Math 140 Final Sample A Solutions. Tyrone Crisp

Size: px
Start display at page:

Download "Math 140 Final Sample A Solutions. Tyrone Crisp"

Transcription

1 Math 4 Final Sample A Solutions Tyrone Crisp (B) Direct substitution gives, so the limit is infinite. When is close to, but greater than,, the numerator is negative while the denominator is positive. So the limit is negative infinity. (E) Calculate the one-sided limits separately. As +, we have =, so the function is equal to and therefore has limit. As, we have =, so the function is equal to and therefore has limit. The one-sided limits are different, so the limit doesn t eist. (C) Direct substitution! 4 (D) Looking only at leading terms gives lim 6 9 = lim 6 = 6. 5 (A) Use the quotient rule and the identity cos + sin = : ( ) d sin ( cos )(cos ) sin (sin ) = = cos d cos ( cos ) (cos ) = cos. 6 (A) Write f() = ( cos(() / ) ) /, and use the chain rule (four times!): f () = ( cos(() / ) ) / ( sin(() / )) ( () / ) = sin cos. 7 (E) We need to know the slope of the tangent, i.e. y. Use implicit differentiation: take derivatives of both sides with respect to, using the product rule on 4y: yy +4(y +y)+ =. Now set = and y = and solve for y. You find y + 8y + 8 = so y = 4. We could 5 now use the point-slope formula, or just notice that answer E is the only one with the correct slope. 8 (A) Let r be the radius, A the area. Given: dr relating A and r: A = πr. Differentiate both sides with respect to t: da knowns and solve for the unknown: da π m /s. = π = /. Find: da when r =. Equation dr = πr. Substitute the = π. So the area is increasing at a rate of 9 (E) Possible local etrema occur at critical numbers. To find the critical numbers, first take the derivative: f () = /, and then find those in the domain of f for which f () is zero or doesn t eist. Now, the domain of f contains all real numbers, so no need to worry about that. We have f () = when / =, i.e. = 8. On the other hand, f () doesn t eist when = (because of the negative power of ). So the critical numbers are and 8. Now, apply the first-derivative test to each critical number. (You could also use the secondderivative test at 8, but not at because the second derivative is not defined at ). When is a little bigger than, the quantity / is a huge positive number, so f () >. When is a little smaller than, the quantity / is a huge negative, so f () <. So, by the first-derivative test, f has a local minimum at. Similarly, we find that f has a local maimum at 8. (A) Vertical asymptotes: First, we must CANCEL COMMON FACTORS. So cancel the ( 6) from top and bottom. There is now only one factor in the denominator, and hence just one vertical asymptote (namely = ).

2 Horizontal asymptotes: since the degree of the numerator is greater than that of the denominator, this function will have infinite limits at ±, and hence no horizontal asymptote. Slant asymptote: the degree of the numerator is eactly one greater than that of the denominator, so the graph will have a slant asymptote. (C) Antidifferentiate a(t), to get v(t) = sin t + C. To find C, use the fact that when t = π/, v =. So = sin(π/) + C, giving C =. So v(t) = sin t +. Now antidifferentiate v(t), to get s(t) = cos(t) + t + D. To find D, use the fact that when t = π/, s = π/. So π = cos π + π + D, giving D =. So s(t) = cos t + t +. (A)Let u = 4, so du = d. Inserting appropriate factors of and, the integral becomes ( ) 4 d = u / du. To get rid of the, notice that = 4 u by our choice of u. So the integral is equal to (4 u)u / du = 4u / u 4/ du = u 4/ 4u / du. (E) Let u = + /, so du = d. Since this is a definite integral, we must also translate the limits of integration. When =, u =. When = 4, u =. So inserting appropriate factors of and, we get 4 ( + ) d = u du = [ u ] [ = ( )] =. 4 (B) Since the denominator consists of a single term, we can simplify the integrand by writing it as a sum: 4 + = 4 + = +. We then have + d = [ ] = [4 4 ] ( ) = Terrible, terrible question. They want you to use the FTC and the chain rule. BUT the FTC is only applicable at those for which the integrand is continuous on the interval between sin and π. Since sin < π for all, this interval always contains the number π/, where the secant function has an infinite discontinuity. The upshot is, that it s not clear whether the function you re supposed to be differentiating is even defined for any value of at all! (It turns out that this function is actually defined for all, because the positive and negative infinite limits of the secant function on either side of π/ cancel each other out in some sense. But in this course, you re not assumed to know about this.) 6 (A) To integrate an absolute value, break the integral up into two pieces. Since { if = if <,

