Chapter 24 Review Prof. Townsend Fall 2016

Size: px
Start display at page:

Download "Chapter 24 Review Prof. Townsend Fall 2016"

Transcription

1 Chapter 4 Review Prof. Townsend Fall 016 Sections covered: 4.1 Tangents and Normals 4.3 Curvilinear Motion 4.4 Related Rates 4.5 Using Derivatives in Curve Sketching 4.7 Applied Maximum and Minimum Problems Note that in the examples below, you don't see much algebra or derivative detail. On the exam, you are free to use your calculator for those calculations. Follow the details of the solutions below to show me your work. For example, state the derivative you are taking, given y = 1 7 x , = 1. That is all you have to do to show your work. For algebra, say 7 something like "using Solver:" or "using Factor:", or "using Expand:". Note don't plug in values of variables until the derivatives are taken. All problems involve differentiation. 4.1 Tangents and Normals Notation: a given point will be denoted as ( x 1, y 1 ) As developed in section 3., the slope of the tangent line at a point is the first derivative of the function evaluated at that point. m = x=x1 Since we will be dealing with both tangents and normal, I will use subscripts to distinguish between them when necessary. m tan = and m = 1. x=x1 m tan We solved four types of problems in this section. A) Find the equation of the line tangent to the given curve, given the point x 1, y 1 ( ). ( ). B) Find the equation of the line normal to the given curve, given the point x 1, y 1 C) Find the equation of the line tangent to the given curve, given the slope m. D) Find the equation of the line normal to the given curve, given the slope m. Following are examples of each. Page 1 of 19

2 A) Find the equation of the line tangent to the curve y = x x +1 at the point x =. 1) Take the derivative of the function. = 4x 1 ) Evaluate the slope. m = x= = 4 ( ) 1 = 7 3) Find the y value at the point. y = ( ) +1 = 7 4) Find the value of the y intercept. b = y 1 mx 1 = 7 7 = 7 5) Write the equation of the tangent line. y = 7x 7 Check by graphing both the original function, y = x x +1, and the equation of the tangent line, y = 3x +1. 1) Define the functions ( F1). ) Window make sure the point of interest is within the window of the graph. I also like to make sure the two axes show. Note that I first graphed the function using ZoomStd to see what it looks like. Finally, I chose the following window after playing with some of the values. 3) The graph. Page of 19

3 If instead of graphing the tangent line as a second equation, I just use F5/A when looking at the graph, I get the following, thus verifying my tangent line equation. B) Find the equation of the line normal to the curve y = x x +1 at the point x =. 1) We know the tangent slope from the above. m tan = x= = 7 Here we want the normal slope. m = 1 m tan = 1 7 ) Find the value of the y intercept. b = y 1 mx 1 = = ) Write the equation of the normal line. y = 1 7 x ) Check by graphing both the original function, y = x x +1, and the equation of the normal line, y = 1 7 x Since the original function and the point are the same 7 as above, I will just include the normal line equation, then graph both. The graph The line does not look perpendicular (normal). So, F/ZoomSqr. Page 3 of 19

4 Now I will add the tangent line (F5/A) to make sure the line is perpendicular. For this example, the above graph looks perpendicular but for some graphs it will not be so obvious. C) Find the equation of the line tangent to the curve y = x x +1 given the slope, m = 7. You are given the slope so you need to find the point on the curve where the slope is 7. 1) Take the derivative of the function. = 4x 1 ) Find the value of x. = 4( x) 1 = m = 7 x = 8 4 = 3) You know both m and x so follow example A above to finish the problem. D) Find the equation of the line normal to the curve y = x x +1 given the slope, m = ) Find the tangent line slope so you can equate it to the derivative to find x. m tan = 1 = 1/ ( 1/ 7) = 7 m ) Find the value of x given the tangent slope. = 4 x 3) You know both m and x so follow example B above to finish the problem. ( ) 1 = m = 7 x = 8 4 = Page 4 of 19

