If y = 6ln(x), then = a. 6 x b. 6 6x (It is not necessary to state restrictions) c. 6 x d. 6 x 15. If y = 5e 2x, then = (It is not necessary to state restrictions) a. 10e 2x b. 10xe x 2 c. 2e 2x d. 10e 2x 16. In the prism shown, FA to... is equal a. a + b + c b. a + b c 17. Given a 18. Given a 19. Given a = ( 1,2, 4) then a = c. a a. 21 c. 7 b. 3 d. 21 = (2, 3, 2) then 3a b c d. a b + c is equal to a. 6, 9, 6 ˆ c. 6, 9,1 ˆ b. 5,0, 6 ˆ d. 5, 9,1 ˆ = (1,2, 3) and b = (0, 2,3) then 2a b is equal to ˆ ˆ a. 2,6, 9 ˆ c. 2,6,3 b. 2,2, 9 ˆ d. 2,6,3 3
4 Name: ID: A 20. Given the following vectors where aä = 10 and b ä = 9 then a b is approximately a b. 45 c d Given the following vectors where aä = 8 and b ä = 9 then a b is approximately a b. 36 c d Given the following vectors where aä = 6 and b ä = 8 then a b is approximately a b c d
5 Name: ID: A 23. In ò 3,the system of equations x y + z = 3 3x 3y + 3z = 9 defines a. a point b. a line c. a plane d. no points 24. Given eqtn #1: 2x + 2y + 2z = 2 and eqtn #2: r = 2, 2, 2 ˆ + t(0,0,1),t ò The point wth coordinates 2, 2, 1ˆ lies on the graphs of... a. both b. neither c. only eqtn #1 d. only eqtn #2 25. The equation r = (0,3, 1) + s(0,0,0) where s ò defines a. a point b. a line c. a plane d. no points 26. The equation r = (3, 2,1) defines a. a point b. a line c. a plane d. no points 27. The equation r = ( 3,2, 1) + s( 3,1,2) + t( 9,1,2) where s,t ò defines a. a point b. a line c. a plane d. no points 28. In ò 3,the system of equations 2x + 2y = 6 6x + 6y = 17 x + 3 = 0 defines a. a point b. a line c. a plane d. no points 29. Given eqtn #1: 2x + 2y + 2z = 5 and eqtn #2: r = 1,2,3 ˆ + t( 1,1, 1),t ò The point wth coordinates 2,3,1ˆ lies on the graphs of... a. both b. neither c. only eqtn #1 d. only eqtn #2 30. Given eqtn #1: 3x 2y + z = 9 and eqtn #2: r = 4,3, 1 ˆ + t(1, 2,2),t ò The point wth coordinates 3,1,1ˆ lies on the graphs of... a. both b. neither c. only eqtn #1 d. only eqtn #2 5
6 MCV4U - Practice Mastery Test #9 & #10 Answer Section MULTIPLE CHOICE 1. ANS: B if f(x) = x 3 3x 2 + 6x 8, then f( 2) = ( 2) 3 3( 2) 2 + 6( 2) 8 0 So...x + 2 is not a factor of x 3 3x 2 + 6x 8 If g(x) = x 3 + 8, then g( 2) = ( 2) = 0 So...x + 2 is a factor of x ANS: B The easiest way to solve these is probably to just guess and check. However, an algebraic solution follows: f(x) = 2-5x f(a) = 2-5a -13 = 2-5a 5a = 15 a = 3 3. ANS: A As x 3 +, x is greater than 3, so 3 x will negative, so exist. 3 x will be undefined, so lim x x does not As x 3, x is less than 3, so 3 x will positive, so lim 3 x = 0 x 3 3 x will be defined and will approach 0, so As x 2, from either side, x is less than 3, so 3 x will positive, 1, so lim x 3 3 x = 1 3 x will be defined, and will approach 1
7 4. ANS: A If f(x) = x, then the expression is of the form lim h 0 f(x + h) f(x) h which is the derivative. 5. ANS: A if s(t) = t 3 + 2t then v(t) = s (t) = 3t ANS: E At a point of inflection, the second derivative will change sign, (it changes from concave up/down to concave down/up), so there are none. There is a cusp at x= ANS: A y = 3x 2 ˆ ( x + 3), so = d dx 3x 2 ˆ ˆ ( x + 3) + d dx ( x + 3) ˆ ˆ 3x2 = ( 6x) ( x + 3) + ( 1) 3x 2 ˆ 8. ANS: D y = ( 3x + 2) x 2 ˆ, so = d dx ( 3x + 2) ˆ x 2 ˆ + d dx x 2 ˆ ˆ ( 3x + 2) = ( 3) x 2 ˆ + ( 2x + 3) ( 3x + 2) 2