3 we should split the integral up at. We get + d = + ( ) d + + d = d = [ ] = 4. 7 (D) First choose which method to use: the disks/washers method will tell us to integrate with respect to (i.e. parallel to the ais of rotation), and this is good since the boundary curve is given in the form y = f(). So use the disks/washers method. We re integrating with respect to, so the limits of integration will be the smallest and largest values of in the solid. These are and 5. So the volume will be V = 5 A() d, where A() represents the area of the vertical cross-section through. This cross-section will be a circle of radius, so we have A() = π( ) = π( ). The volume is therefore 5 [ ] 5 [ ( )] 5 4 V = π( ) d = π = π 5 = 5π. 8 (A) First notice that the graph will have vertical asymptotes at = ±. We can eliminate (c), which has an etraneous vertical asymptote at =. Now look at the behaviour of f near the vertical asymptotes. For eample, we have lim f() = and lim f() =, + which rules out (b), (d) and (e). The only option left is (a). (Note: this is certainly not the only way to solve this problem.) 9 (B:False) For f to be continuous at a, we require also that f(a) be defined and equal to the limit. (B:False) Just about any function f you can think of will give a counter eample. The point is that the integral behaves nicely with respect to additive operations (like adding two functions together), but NOT with respect to multiplicative operations (like multiplying two functions, taking the square, etc.). (A:True) The thing inside the parentheses doesn t depend on, so it is constant with respect to and its derivative is therefore zero. The slope of the tangent line is dy = d 5 4. The problem is to find the value(s) of which maimise this quantity: i.e., we re trying to maimise the function f() = dy = d 5 4. So, we look at the derivative of f (which will be the second derivative of y): f () = d y d = 4 6. Possible maima occur when f () =, or f () doesn t eist. Since f () eists for all, there are no critical numbers of the second kind. We can factor f () = 6(4 ), so the critical numbers are, and. Apply either the first- or the second-derivative test to each of these. I ll use the second-derivative test, which involves looking at (Note that this is now the third derivative of y.) f () = 4 8.

4 4 When =, f () = 4 >, so f has a local minimum at. When = ±, f () < and so f has local maima at = ±. Now, f is an even function, so f() = f( ). We now want to argue that this common local maimum value is actually the global maimum of f. But f is a polynomial, with lim f() = lim f() =. This means that f has a global maimum value, which must then occur at a local maimum. We found that the only local maima are at = ±, and that f has the same value at these two points, so these values of give the global maimum. So, the slope of the tangent line to the given graph is steepest at = ±. Draw a picture. The curves intersect at = and =. Since neither curve is on top of the other throughout the whole interval [, ], we must compute the area as a sum of two integrals. On the interval [, ], the line is on top of the parabola, so the area of that piece is found by integrating. On the interval [, ], the parabola is above the line, so the area of that piece is found by integrating. So the area is A = d+ [ d = ] + [ ] = ( ) =. 4 (a) I would first just sketch y = 8/ and y =. They intersect at one point, and in order to make our picture accurate we need to know if the line = goes to the left or to the right of this point. So, find the coordinates of the intersection point, by solving 8 =. The solution is =. So the intersection point is (, 4), and the line = therefore lies to the left of the intersection point. Strictly speaking, there are now two more intersection points: the point (, 8) where the line intersects y = 8/, and the point (, ) where the line intersects the parabola. You should label these as well. 4 (b) Choose which method to use: For disks/washers I d need to integrate parallel to the ais of rotation, i.e. with respect to. For cylindrical shells I d need to integrate perpendicular to the ais of rotation, i.e. with respect to y. My boundary curves are both given in the form y = f() i.e. is set up as the independent variable so I prefer to integrate with respect to. So I ll use disks/washers. Integrating with respect to, the limits of integration will be the smallest and largest -values in the solid. In part (a), we found that these are respectively and. So the volume is A() d. A() is the area of the annulus obtained by taking a vertical slice through. This annulus has outer radius R = 8 (the distance from the ais of rotation up to the top curve), and inner radius r = (the distance from the ais of rotation to the bottom curve). So ( ) 8 A() = π π( ) = π ( 64 4). The volume is therefore given by V = π(64 4 ) d. 4 (c) Choose the method: The boundary curves are still in the form y = f(), so I still want to integrate with respect to. Since we re now rotating around a vertical ais, it is now the cylindrical shells method that involves integrating with respect to. So I use cylindrical shells.