5 4.3 Curvilinear Motion In PHY 10 you learned that x = x 0 + v 0 t + 1 at and v = v 0 + at A more general form is given an equation for x, v x = and given an equation for y, v y =. The acceleration is no longer constant, it is the derivative of the velocity. a x = dv x and a y = dv y In class we solved two forms of problems. A) Given x and y as functions of time, i.e. x t acceleration. ( ) and y t B) Given y as functions of x and v x, find the velocity. Examples: ( ), find the velocity and A) Find the velocity and acceleration given x = 1 t +1 and y = t 3 +1 at t =. 1) Straight TI-89 solution - velocity. a) Find the derivative The notation is v! = d( [ x, y],t). Note that the TI removes the comma from the vector brackets when it shows the result. b) Plug in the time Page 5 of 19

6 c) Convert the answer to polar coordinates. You can see from the bottom of the screen that the angle mode is DEG i.e. degrees. ) Straight TI-89 solution - acceleration. a) Find the derivative of the velocity found above or enter the following The notation is a! = d b) Plug in the time ( ) = d x, y v x,v y,t ([ ],t,). c) Convert the answer to polar coordinates. Page 6 of 19

7 1) By hand solution - velocity. a) Find the derivative v x = = d 1 t +1 = d ( t +1) 1 = t +1 ( ) ( ) = 1 so the third factor in the power rule, Note that d t +1 v y = = d ( t 3 +1) = d ( t 3 +1) 1/ = 1 t 3 +1 ( ) 1/ ( 3t ) = d du un = nu n 1, was 1. 3t t 3 +1 b) Plug in the time v x = ( +1) = 1 3 = 1 9 v y = 3( ) 3 +1 = 3 9 = c) Convert the answer to polar coordinates. v = v x + v y = =.003 tan(θ v ) = v y v x θ v = tan 1 v y v x = tan 1 1/ 9 = tan 1 18 ( ) = 86.8! But the velocity is in the second quadrant since the x component is negative and the, component is positive. Whenever the x component is negative, add 180! to the calculator answer. θ v = 86.8! +180! = 93.18! ) By hand solution - acceleration. a) Find the derivative of the velocity found above. v x = ( t +1) a x = dv x { } = = ( t +1) 3 ( t +1) 3 Note I prefer the product rule to the quotient rule as u and v are interchangeable in the product rule and not in the quotient rule. ( ) 1/ v y = 3 t t 3 +1 Page 7 of 19

8 a y = dv y a y = 3 t 3 +1 = 3 t t 3 +1 ( ) 1/ + t 1 ( ) 3/ t ( t 3 +1 ) + t 1 ( ) ( ) 3/ a y = 3t t t 3 +1 ( ) 3/ ( 3t ) t 3 +1 ( 3t ) = 3 t 3 +1 t ( ) 3/ 4 + t b) Plug in the time, t=. a x = ( +1) = 3 7 ( ) ( ) ( ) a y = = 3 1 3/ ( 9) = 3 1 3/ 7 ( ) ( ) = 3 c) Convert the answer to polar coordinates using a = a x + a y = θ a = tan 1 a y a x = / 3 tan 1 / 7 = tan 1 9 ( ) = 83.7! a x is positive so do not add 180! to the calculator answer. Page 8 of 19

9 B) Given v x and y as functions of x, find the y component of the velocity. Find the velocity given y = x 3 + x and v x = 3 at t =. Since, in general, we want the velocity, v! =, but y is given as a function of x, not t, so we need to use the chain rule. v y = = = v x Solution. v y = v = d x ( x3 + x) v x = ( 6x +1) ( 3) = 18x + 3 Since v x is constant, we know that x = v x t = 3t. v y = 18 3t ( ) + 3 = 18( 6) + 3 = 651. Note, if v x is not constant, we need integration (MTH 41) to find x as a function of time. Once you have both components of velocity, you can find their polar representation by methods used above. 4.4 Related Rates Every problem in related rates is essentially the same. You are given one rate (either or ) and the value of x. To find the other rate use the chain rule: = Plug in the value of x into the derivative. For volume and surface area formulas, look in the front cover of your book. A couple of good examples you might want to download are found at try_formulas_d_3d_perimeter_area_volume.pdf and In the following examples, I will switch the notation to x and y alongside using the actual variables so you can see the pattern better. The examples below are from pages Page 9 of 19

10 Actual Problem V = I r = I r r = 0.040in dv = dv dr dr di = 0.00 A s x and y version y = x r = x r = 0.00 r = 0.040in = dv di = r dv = r di di = r di = ( 0.040) 0.00 ( ) = V s = r = r = r = ( 0.040) 0.00 ( ) = Actual Problem V = πr h dr mm cm = = min min r = 9.5 cm = 4.75cm h = 15.0cm x and y version y = π x h x = 4.75cm = cm min h = 15.0cm Page 10 of 19

