CALCULUS I. Practice Problems. Paul Dawkins

Transcription

1 CALCULUS I Practice Problems Paul Dawkins

2 Table of Contents Preface... iii Outline... iii Review... Introduction... Review : Functions... Review : Inverse Functions... 6 Review : Trig Functions... 6 Review : Solving Trig Equations... 7 Review : Solving Trig Equations with Calculators, Part I... 9 Review : Solving Trig Equations with Calculators, Part II...0 Review : Eponential Functions...0 Review : Logarithm Functions... Review : Eponential and Logarithm Equations... Review : Common Graphs... Limits... 6 Introduction...6 Rates of Change and Tangent Lines...7 The Limit...9 One-Sided Limits... Limit Properties... Computing Limits... Infinite Limits...5 Limits At Infinity, Part I...6 Limits At Infinity, Part II...7 Continuity...8 The Definition of the Limit... Derivatives... Introduction... The Definition of the Derivative... Interpretations of the Derivative... Differentiation Formulas...5 Product and Quotient Rule...7 Derivatives of Trig Functions...8 Derivatives of Eponential and Logarithm Functions...9 Derivatives of Inverse Trig Functions...9 Derivatives of Hyperbolic Functions...0 Chain Rule...0 Implicit Differentiation... Related Rates... Higher Order Derivatives...5 Logarithmic Differentiation...6 Applications of Derivatives... 7 Introduction...7 Rates of Change...8 Critical Points...8 Minimum and Maimum Values...9 Finding Absolute Etrema...5 The Shape of a Graph, Part I...5 The Shape of a Graph, Part II...55 The Mean Value Theorem...57 Optimization...58 More Optimization Problems Paul Dawkins i

3 Indeterminate Forms and L Hospital s Rule...59 Linear Approimations...60 Differentials...6 Newton s Method...6 Business Applications...6 Integrals... 6 Introduction...6 Indefinite Integrals...6 Computing Indefinite Integrals...6 Substitution Rule for Indefinite Integrals...66 More Substitution Rule...68 Area Problem...69 The Definition of the Definite Integral...69 Computing Definite Integrals...70 Substitution Rule for Definite Integrals...7 Applications of Integrals... 7 Introduction...7 Average Function Value...7 Area Between Curves...7 Volumes of Solids of Revolution / Method of Rings...75 Volumes of Solids of Revolution / Method of Cylinders...76 More Volume Problems...76 Work Paul Dawkins ii

4 Preface Here are a set of practice problems for my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site.. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or,. Go to the download page for the site and select the section you d like solutions for and a link will be provided there.. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Outline Here is a list of sections for which problems have been written. Review Review : Functions Review : Inverse Functions Review : Trig Functions Review : Solving Trig Equations Review : Solving Trig Equations with Calculators, Part I Review : Solving Trig Equations with Calculators, Part II Review : Eponential Functions Review : Logarithm Functions Review : Eponential and Logarithm Equations Review : Common Graphs 007 Paul Dawkins iii

5 Limits Tangent Lines and Rates of Change The Limit One-Sided Limits Limit Properties Computing Limits Infinite Limits Limits At Infinity, Part I Limits At Infinity, Part II Continuity The Definition of the Limit - No problems written yet. Derivatives The Definition of the Derivative Interpretation of the Derivative Differentiation Formulas Product and Quotient Rule Derivatives of Trig Functions Derivatives of Eponential and Logarithm Functions Derivatives of Inverse Trig Functions Derivatives of Hyperbolic Functions Chain Rule Implicit Differentiation Related Rates Higher Order Derivatives Logarithmic Differentiation Applications of Derivatives Rates of Change Critical Points Minimum and Maimum Values Finding Absolute Etrema The Shape of a Graph, Part I The Shape of a Graph, Part II The Mean Value Theorem Optimization Problems More Optimization Problems L Hospital s Rule and Indeterminate Forms Linear Approimations Differentials Newton s Method Business Applications Integrals 007 Paul Dawkins iv

6 Indefinite Integrals Computing Indefinite Integrals Substitution Rule for Indefinite Integrals More Substitution Rule Area Problem Definition of the Definite Integral Computing Definite Integrals Substitution Rule for Definite Integrals Applications of Integrals Average Function Value Area Between Two Curves Volumes of Solids of Revolution / Method of Rings Volumes of Solids of Revolution / Method of Cylinders More Volume Problems Work 007 Paul Dawkins v

7 007 Paul Dawkins

8 Review Introduction Here are a set of practice problems for the Review chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site.. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or, 5. Go to the download page for the site and select the section you d like solutions for and a link will be provided there. 6. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. Review : Functions Review : Inverse Functions Review : Trig Functions Review : Solving Trig Equations Review : Solving Trig Equations with Calculators, Part I Review : Solving Trig Equations with Calculators, Part II Review : Eponential Functions Review : Logarithm Functions Review : Eponential and Logarithm Equations Review : Common Graphs 007 Paul Dawkins