5 Again integrating with respect to, the limits of integration are still the largest and smallest values of in the region, which are still and respectively. So the volume is S() d, where S() is the surface area of the cylinder through. This cylinder has radius (the distance from the ais of rotation out to the edge of the cylinder), and it has height 8 (top y-coordinate minus bottom y-coordinate). So ( ) 8 S() = π = π(8 ), and the volume is therefore given by V = π(8 ) d. 5

Math 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.

Math 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line. Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through

More information

Math 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.

Math 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line. Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through

More information

Review: Limits of Functions - 10/7/16

Review: Limits of Functions - 10/7/16 Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition 1.0.1 We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left

More information

Review sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston

Review sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar

More information

November 13, 2018 MAT186 Week 8 Justin Ko

November 13, 2018 MAT186 Week 8 Justin Ko 1 Mean Value Theorem Theorem 1 (Mean Value Theorem). Let f be a continuous on [a, b] and differentiable on (a, b). There eists a c (a, b) such that f f(b) f(a) (c) =. b a Eample 1: The Mean Value Theorem

More information

4.3 - How Derivatives Affect the Shape of a Graph

4.3 - How Derivatives Affect the Shape of a Graph 4.3 - How Derivatives Affect the Shape of a Graph 1. Increasing and Decreasing Functions Definition: A function f is (strictly) increasing on an interval I if for every 1, in I with 1, f 1 f. A function

More information

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13 Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................

More information

Math 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it.

Math 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it. Math 80, Final Eam, Spring 008 Problem Solution. For each of the following limits, determine whether the limit eists and, if so, evaluate it. + (a) lim 0 (b) lim ( ) 3 (c) lim Solution: (a) Upon substituting

More information

Answers for Ch. 6 Review: Applications of the Integral

Answers for Ch. 6 Review: Applications of the Integral Answers for Ch. 6 Review: Applications of the Integral. The formula for the average value of a function, which you must have stored in your magical mathematical brain, is b b a f d. a d / / 8 6 6 ( 8 )

More information

(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,

(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6, Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),

More information

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2 Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

More information

MATH 1325 Business Calculus Guided Notes

MATH 1325 Business Calculus Guided Notes MATH 135 Business Calculus Guided Notes LSC North Harris By Isabella Fisher Section.1 Functions and Theirs Graphs A is a rule that assigns to each element in one and only one element in. Set A Set B Set

More information

Solutions to Math 41 Exam 2 November 10, 2011

Solutions to Math 41 Exam 2 November 10, 2011 Solutions to Math 41 Eam November 10, 011 1. (1 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it is or.

More information

10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.

10550 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 information

Module 2: Reflecting on One s Problems

Module 2: Reflecting on One s Problems MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations

More information

MATH section 3.4 Curve Sketching Page 1 of 29

MATH section 3.4 Curve Sketching Page 1 of 29 MATH section. Curve Sketching Page of 9 The step by step procedure below is for regular rational and polynomial functions. If a function contains radical or trigonometric term, then proceed carefully because

More information

AP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015

AP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015 AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted

More information

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes

More information

Integrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61

Integrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61 Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up

More information

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work.

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Spring 008 Test #1 Find the following its. For each one, if it does not eist, tell why not. Show all necessary work. 1.) 4.) + 4 0 1.) 0 tan 5.) 1 1 1 1 cos 0 sin 3.) 4 16 3 1 6.) For

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this

More information

Objectives List. Important Students should expect test questions that require a synthesis of these objectives.

Objectives List. Important Students should expect test questions that require a synthesis of these objectives. MATH 1040 - of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List

More information

Calculus 1: Sample Questions, Final Exam

Calculus 1: Sample Questions, Final Exam Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)

More information

Solutions to Math 41 Final Exam December 9, 2013

Solutions to Math 41 Final Exam December 9, 2013 Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

Math 2250 Exam #3 Practice Problem Solutions 1. Determine the absolute maximum and minimum values of the function f(x) = lim.