11 dv = dv dr dr h is constant = dv dr = πh dr dr = πhr dv = dv dr dr = ( πhr) dr = ( π ) ( 0.010) = cm3 min = ( π xh ) = = ( πhx) = ( π ) ( 0.010) = cm3 min Actual Problem dw D = w + h = 0.5 in s h = 4.0in dd = dd dw dw x and y version y = x + h = 0.5 in s h = 4.0in = dd dw = d dw w + h = d ( dw w + h ) 1/ = = d x + h = d ( x + h ) 1/ = 1 ( w + h ) 1/ ( w) dw = dd = w dw w + h w dw w + h = 6.5 ( 0.5) = ( x + h ) 1/ ( x) = = x x + h x x + h = 6.5 ( 0.5) = in s 0.13 in s Page 11 of 19

12 Actual Problem T v = = T dt! =.0 C! h =.0 K since! K =! C + 73 h dv = dv dt dt dv dt = 331 d T 73 dt = 331 dt 1/ 73 dt = T 1/ 331 = T 73 dv = T dt = ( ) = x and y version y = 331 =.0 = = x 73 = x 1/ = 331 x 73 = = m s h = m s h 1h 3600s = 4.144E3 m s 4.5 Using Derivatives in Curve Sketching d y >0 >0 =0 =0 or <0 <0 The maxima and minima are found where the slope is zero and the concavity is positive for minima and negative for maxima. Page 1 of 19

13 4.7 Applied Maximum and Minimum Problems There are two types of max/min problems we have met. 1) The first is y is a function of x, find the max/min value of y. ) The second involves two equations. u is a function of x and y, and v is a function of x and y. Find the max/min of one of them given the value of the other one. The examples below are from pages Type 1 Max/Min The first max/min type is where y is a function of x, find the max/min value of y. i) The equation: y = k( x 4 5Lx 3 + 3L x ) where both k and L are constant. ii) Find the two derivatives of y. = k 8x3 15Lx + 6L x ( ) d y = k 4x 30Lx + 6L ( ) iii) Solve for the location of the minimum. = k ( 8x3 15Lx + 6L x) = 0 x = 0 clearly not the answer physically. x = 1.3L x cannot be larger than L. x = 0.578L must be the answer iv) Check the second derivative. d y = k 4 ( 0.578L ) 30L 0.578L so x = 0.578L ( ( ) + 6L ) = 3.3kL < 0 maximum Page 13 of 19

14 144r i) The equation: P = ( r + 0.6) ii) Find the two derivatives of P. dp 144 = ( r 0.6 ) dr r ( ) 3 ( ) ( ) 4 d P 88 r 1. = dr r iii) Solve for the location of the minimum. dp 144 = ( r 0.6 ) dr r ( ) 3 = 0 r = 0.6 Plug in r into the second derivative to make sure the solution is for a maximum or just observe that if r<1., d P 88 = ( r 1. ) dr r maximum. then is negative so the solution is at a ( ) 4 88 = ( 0.6 ) ( ) < 0 maximum 4 d P dr Page 14 of 19

15 Type Max/Min The second max/min type involves two equations. u is a function of x and y, and v is a function of x and y. Find the max/min of one of them given the value of the other one. i) The equations: V = 3x + y 6 = 3x + y The second equation has the number so solve for x or y and plug it into the first equation. y = 3 3 x V = 3x x ii) Find the two derivatives of V. dv = 15x 18 d V = 15 > 0 minimum = 15 x 18x +18 iii) Solve for the location of the minimum. dv = 15x 18 = 0 x = = 1. y = ( ) = 1. iv) Graph to check your answer. Since the point of interest is x=1.. What is y? Note this is the y value representing V for the graph, not the y of the problem. Page 15 of 19

16 Go back to the HOME screen to find y. The calculator knows the function y1 since you just typed it in so perform the following calculation. So we need YMAX at least 7.. I will choose 15. I will pick the following window. Now graph then take the minimum (F5/3). Page 16 of 19