8 9. ANS: D k(x) = f û g(x), then k(x) = f g(x) ˆ so k(x) = f g(x) ˆ = f( x + 2) = 4 ( x + 2) = 2 x For k(x) to be defined, 2 x 0 x 2 Ï The domain of y=k(x) is Ô Ì x x 2, x ò Ô ÓÔ Ô 10. ANS: A y = 3x 2 5 ˆ, so d 3x 2 5 ˆ = d 3x 2 ˆ d 3x 2 ˆ = 5 3x 2 4 ˆ ( 6x) 3
9 11. ANS: D We know that tanx = sinx tanx = = = = sinx sinx 1 sinx 1 sinx cos 2 x for all x. Substituting this in, we get Another method is to rearrange tanx = sinx (by multiplying both sides by 1 ) tanx = sinx 1 (tanx) = 1 tanx cosx = sinx cos 2 x sinx ˆ 12. ANS: B y = cos 4x 2 ˆ, so d cos 4x 2 ˆ = d cos 4x 2 ˆ d 4x 2 = d 4x 2 ˆ = sin 4x 2 ˆ 8x = 8x sin 4x 2 ˆ (chain rule) 4
10 13. ANS: A Each term is 3 times the previous term (check by dividing consecutive terms), so if we take 270 and multiply it by 3, we get ANS: C d ln(x) = 6 ˆ + d 1 = 6 1 x + 0 = 6 x 15. ANS: D 2xˆ d e = 5 d (2x) d ( 2x) = 5e 2x 2 = 10e 2x 16. ANS: C In the prism shown, to get from F to A, follow a path along the edges. Go from F to E, then E to H, and then H to A, so... FA = FE + EH + HA = a b c because FE is in the opposite direction to a, EH is in the opposite direction to b and HA is in the opposite direction to c 5
11 17. ANS: A Starting at the origin, the 1st component of the vector comes out towards us (if positive) or goes back into the page (if negative). In this case the 1 is shown in green. The 2nd component of the vector moves left or right from there. In this case the 2 is shown in blue. These two vectors form a right triangle with hypotenuse of length ( 1) 2 + ( 2) 2 When we move up or down using the 3rd component, the former hypotenuse forms a right triangle with the red vector. The new hypotenuse has length ( 1) 2 + ( 2) 2 + ( 4) 2 = 21 So the length of the vector in simplest form is ANS: A 3a + b = 3 2, 3, 2ˆ = 3(2),3( 3),3( 2) ˆ = 6, 9, 6ˆ 6
12 19. ANS: A 2a b = 2 1,2, 3ˆ 0, 2,3ˆ = 2(1),2(2),2( 3) ˆ + (0), ( 2), (3) ˆ = 2,6, 9ˆ 20. ANS: A Because we know the angle between the vectors, to find a a b = a b cos(θ) = (10)(9) cos(30 ) ANS: A Because we know the angle between the vectors, to find a a b = a b cos(θ) = (8)(9)cos(150 ) 62.4 b b, use the formula..., use the formula ANS: B Because we know the angle between the vectors, to find a b = a b sin(θ) = (6)(8) sin(35 ) 27.5 a b use the formula... 7
13 23. ANS: C Think of equations as restrictions on the freedom variables naturally have. Both of these equations define a plane in ò 3, so they would normally intersect in a line. However, in this case the 2nd equation is exactly 3 the first, so any point that satisfies the 1st equation will also satisfy the 2nd. This means that the planes intersect in a plane - with equation x y + z = 3 (or 3x 3y + 3z = 9). 24. ANS: A By substitution, we can see that 2, 2, 1ˆ satisfies eqtn#1. I.e., 2(2) + 2( 2) + 2( 1) = 2. If we use t=1, we see that the vector eqtn gives r = 2, 2, 2 ˆ + (1)(0,0,1) or r = 2, 2, 1 ˆ. So the vector eqtn also generates the point at 2, 2, 1ˆ. So... it lies on both. 25. ANS: A The equation r = (0,3, 1) + s(0,0,0) takes us from the origin out to the point (0,3, 1) and then travels from there along multiples of (0,0,0)... in other words it doesn t move from the point at (0,3, 1). So... this equation defines a point. 26. ANS: A The equation r = (3, 2,1) takes us from the origin out to the point (3, 2,1) and then stays there! So... this equation defines a point. 27. ANS: C The equation r = ( 3,2, 1) + s( 3,1,2) + t( 9,1,2) takes us from the origin out to the point ( 3,2, 1) and then travels from there along multiples of ( 3,1,2) and along multiples of ( 9,1,2). Note that the direction vectors in this equation are NOT parallel because ( 9,1,2) is not a multiple of ( 3,1,2). This means that adding multiples of ( 9,1,2) moves us off the line already formed by travelling along multiples of ( 3,1,2).... in other words the equation defines all the points on the plane through ( 3,2, 1) with direction vectors ( 3,1,2) and ( 9,1,2). 8