Math 2250 Exam #3 Practice Problem Solutions 1. Determine the absolute maximum and minimum values of the function f(x) = lim. Math 50 Eam #3 Practice Problem Solutions. Determine the absolute maimum and minimum values of the function f() = +. f is defined for all. Also, so f doesn t go off to infinity. Now, to find the critical

More information

PACKET Unit 4 Honors ICM Functions and Limits 1

PACKET Unit 4 Honors ICM Functions and Limits 1 PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.

More information

Math 31A Discussion Session Week 1 Notes January 5 and 7, 2016

Math 31A Discussion Session Week 1 Notes January 5 and 7, 2016 Math 31A Discussion Session Week 1 Notes January 5 and 7, 2016 This week we re discussing two important topics: its and continuity. We won t give a completely rigorous definition of either, but we ll develop

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

MATH140 Exam 2 - Sample Test 1 Detailed Solutions

MATH140 Exam 2 - Sample Test 1 Detailed Solutions www.liontutors.com 1. D. reate a first derivative number line MATH140 Eam - Sample Test 1 Detailed Solutions cos -1 0 cos -1 cos 1 cos 1/ p + æp ö p æp ö ç è 4 ø ç è ø.. reate a second derivative number

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Math 1500 Fall 2010 Final Exam Review Solutions

Math 1500 Fall 2010 Final Exam Review Solutions Math 500 Fall 00 Final Eam Review Solutions. Verify that the function f() = 4 + on the interval [, 5] satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that

More information

1993 AP Calculus AB: Section I

1993 AP Calculus AB: Section I 99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among

More information

Final practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90

Final practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90 Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x

More information

Understanding Part 2 of The Fundamental Theorem of Calculus

Understanding Part 2 of The Fundamental Theorem of Calculus Understanding Part of The Fundamental Theorem of Calculus Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is

More information

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal) TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents

More information

3.3 Limits and Infinity

3.3 Limits and Infinity Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult

More information

Calculus I Review Solutions

Calculus 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 information

Curriculum Framework Alignment and Rationales for Answers

Curriculum Framework Alignment and Rationales for Answers The multiple-choice section on each eam is designed for broad coverage of the course content. Multiple-choice questions are discrete, as opposed to appearing in question sets, and the questions do not

More information

Work the following on notebook paper. You may use your calculator to find

Work the following on notebook paper. You may use your calculator to find CALCULUS WORKSHEET ON 3.1 Work the following on notebook paper. You may use your calculator to find f values. 1. For each of the labeled points, state whether the function whose graph is shown has an absolute

More information

MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation

MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between

More information

Solutions to Math 41 First Exam October 12, 2010

Solutions to Math 41 First Exam October 12, 2010 Solutions to Math 41 First Eam October 12, 2010 1. 13 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it

More information

Properties of Derivatives

Properties of Derivatives 6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve

More information

Math 20C Homework 2 Partial Solutions

Math 20C Homework 2 Partial Solutions Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

More information

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions. October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L

More information

Math 19, Homework-1 Solutions

Math 19, Homework-1 Solutions SSEA Summer 207 Math 9, Homework- Solutions. Consider the graph of function f shown below. Find the following its or eplain why they do not eist: (a) t 2 f(t). = 0. (b) t f(t). =. (c) t 0 f(t). (d) Does

More information

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26. Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

More information

+ 2 on the interval [-1,3]

+ 2 on the interval [-1,3] Section.1 Etrema on an Interval 1. Understand the definition of etrema of a function on an interval.. Understand the definition of relative etrema of a function on an open interval.. Find etrema on a closed

More information

k y = where k is the constant of variation and

k y = where k is the constant of variation and Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

More information

Math 75B Practice Problems for Midterm II Solutions Ch. 16, 17, 12 (E), , 2.8 (S)

Math 75B Practice Problems for Midterm II Solutions Ch. 16, 17, 12 (E), , 2.8 (S) Math 75B Practice Problems for Midterm II Solutions Ch. 6, 7, 2 (E),.-.5, 2.8 (S) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual

More information

Math 180, Exam 2, Spring 2013 Problem 1 Solution

Math 180, Exam 2, Spring 2013 Problem 1 Solution Math 80, Eam, Spring 0 Problem Solution. Find the derivative of each function below. You do not need to simplify your answers. (a) tan ( + cos ) (b) / (logarithmic differentiation may be useful) (c) +

More information

Rational Functions. p x q x. f x = where p(x) and q(x) are polynomials, and q x 0. Here are some examples: x 1 x 3.