17 i) The equations. We need to create two formulas. To do this we need to label some more sides. First note that u = v since the wih of the box will be the same as the length of the lid. The full wih is W = x + v + x + u = x + v = 15 The full height is H = x + w + x = x + w = 10 The volume of the box is V = x w v What do we actually need? Since V = x w v, we need w in terms of x from x + w = 10 and v in terms of x from x + v = 15. The equations are then V = x w v v = 15 Plugging into V we get V = x 10 x ( ) 15 x x w = 10 x = x 3 5x + 75x Page 17 of 19

18 where I used the TI expand function to get the product. ii) Find the two derivatives of V. dv = 6x 50x + 75 d V = 1x 50 iii) Solve for the location of the minimum. dv = 6x 50x + 75 = 0 Which one is it? Plug the answers into one of the equations above. w = ( ) =.74 A dimension cannot be negative so x = iv) Check the second derivative. Is this a maximum or a minimum? d V = It's a maximum. ( ) 50 = 6.5 < 0 The question was to find x so we are done. Page 18 of 19

19 i) The equations: Height: d where d is the diameter of the semicircles Perimeter: P = x + πr = x + πd = 400 where r = d Area: A = xd + πr Solve the perimeter equation for d: πd = 400 x d = 400 x π Plug into the area equation of just the rectangle 400 x A = xd = x π = ( 00x x ) π ii) Find the two derivatives of A. da = ( 00 x ) π = 4 ( π 100 x) d A dr = 4 π < 0 maximum iii) Solve for the location of the maximum. da = 4 ( π 100 x) = 0 x = 100m Page 19 of 19

20

Leaving Cert Differentiation

Leaving Cert Differentiation Leaving Cert Differentiation Types of Differentiation 1. From First Principles 2. Using the Rules From First Principles You will be told when to use this, the question will say differentiate with respect

More information

Exam 2 Solutions October 12, 2006

Exam 2 Solutions October 12, 2006 Math 44 Fall 006 Sections and P. Achar Exam Solutions October, 006 Total points: 00 Time limit: 80 minutes No calculators, books, notes, or other aids are permitted. You must show your work and justify

More information

MAT 122 Homework 7 Solutions

MAT 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 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

M408 C Fall 2011 Dr. Jeffrey Danciger Exam 2 November 3, Section time (circle one): 11:00am 1:00pm 2:00pm

M408 C Fall 2011 Dr. Jeffrey Danciger Exam 2 November 3, Section time (circle one): 11:00am 1:00pm 2:00pm M408 C Fall 2011 Dr. Jeffrey Danciger Exam 2 November 3, 2011 NAME EID Section time (circle one): 11:00am 1:00pm 2:00pm No books, notes, or calculators. Show all your work. Do NOT open this exam booklet

More information

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018 DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle

More information

3.1 Derivative Formulas for Powers and Polynomials

3.1 Derivative Formulas for Powers and Polynomials 3.1 Derivative Formulas for Powers and Polynomials First, recall that a derivative is a function. We worked very hard in 2.2 to interpret the derivative of a function visually. We made the link, in Ex.

More information

Learning Target: I can sketch the graphs of rational functions without a calculator. a. Determine the equation(s) of the asymptotes.

Learning Target: I can sketch the graphs of rational functions without a calculator. a. Determine the equation(s) of the asymptotes. Learning Target: I can sketch the graphs of rational functions without a calculator Consider the graph of y= f(x), where f(x) = 3x 3 (x+2) 2 a. Determine the equation(s) of the asymptotes. b. Find the

More information

SOLUTIONS FOR PRACTICE FINAL EXAM

SOLUTIONS 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 information

Answer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2

Answer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2 Answer Key Calculus I Math 141 Fall 2003 Professor Ben Richert Exam 2 November 18, 2003 Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem

More information

Review for the Final Exam

Review for the Final Exam Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x

More information

MIDTERM 2. Section: Signature:

MIDTERM 2. Section: Signature: MIDTERM 2 Math 3A 11/17/2010 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. When you use a major theorem (like

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

MATH 162 R E V I E W F I N A L E X A M FALL 2016

MATH 162 R E V I E W F I N A L E X A M FALL 2016 MATH 6 R E V I E W F I N A L E X A M FALL 06 BASICS Graphs. Be able to graph basic functions, such as polynomials (eg, f(x) = x 3 x, x + ax + b, x(x ) (x + ) 3, know about the effect of multiplicity of

More information

= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?

= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds? Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral

More information

AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.

AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. Name: Date: Period: I. Limits and Continuity Definition of Average

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

AB 1: Find lim. x a.

AB 1: Find lim. x a. AB 1: Find lim x a f ( x) AB 1 Answer: Step 1: Find f ( a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f ( x). Do any cancellations and go back to Step 1. If you

More information

Polynomial functions right- and left-hand behavior (end behavior):

Polynomial functions right- and left-hand behavior (end behavior): Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify

More information

Sections 4.1 & 4.2: Using the Derivative to Analyze Functions

Sections 4.1 & 4.2: Using the Derivative to Analyze Functions Sections 4.1 & 4.2: Using the Derivative to Analyze Functions f (x) indicates if the function is: Increasing or Decreasing on certain intervals. Critical Point c is where f (c) = 0 (tangent line is horizontal),

More information

Chapter 3: Derivatives and Graphing

Chapter 3: Derivatives and Graphing Chapter 3: Derivatives and Graphing 127 Chapter 3 Overview: Derivatives and Graphs There are two main contexts for derivatives: graphing and motion. In this chapter, we will consider the graphical applications

More information

Second Midterm Exam Name: Practice Problems Septmber 28, 2015

Second Midterm Exam Name: Practice Problems Septmber 28, 2015 Math 110 4. Treibergs Second Midterm Exam Name: Practice Problems Septmber 8, 015 1. Use the limit definition of derivative to compute the derivative of f(x = 1 at x = a. 1 + x Inserting the function into

More information

Characteristics of Linear Functions (pp. 1 of 8)

Characteristics of Linear Functions (pp. 1 of 8) Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent

More information

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY 2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you

More information

Final Exam Review Exercise Set A, Math 1551, Fall 2017

Final Exam Review Exercise Set A, Math 1551, Fall 2017 Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

Math 210 Midterm #2 Review

Math 210 Midterm #2 Review Math 210 Mierm #2 Review Related Rates In general, the approach to a related rates problem is to first determine which quantities in the problem you care about or have relevant information about. Then

More information

Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P (x) = 3, Q(x) = 4x 7, R(x) = x 2 + x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 + 2x +

More information

d (5 cos 2 x) = 10 cos x sin x x x d y = (cos x)(e d (x 2 + 1) 2 d (ln(3x 1)) = (3) (M1)(M1) (C2) Differentiation Practice Answers 1.

d (5 cos 2 x) = 10 cos x sin x x x d y = (cos x)(e d (x 2 + 1) 2 d (ln(3x 1)) = (3) (M1)(M1) (C2) Differentiation Practice Answers 1. . (a) y x ( x) Differentiation Practice Answers dy ( x) ( ) (A)(A) (C) Note: Award (A) for each element, to a maximum of [ marks]. y e sin x d y (cos x)(e sin x ) (A)(A) (C) Note: Award (A) for each element.

More information

MTH 132 Solutions to Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 11.

MTH 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 information

a 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).

a 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 information

NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.

NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity. AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW 1. Find lim 7x 6x x 7 x 9. 1 B) 0 C) D). Find the points of discontinuity of the function y of discontinuity. x 9x 0. For each discontinuity identify the type

More information

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and

More information

Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?

Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x? . What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when

More information

MATHEMATICS FOR ENGINEERING TUTORIAL 4 MAX AND MIN THEORY

MATHEMATICS FOR ENGINEERING TUTORIAL 4 MAX AND MIN THEORY MATHEMATICS FOR ENGINEERING TUTORIAL 4 MAX AND MIN THEORY This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example.

More information

MTH Calculus with Analytic Geom I TEST 1

MTH Calculus with Analytic Geom I TEST 1 MTH 229-105 Calculus with Analytic Geom I TEST 1 Name Please write your solutions in a clear and precise manner. SHOW your work entirely. (1) Find the equation of a straight line perpendicular to the line

More information

Math Exam 02 Review

Math Exam 02 Review Math 10350 Exam 02 Review 1. A differentiable function g(t) is such that g(2) = 2, g (2) = 1, g (2) = 1/2. (a) If p(t) = g(t)e t2 find p (2) and p (2). (Ans: p (2) = 7e 4 ; p (2) = 28.5e 4 ) (b) If f(t)

More information

March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work.