14 28. ANS: D Think of equations as restrictions on the freedom variables naturally have. All of these equations define a plane in ò 3, so they would normally intersect in a point. However, in this case the left side of the 2nd equation is exactly 3 the left side of the first, so any point that satisfies the 1st equation would satisfy 3 2x + 2yˆ = 3( 6), or 6x + 6y = 18. This means that any point on the 1st plane could NOT be on the second plane. Although the third equation defines a plane that does intersect each of the other two planes, this means that this system defines a set of NO points, because there are no points in common to all three planes. 29. ANS: B By substitution, we can see that 2,3,1ˆ does NOT satisfy eqtn#1. I.e., 2( 2) + 2(3) + 2(1) = 4 NOT 5. If we use t=1, we see that the vector eqtn gives r = 1,2,3 ˆ + (1)( 1,1, 1) or r = 2,3,2 ˆ. So the vector eqtn gives us the point at 2,3,2ˆ, but not 2,3,1ˆ. So... 2,3,1ˆ does not lie on the graphs of either eqtn. 30. ANS: D By substitution, we can see that 3,1,1ˆ does NOT satisfy eqtn#1. I.e., 3( 3) 2(1) + (1) = 8 NOT 9. If we use t=1, we see that the vector eqtn gives r = 4,3, 1 ˆ + (1)(1, 2,2) or r = 3,1,1 ˆ. So the vector eqtn does give us the point at 3,1,1ˆ. So... 3,1,1ˆ only lies on the graph of eqtn #2 9
MCV4U - Practice Mastery Test #1
Name: Class: Date: ID: A MCVU - Practice Mastery Test # Multiple Choice Identify the choice that best completes the statement or answers the question.. Solve a + b = a b = 5 a. a= & b=- b. a=- & b= c.
More informationMCR3U - Practice Mastery Test #6
Name: Class: Date: MCRU - Practice Mastery Test #6 Multiple Choice Identify the choice that best completes the statement or answers the question.. Factor completely: 4x 2 2x + 9 a. (2x ) 2 b. (4x )(x )
More informationTest 3 Review. fx ( ) ( x 2) 4/5 at the indicated extremum. y x 2 3x 2. Name: Class: Date: Short Answer
Name: Class: Date: ID: A Test 3 Review Short Answer 1. Find the value of the derivative (if it exists) of fx ( ) ( x 2) 4/5 at the indicated extremum. 7. A rectangle is bounded by the x- and y-axes and
More informationSummer Review for Students Entering AP Calculus AB
Summer Review for Students Entering AP Calculus AB Class: Date: AP Calculus AB Summer Packet Please show all work in the spaces provided The answers are provided at the end of the packet Algebraic Manipulation
More informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More information. CALCULUS AB. Name: Class: Date:
Class: _ Date: _. CALCULUS AB SECTION I, Part A Time- 55 Minutes Number of questions -8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using
More informationAP Calculus AB Unit 3 Assessment
Class: Date: 2013-2014 AP Calculus AB Unit 3 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationTest for Increasing and Decreasing Theorem 5 Let f(x) be continuous on [a, b] and differentiable on (a, b).