Rational Functions. p x q x. f x = where p(x) and q(x) are polynomials, and q x 0. Here are some examples: x 1 x 3. Rational Functions In mathematics, rational means in a ratio. A rational function is a ratio of two polynomials. Rational functions have the general form p x q x, where p(x) and q(x) are polynomials, and

More information

Math 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below.

Math 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below. Math Final Eam - Practice Problem Solutions. A function f is graphed below. f() 8 7 7 8 (a) Find f(), f( ), f(), and f() f() = ;f( ).;f() is undefined; f() = (b) Find the domain and range of f Domain:

More information

Precalculus Notes: Functions. Skill: Solve problems using the algebra of functions.

Precalculus Notes: Functions. Skill: Solve problems using the algebra of functions. Skill: Solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical Using Numerical Values: Look for a common difference. If the first difference

More information

Calculus 1st Semester Final Review

Calculus 1st Semester Final Review Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ), c /, > 0 Find the limit: lim 6+

More information

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23 Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical

More information

4.5 Rational functions.

4.5 Rational functions. 4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations

More information

MATH 101 Midterm Examination Spring 2009

MATH 101 Midterm Examination Spring 2009 MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.

More information

Chapter 5: Limits, Continuity, and Differentiability

Chapter 5: Limits, Continuity, and Differentiability Chapter 5: Limits, Continuity, and Differentiability 63 Chapter 5 Overview: Limits, Continuity and Differentiability Derivatives and Integrals are the core practical aspects of Calculus. They were the

More information

Solutions to the Exercises of Chapter 8

Solutions to the Exercises of Chapter 8 8A Domains of Functions Solutions to the Eercises of Chapter 8 1 For 7 to make sense, we need 7 0or7 So the domain of f() is{ 7} For + 5 to make sense, +5 0 So the domain of g() is{ 5} For h() to make

More information

Review of elements of Calculus (functions in one variable)

Review of elements of Calculus (functions in one variable) Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove

More information

It s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]

It s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4] It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)

More information

Midterm Exam #1. (y 2, y) (y + 2, y) (1, 1)

Midterm Exam #1. (y 2, y) (y + 2, y) (1, 1) Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are

More information

Troy High School AP Calculus Summer Packet

Troy High School AP Calculus Summer Packet Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by

More information

In economics, the amount of a good x demanded is a function of the price of that good. In other words,

In economics, the amount of a good x demanded is a function of the price of that good. In other words, I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with eactly one member of Y. That is, some goes in, and some y comes out. These notations are used to

More information

(d by dx notation aka Leibniz notation)

(d by dx notation aka Leibniz notation) n Prerequisites: Differentiating, sin and cos ; sum/difference and chain rules; finding ma./min.; finding tangents to curves; finding stationary points and their nature; optimising a function. Maths Applications:

More information

MATHEMATICS 200 April 2010 Final Exam Solutions

MATHEMATICS 200 April 2010 Final Exam Solutions MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives

More information

WEEK 7 NOTES AND EXERCISES

WEEK 7 NOTES AND EXERCISES WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain

More information

( ) 7 ( 5x 5 + 3) 9 b) y = x x

( ) 7 ( 5x 5 + 3) 9 b) y = x x New York City College of Technology, CUNY Mathematics Department Fall 0 MAT 75 Final Eam Review Problems Revised by Professor Kostadinov, Fall 0, Fall 0, Fall 00. Evaluate the following its, if they eist:

More information

Unit IV Derivatives 20 Hours Finish by Christmas

Unit IV Derivatives 20 Hours Finish by Christmas Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one

More information

Unit IV Derivatives 20 Hours Finish by Christmas

Unit IV Derivatives 20 Hours Finish by Christmas Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one

More information

1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x)

1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x) APPM 5 Final Eam (5 pts) Fall. The following problems are not related: (a) (5 pts, 5 pts ea.) Find the following limits or show that they do not eist: (i) lim e (ii) lim arcsin() (b) (5 pts) Find and classify