March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work. March 5, 2009 Name The problems count as marked. The total number of points available is 131. Throughout this test, show your work. 1. (12 points) Consider the cubic curve f(x) = 2x 3 + 3x + 2. (a) What

More information

Chapter 12 Overview: Review of All Derivative Rules

Chapter 12 Overview: Review of All Derivative Rules Chapter 12 Overview: Review of All Derivative Rules The emphasis of the previous chapters was graphing the families of functions as they are viewed (mostly) in Analytic Geometry, that is, with traits.

More information

AP Calculus Worksheet: Chapter 2 Review Part I

AP Calculus Worksheet: Chapter 2 Review Part I AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative

More information

APPLICATION OF DERIVATIVES

APPLICATION OF DERIVATIVES 94 APPLICATION OF DERIVATIVES Chapter 6 With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature. WHITEHEAD 6. Introduction In Chapter 5, we have learnt

More information

Calculus Example Exam Solutions

Calculus 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 information

Precalc Crash Course

Precalc Crash Course Notes for CETA s Summer Bridge August 2014 Prof. Lee Townsend Townsend s Math Mantra: What s the pattern? What s the rule? Apply the rule. Repeat. Precalc Crash Course Classic Errors 1) You can cancel

More information

Math Essentials of Calculus by James Stewart Prepared by Jason Gaddis

Math Essentials of Calculus by James Stewart Prepared by Jason Gaddis Math 231 - Essentials of Calculus by James Stewart Prepared by Jason Gaddis Chapter 3 - Applications of Differentiation 3.1 - Maximum and Minimum Values Note We continue our study of functions using derivatives.

More information

Spring /06/2009

Spring /06/2009 MA 123 Elementary Calculus FINAL EXAM Spring 2009 05/06/2009 Name: Sec.: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or

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

1 + x 2 d dx (sec 1 x) =

1 + x 2 d dx (sec 1 x) = Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating

More information

Technical Calculus I Homework. Instructions

Technical Calculus I Homework. Instructions Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the

More information

Multiple Choice. at the point where x = 0 and y = 1 on the curve

Multiple Choice. at the point where x = 0 and y = 1 on the curve Multiple Choice 1.(6 pts.) A particle is moving in a straight line along a horizontal axis with a position function given by s(t) = t 3 12t, where distance is measured in feet and time is measured in seconds.

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

Math 241 Final Exam, Spring 2013

Math 241 Final Exam, Spring 2013 Math 241 Final Exam, Spring 2013 Name: Section number: Instructor: Read all of the following information before starting the exam. Question Points Score 1 5 2 5 3 12 4 10 5 17 6 15 7 6 8 12 9 12 10 14

More information

GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion

GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion 1. You are hunting for apples, aka easy questions. Do not go in numerical order; that is a trap! 2. Do all Level 1s first. Then 2. Then

More information

4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity 4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

More information

Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam

Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam This outcomes list summarizes what skills and knowledge you should have reviewed and/or acquired during this entire quarter in Math 121, and what

More information

MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS. 1 2 x + 1. y = + 1 = x 1/ = 1. y = 1 2 x 3/2 = 1. into this equation would have then given. y 1.

MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS. 1 2 x + 1. y = + 1 = x 1/ = 1. y = 1 2 x 3/2 = 1. into this equation would have then given. y 1. MIDTERM 2 REVIEW: ADDITIONAL PROBLEMS ) If x + y =, find y. IMPLICIT DIFFERENTIATION Solution. Taking the derivative (with respect to x) of both sides of the given equation, we find that 2 x + 2 y y =

More information

CHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:

CHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg: CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed

More information

Math 170 Calculus I Final Exam Review Solutions

Math 170 Calculus I Final Exam Review Solutions Math 70 Calculus I Final Eam Review Solutions. Find the following its: (a (b (c (d 3 = + = 6 + 5 = 3 + 0 3 4 = sin( (e 0 cos( = (f 0 ln(sin( ln(tan( = ln( (g (h 0 + cot( ln( = sin(π/ = π. Find any values

More information

AP Calculus Chapter 3 Testbank (Mr. Surowski)

AP Calculus Chapter 3 Testbank (Mr. Surowski) AP Calculus Chapter 3 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.). If f(x) = 0x 4 3 + x, then f (8) = (A) (B) 4 3 (C) 83 3 (D) 2 3 (E) 2