Definition of Increasing and Decreasing A function f(x) is increasing on an interval if for any two numbers x 1 and x in the interval with x 1 < x, then f(x 1 ) < f(x ). As x gets larger, y = f(x) gets
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
More information14 Increasing and decreasing functions
14 Increasing and decreasing functions 14.1 Sketching derivatives READING Read Section 3.2 of Rogawski Reading Recall, f (a) is the gradient of the tangent line of f(x) at x = a. We can use this fact to
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationState Precalculus/Trigonometry Contest 2008
State Precalculus/Trigonometry Contest 008 Select the best answer for each of the following questions and mark it on the answer sheet provided. Be sure to read all the answer choices before making your
More informationMATH 1902: Mathematics for the Physical Sciences I
MATH 1902: Mathematics for the Physical Sciences I Dr Dana Mackey School of Mathematical Sciences Room A305 A Email: Dana.Mackey@dit.ie Dana Mackey (DIT) MATH 1902 1 / 46 Module content/assessment Functions
More informationMATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed
MATH 11 FINAL EXAM FALL 01 Part I, No Calculators Allowed 1. Evaluate the limit: lim x x x + x 1. (a) 0 (b) 0.5 0.5 1 Does not exist. Which of the following is the derivative of g(x) = x cos(3x + 1)? (a)
More informationTest one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More information+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent
dx = (A) 3 sin(3x ) + C 1. cos ( 3x) 1 (B) sin(3x ) + C 3 1 (C) sin(3x ) + C 3 (D) sin( 3x ) + C (E) 3 sin(3x ) + C 6 3 2x + 6x 2. lim 5 3 x 0 4x + 3x (A) 0 1 (B) 2 (C) 1 (D) 2 (E) Nonexistent is 2 x 3x
More informationMath Fall 08 Final Exam Review
Math 173.7 Fall 08 Final Exam Review 1. Graph the function f(x) = x 2 3x by applying a transformation to the graph of a standard function. 2.a. Express the function F(x) = 3 ln(x + 2) in the form F = f
More informationLimits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4
Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationMath 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 informationThere 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 informationMath 131 Final Exam Spring 2016
Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing
More informationMATH 408N PRACTICE FINAL
05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationM152: 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 information1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationMath2413-TestReview2-Fall2016
Class: Date: Math413-TestReview-Fall016 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the value of the derivative (if it exists) of the function
More informationMLC Practice Final Exam
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 all your work on the standard response
More informationCalculus Example Exam Solutions
Calculus Example Exam Solutions. Limits (8 points, 6 each) Evaluate the following limits: p x 2 (a) lim x 4 We compute as follows: lim p x 2 x 4 p p x 2 x +2 x 4 p x +2 x 4 (x 4)( p x + 2) p x +2 = p 4+2
More informationWithout fully opening the exam, check that you have pages 1 through 11.