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

Secondary Math 2 Honors Unit 4 Graphing Quadratic Functions

Secondary Math 2 Honors Unit 4 Graphing Quadratic Functions SMH Secondary Math Honors Unit 4 Graphing Quadratic Functions 4.0 Forms of Quadratic Functions Form: ( ) f = a + b + c, where a 0. There are no parentheses. f = 3 + 7 Eample: ( ) Form: f ( ) = a( p)( q),

More information

3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY

3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY MATH00 (Calculus).1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY Name Group No. KEYWORD: increasing, decreasing, constant, concave up, concave down, and inflection point Eample 1. Match the

More information

1 Exponential Functions Limit Derivative Integral... 5

1 Exponential Functions Limit Derivative Integral... 5 Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................

More information

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if

More information

Department of Mathematical x 1 x 2 1

Department of Mathematical x 1 x 2 1 Contents Limits. Basic Factoring Eample....................................... One-Sided Limit........................................... 3.3 Squeeze Theorem.......................................... 4.4

More information

2 3 x = 6 4. (x 1) 6

2 3 x = 6 4. (x 1) 6 Solutions to Math 201 Final Eam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( 1) 1 3 = 4 both sides 6: 6 1 1 (

More information

4.3 Mean-Value Theorem and Monotonicity

4.3 Mean-Value Theorem and Monotonicity .3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such

More information

V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

More information

Math 229 Mock Final Exam Solution

Math 229 Mock Final Exam Solution Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it

More information

PTF #AB 07 Average Rate of Change

PTF #AB 07 Average Rate of Change The average rate of change of f( ) over the interval following: 1. y dy d. f() b f() a b a PTF #AB 07 Average Rate of Change ab, can be written as any of the. Slope of the secant line through the points

More information

Review Guideline for Final

Review Guideline for Final Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.

More information

Chapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the

Chapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x

More information

Chapter 4 Notes, Calculus I with Precalculus 3e Larson/Edwards

Chapter 4 Notes, Calculus I with Precalculus 3e Larson/Edwards 4.1 The Derivative Recall: For the slope of a line we need two points (x 1,y 1 ) and (x 2,y 2 ). Then the slope is given by the formula: m = y x = y 2 y 1 x 2 x 1 On a curve we can find the slope of a

More information

Solutions to the Exercises of Chapter 5

Solutions to the Exercises of Chapter 5 Solutions to the Eercises of Chapter 5 5A. Lines and Their Equations. The slope is 5 5. Since (, is a point on the line, y ( ( is an ( 6 8 8 equation of the line in point-slope form. This simplifies to

More information

MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS

MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 is a fast-paced and thorough tour of precalculus mathematics, where the choice of topics is primarily motivated by the conceptual and technical knowledge

More information

Slide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function

Slide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function Slide 1 2.2 The derivative as a Function Slide 2 Recall: The derivative of a function number : at a fixed Definition (Derivative of ) For any number, the derivative of is Slide 3 Remark is a new function

More information

Chapter 6 Overview: Applications of Derivatives

Chapter 6 Overview: Applications of Derivatives Chapter 6 Overview: Applications of Derivatives There are two main contets for derivatives: graphing and motion. In this chapter, we will consider the graphical applications of the derivative. Much of

More information

Solutions to Math 41 First Exam October 18, 2012

Solutions to Math 41 First Exam October 18, 2012 Solutions to Math 4 First Exam October 8, 202. (2 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether it

More information

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs Sample Questions to the Final Eam in Math 1111 Chapter Section.1: Basics of Functions and Their Graphs 1. Find the range of the function: y 16. a.[-4,4] b.(, 4],[4, ) c.[0, ) d.(, ) e.. Find the domain

More information

AP Calculus BC Summer Assignment 2018

AP Calculus BC Summer Assignment 2018 AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2

3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2 AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using

More information

AP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:

AP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name: AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In

More information

MATH141: Calculus II Exam #1 review 6/8/2017 Page 1

MATH141: Calculus II Exam #1 review 6/8/2017 Page 1 MATH: Calculus II Eam # review /8/7 Page No review sheet can cover everything that is potentially fair game for an eam, but I tried to hit on all of the topics with these questions, as well as show you

More information