More information

Math M111: Lecture Notes For Chapter 3

Math M111: Lecture Notes For Chapter 3 Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following

More information

6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12

6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12 AMS/ECON 11A Class Notes 11/6/17 UCSC *) Higher order derivatives Example. If f = x 3 x + 5x + 1, then f = 6x 8x + 5 Observation: f is also a differentiable function... d f ) = d 6x 8x + 5 ) = 1x 8 dx

More information

Math. 151, WebCalc Sections December Final Examination Solutions

Math. 151, WebCalc Sections December Final Examination Solutions Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for

More information

SOLUTIONS TO MIXED REVIEW

SOLUTIONS TO MIXED REVIEW Math 16: SOLUTIONS TO MIXED REVIEW R1.. Your graphs should show: (a) downward parabola; simple roots at x = ±1; y-intercept (, 1). (b) downward parabola; simple roots at, 1; maximum at x = 1/, by symmetry.

More information

Math 131 Exam 2 Spring 2016

Math 131 Exam 2 Spring 2016 Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0

More information

AP Calculus AB 2015 Free-Response Questions

AP Calculus AB 2015 Free-Response Questions AP Calculus AB 015 Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online

More information

APPM 1350 Exam 2 Fall 2016

APPM 1350 Exam 2 Fall 2016 APPM 1350 Exam 2 Fall 2016 1. (28 pts, 7 pts each) The following four problems are not related. Be sure to simplify your answers. (a) Let f(x) tan 2 (πx). Find f (1/) (5 pts) f (x) 2π tan(πx) sec 2 (πx)

More information

Math 116 Second Midterm November 14, 2012

Math 116 Second Midterm November 14, 2012 Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

More information

Solutions to O Level Add Math paper

Solutions to O Level Add Math paper Solutions to O Level Add Math paper 04. Find the value of k for which the coefficient of x in the expansion of 6 kx x is 860. [] The question is looking for the x term in the expansion of kx and x 6 r

More information

Math 147 Exam II Practice Problems

Math 147 Exam II Practice Problems Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab

More information

Spring Homework Part B Packet. MAT Calculus I

Spring Homework Part B Packet. MAT Calculus I Class: MAT 201-02 Spring 2015 Homework Part B Packet What you will find in this packet: Assignment Directions Class Assignments o Reminders to do your Part A problems (https://www.webassign.net) o All

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

Learning Objectives for Math 165

Learning 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 information

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work.

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Fall 7 Test # In #-5, find the indicated limits. For each one, if it does not eist, tell why not. Show all necessary work. lim sin.) lim.) 3.) lim 3 3-5 4 cos 4.) lim 5.) lim sin 6.)

More information

Mon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers

Mon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers Mon 3 Nov 2014 Tuesday 4 Nov: Quiz 8 (4.2-4.4) Friday 7 Nov: Exam 2!!! In class Covers 3.9-4.5 Today: 4.5 Wednesday: REVIEW Linear Approximation and Differentials In section 4.5, you see the pictures on

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

Limit. Chapter Introduction

Limit. Chapter Introduction Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to

More information

MTH 132 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 11.

MTH 132 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 1 through 11. Show all your work on the standard

More information

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.

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 1A: Homework 6 Solutions

Math 1A: Homework 6 Solutions Math A: Homework Solutions July 30. Sketch the graphs of the following functions. Ensure that your work includes: Domain Intercepts Symmetry Asymptotes and End-behaviour Intervals of Increase/Decrease/Local

More information

Calculus BC: Section I

Calculus BC: Section I Calculus BC: Section I Section I consists of 45 multiple-choice questions. Part A contains 28 questions and does not allow the use of a calculator. Part B contains 17 questions and requires a graphing

More information

sec x dx = ln sec x + tan x csc x dx = ln csc x cot x

sec x dx = ln sec x + tan x csc x dx = ln csc x cot x Name: Instructions: The exam will have eight problems. Make sure that your reasoning and your final answers are clear. Include labels and units when appropriate. No notes, books, or calculators are permitted

More information

4.4: Optimization. Problem 2 Find the radius of a cylindrical container with a volume of 2π m 3 that minimizes the surface area.