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 all your work on the standard response
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationAP Calculus ---Notecards 1 20
AP Calculus ---Notecards 1 20 NC 1 For a it to exist, the left-handed it must equal the right sided it x c f(x) = f(x) = L + x c A function can have a it at x = c even if there is a hole in the graph at
More informationPower 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 informationTopics and Concepts. 1. Limits
Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits
More informationMath 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
More informationn=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 information2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where
AP Review Chapter Name: Date: Per: 1. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the
More informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More informationMath 251, Spring 2005: Exam #2 Preview Problems
Math 5, Spring 005: Exam # Preview Problems. Using the definition of derivative find the derivative of the following functions: a) fx) = e x e h. Use the following lim =, e x+h = e x e h.) h b) fx) = x
More informationAP Calculus Multiple Choice Questions - Chapter 5
1 If f'(x) = (x - 2)(x - 3) 2 (x - 4) 3, then f has which of the following relative extrema? I. A relative maximum at x = 2 II. A relative minimum at x = 3 III. A relative maximum at x = 4 a I only b III
More informationCalculus & Analytic Geometry I
Functions Form the Foundation What is a function? A function is a rule that assigns to each element x (called the input or independent variable) in a set D exactly one element f(x) (called the ouput or
More informationMAT 122 Homework 7 Solutions
MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function
More informationTrig Identities, Solving Trig Equations Answer Section
Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ:
More informationTest 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a)
MATH 2250 Calculus I Eric Perkerson Test 3 Review Sections Covered: 3.11, 4.1 4.6. Topics Covered: Linearization, Extreme Values, The Mean Value Theorem, Consequences of the Mean Value Theorem, Concavity
More informationDecember Exam Summary
December Exam Summary 1 Lines and Distances 1.1 List of Concepts Distance between two numbers on the real number line or on the Cartesian Plane. Increments. If A = (a 1, a 2 ) and B = (b 1, b 2 ), then
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationSECTION 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 informationMath 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator
Math Test - Review Use differentials to approximate the following. Compare your answer to that of a calculator.. 99.. 8. 6. Consider the graph of the equation f(x) = x x a. Find f (x) and f (x). b. Find
More informationPrecalculus Summer Assignment 2015
Precalculus Summer Assignment 2015 The following packet contains topics and definitions that you will be required to know in order to succeed in CP Pre-calculus this year. You are advised to be familiar
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More informationBrief 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 informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationMath 112, Precalculus Mathematics Solutions to Sample for the Final Exam.
Math 11, Precalculus Mathematics Solutions to Sample for the Final Exam. Phone and calculator use is not allowed on this exam. You may use a standard one sided sheet of note paper. The actual final exam
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationM155 Exam 2 Concept Review
M155 Exam 2 Concept Review Mark Blumstein DERIVATIVES Product Rule Used to take the derivative of a product of two functions u and v. u v + uv Quotient Rule Used to take a derivative of the quotient of
More informationMATH 019: Final Review December 3, 2017
Name: MATH 019: Final Review December 3, 2017 1. Given f(x) = x 5, use the first or second derivative test to complete the following (a) Calculate f (x). If using the second derivative test, calculate
More informationChapter 4: More Applications of Differentiation
Chapter 4: More Applications of Differentiation Autumn 2017 Department of Mathematics Hong Kong Baptist University 1 / 68 In the fall of 1972, President Nixon announced that, the rate of increase of inflation
More information2.8 Linear Approximation and Differentials
2.8 Linear Approximation Contemporary Calculus 1 2.8 Linear Approximation and Differentials Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use
More informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationBlue Pelican Calculus First Semester
Blue Pelican Calculus First Semester Student Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function
More informationCalculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić. Name: Show all your work!
Calculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić Name: Show all your work! 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 2 f(x)
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More information(b) x = (d) x = (b) x = e (d) x = e4 2 ln(3) 2 x x. is. (b) 2 x, x 0. (d) x 2, x 0
1. Solve the equation 3 4x+5 = 6 for x. ln(6)/ ln(3) 5 (a) x = 4 ln(3) ln(6)/ ln(3) 5 (c) x = 4 ln(3)/ ln(6) 5 (e) x = 4. Solve the equation e x 1 = 1 for x. (b) x = (d) x = ln(5)/ ln(3) 6 4 ln(6) 5/ ln(3)
More informationWeek #6 - Taylor Series, Derivatives and Graphs Section 10.1
Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by
More information(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
More informationWithout fully opening the exam, check that you have pages 1 through 11.
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 all your work on the standard response
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationTable of Contents. Module 1
Table of Contents Module Order of operations 6 Signed Numbers Factorization of Integers 7 Further Signed Numbers 3 Fractions 8 Power Laws 4 Fractions and Decimals 9 Introduction to Algebra 5 Percentages
More informationc) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0
Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to
More informationCalculus. Summer Assignment
Summer Review Packet for All Students Enrolling in Calculus #160 Next Year. Name: To earn credit, show all necessary work to support your answer in the space provided. Calculus Summer Assignment Name This
More informationSection 3.5: Implicit Differentiation
Section 3.5: Implicit Differentiation In the previous sections, we considered the problem of finding the slopes of the tangent line to a given function y = f(x). The idea of a tangent line however is not
More informationMTH 132 Solutions to Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 11.
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 all your work on the standard response
More informationAnalytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7
Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions /19/7 Question 1 Write the following as an integer: log 4 (9)+log (5) We have: log 4 (9)+log (5) = ( log 4 (9)) ( log (5)) = 5 ( log
More informationWithout fully opening the exam, check that you have pages 1 through 11.
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 all your work on the standard response
More informationSection 4.2 Logarithmic Functions & Applications
34 Section 4.2 Logarithmic Functions & Applications Recall that exponential functions are one-to-one since every horizontal line passes through at most one point on the graph of y = b x. So, an exponential
More information1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim
Spring 10/MAT 250/Exam 1 Name: Show all your work. 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 1 +f(x) = lim x 3 f(x) = lim x
More informationMath 112, Precalculus Mathematics Sample for the Final Exam.
Math 11, Precalculus Mathematics Sample for the Final Exam. Phone use is not allowed on this exam. You may use a standard two sided sheet of note paper and a calculator. The actual final exam consists
More informationPRELIM 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 informationAnnouncements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 5.2 5.7, 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems
More informationSummer Math Packet for AP Calculus BC
Class: Date: Summer Math Packet for AP Calculus BC 018-19 1. Find the smallest value in the range of the function f (x) = x + 4x + 40. a. 4 b. 5 c. 6 d. 7 e. 8 f. 16 g. 4 h. 40. Find the smallest value
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationOne of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.
2.24 Tanz and the Reciprocals Derivatives of Other Trigonometric Functions One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the
More informationMore 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 informationCalculus I Practice Problems 8: Answers
Calculus I Practice Problems : Answers. Let y x x. Find the intervals in which the function is increasing and decreasing, and where it is concave up and concave down. Sketch the graph. Answer. Differentiate
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More informationNote: Final Exam is at 10:45 on Tuesday, 5/3/11 (This is the Final Exam time reserved for our labs). From Practice Test I
MA Practice Final Answers in Red 4/8/ and 4/9/ Name Note: Final Exam is at :45 on Tuesday, 5// (This is the Final Exam time reserved for our labs). From Practice Test I Consider the integral 5 x dx. Sketch
More informationExam 3. MA Exam 3 Fall Solutions. 1. (10 points) Find where the following functions are increasing and decreasing.
Exam 3 Solutions. (0 points) Find where the following functions are increasing and decreasing. (a) (5 points) f(x) = x2 + 2x 3, x 3 2. x2 f(x) = + 2x 3 f (x) = 2x( x3 ) ( + 2x 3 ) 2 Critical points: x
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationCalculus I Practice Final Exam A
Calculus I Practice Final Exam A This practice exam emphasizes conceptual connections and understanding to a greater degree than the exams that are usually administered in introductory single-variable
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationWeBWorK demonstration assignment
WeBWorK demonstration assignment.( pt) Match the statements defined below with the letters labeling their equivalent expressions. You must get all of the answers correct to receive credit.. x is less than
More informationMA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September Multiple Choice Answers. Question
MA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September 2017 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationÊ 7, 45 Ê 7 Ë 7 Ë. Time: 100 minutes. Name: Class: Date:
Class: Date: Time: 100 minutes Test1 (100 Trigonometry) Instructor: Koshal Dahal SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE FULL CREDIT. 1. Find the terminal point P (x, y) on the unit
More information