4.4: Optimization. Problem 2 Find the radius of a cylindrical container with a volume of 2π m 3 that minimizes the surface area. 4.4: Optimization Problem 1 Suppose you want to maximize a continuous function on a closed interval, but you find that it only has one local extremum on the interval which happens to be a local minimum.

More information

MATH 408N PRACTICE FINAL

MATH 408N PRACTICE FINAL 2/03/20 Bormashenko MATH 408N PRACTICE FINAL Show your work for all the problems. Good luck! () Let f(x) = ex e x. (a) [5 pts] State the domain and range of f(x). Name: TA session: Since e x is defined

More information

Ch 4 Differentiation

Ch 4 Differentiation Ch 1 Partial fractions Ch 6 Integration Ch 2 Coordinate geometry C4 Ch 5 Vectors Ch 3 The binomial expansion Ch 4 Differentiation Chapter 1 Partial fractions We can add (or take away) two fractions only

More information

Calculus AB Topics Limits Continuity, Asymptotes

Calculus AB Topics Limits Continuity, Asymptotes Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3

More information

Math3A Exam #02 Solution Fall 2017

Math3A Exam #02 Solution Fall 2017 Math3A Exam #02 Solution Fall 2017 1. Use the limit definition of the derivative to find f (x) given f ( x) x. 3 2. Use the local linear approximation for f x x at x0 8 to approximate 3 8.1 and write your

More information

See animations and interactive applets of some of these at. Fall_2009/Math123/Notes

See animations and interactive applets of some of these at.   Fall_2009/Math123/Notes MA123, Chapter 7 Word Problems (pp. 125-153) Chapter s Goal: In this chapter we study the two main types of word problems in Calculus. Optimization Problems. i.e., max - min problems Related Rates See

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics

King s Year 12 Medium Term Plan for LC1- A-Level Mathematics King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives

More information

4/16/2015 Assignment Previewer

4/16/2015 Assignment Previewer Practice Exam # 3 (3.10 4.7) (5680271) Due: Thu Apr 23 2015 11:59 PM PDT Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1. Question Details SCalcET7 3.11.023. [1644808] Use the definitions

More information

MLC Practice Final Exam

MLC 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 information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form 0 Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

More information

Table of Contents. Module 1

Table 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 information

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions Spring 5, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card,

More information

EDEXCEL ANALYTICAL METHODS FOR ENGINEERS H1 UNIT 2 - NQF LEVEL 4 OUTCOME 3 - CALCULUS TUTORIAL 2 MAXIMA AND MINIMA

EDEXCEL ANALYTICAL METHODS FOR ENGINEERS H1 UNIT 2 - NQF LEVEL 4 OUTCOME 3 - CALCULUS TUTORIAL 2 MAXIMA AND MINIMA EDEXCEL ANALYTICAL METHODS FOR ENGINEERS H1 UNIT - NQF LEVEL 4 OUTCOME 3 - CALCULUS TUTORIAL MAXIMA AND MINIMA The calculus: the concept of the limit and continuity; definition of the derivative; derivatives

More information

Solutions to Math 41 Final Exam December 10, 2012

Solutions to Math 41 Final Exam December 10, 2012 Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)

More information

2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root

2. If the discriminant of a quadratic equation is zero, then there (A) are 2 imaginary roots (B) is 1 rational root Academic Algebra II 1 st Semester Exam Mr. Pleacher Name I. Multiple Choice 1. Which is the solution of x 1 3x + 7? (A) x -4 (B) x 4 (C) x -4 (D) x 4. If the discriminant of a quadratic equation is zero,

More information

Final Exam Review Packet

Final Exam Review Packet 1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics

More information

Final Exam Review Packet

Final Exam Review Packet 1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics

More information

Ï ( ) Ì ÓÔ. Math 2413 FRsu11. Short Answer. 1. Complete the table and use the result to estimate the limit. lim x 3. x 2 16x+ 39

Ï ( ) Ì ÓÔ. Math 2413 FRsu11. Short Answer. 1. Complete the table and use the result to estimate the limit. lim x 3. x 2 16x+ 39 Math 43 FRsu Short Answer. Complete the table and use the result to estimate the it. x 3 x 3 x 6x+ 39. Let f x x.9.99.999 3.00 3.0 3. f(x) Ï ( ) Ô = x + 5, x Ì ÓÔ., x = Determine the following it. (Hint:

More information