9 Review : Functions For problems the given functions perform the indicated function evaluations. f 5. ( ) (a) f ( ) (b) f ( 0) (c) f ( ) (d) f ( 6 t) (e) f ( 7 ) (f) f ( + h). g( t) t t + 6 (a) g ( 0) (b) g ( ) (c) g ( 0) g (e) g( t h) (d) ( ) + (f) g( t t+ ) h z z. ( ) (a) h ( 0) (b) h( ) (c) h ( ) (d) ( 9 ) h z z (f) h( z+ k). R( ) h z (e) ( ) + + (a) R ( 0) (b) R ( 6) (c) R( 9) (d) ( ) R (f) R( ) R + (e) ( ) The difference quotient of a function f ( ) is defined to be, f ( + h) f ( ) h For problems 5 9 compute the difference quotient of the given function. 5. f ( ) 9 g 6 6. ( ) f t t t ( ) 007 Paul Dawkins

10 8. y( z) 9. At ( ) z + t t For problems 0 7 determine all the roots of the given function. f 0. ( ) 5 R y y + y 5. ( ) ht 8 t t. ( ) g + 7. ( ) W ( ) 5. ( ) 5 f t t 7t 8t 6. h( z) 7. g( w) z z 5 z 8 w w + w+ w For problems 8 find the domain and range of the given function. Y t t t+ 8. ( ) g z z z ( ) f z + z + 0. ( ). h( y) +. M ( ) y 007 Paul Dawkins

11 For problems find the domain of the given function.. f ( w) R z w w+ w 7 5 z + 0z + 9z. ( ) 5. g( t) g 6t t 7 t t 5 6. ( ) h 0 7. ( ) P t 5t + t t 8t 8. ( ) 9. f ( z) z + z h( y) y+ 9 y A 6 9. ( ) Q y y + y. ( ) For problems 6 compute ( f g)( ) and ( g f )( ) functions.. f ( ), g( ) f ( ) 5+, ( ) g 5. ( ) +, ( ) f g 8 for each of the given pair of 007 Paul Dawkins 5

12 6. f ( ) +, ( ) g 5 + Review : Inverse Functions For each of the following functions find the inverse of the function. Verify your inverse by computing one or both of the composition as discussed in this section.. f ( ) h( ) 9 R. ( ) g ( ) ( ) 5 5. W( ) f ( ) h( ) 8. f ( ) Review : Trig Functions Determine the eact value of each of the following without using a calculator. Note that the point of these problems is not really to learn how to find the value of trig functions but instead to get you comfortable with the unit circle since that is a very important skill that will be needed in solving trig equations.. 5π cos Paul Dawkins 6

13 π sin 7π sin π cos π tan π sec 6 8π cos π 8. tan 9. 5π tan 0.. π sin 9π sec Review : Solving Trig Equations Without using a calculator find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation.. sin ( t ) 007 Paul Dawkins 7

14 . sin ( t ) in. cos + 0 π 0,. cos + 0 in [ 7 π,7π] 5. cos( 6 ) π z in 0, 6. y sin + 0 in 7 π, tan ( ) 5 in π π, 8. ( ) 6 9sin 7 in 9π π, t 9. tan + 5 in [ 0, π ] csc 9z 7 5 in 0. ( ) π π, 9. cos 6 5 in 0π 5 π, y cos 7 in [ 0 π,5 π ] 007 Paul Dawkins 8

15 Review : Solving Trig Equations with Calculators, Part I Find the solution(s) to the following equations. If an interval is given then find only those solutions that are in the interval. If no interval is given then find all solutions to the equation. These will require the use of a calculator so use at least decimal places in your work.. 7cos( ) cos 0 in [ 0,8 ] t. 6 sin sin ( 6z ) + in [ 0, ] 5. z z 9cos + sin in [ 0,0] w w 6. tan tan in [ 50,0] z 7. 7 sec in [ 0, 5 ] 8. sin ( 7y) sin ( 7y) + + in, 9. 5 tan ( 8) 0 in [,] t csc in [ 0,5 ]. cos + 8 in [ 0,00 ] sec. + ( t) in [,6] 007 Paul Dawkins 9

16 Review : Solving Trig Equations with Calculators, Part II Find all the solution(s) to the following equations. These will require the use of a calculator so use at least decimal places in your work.. ( t ) sin + 7. ( z) sec 9 0. ( ) ( ) ( ) sin + 5sin + tan 0. y y y cos sin + cos ( ) ( ) 7 cos cos 0 6. w w + tan tan 7. csc ( t) + 6 5csc( t) ysec 7 y y 8. ( ) 9. 0 sin ( + ) 7sin ( + ) 6t tan 5 0t 0. ( t ) Review : Eponential Functions Sketch the graphs of each of the following functions. f +. ( ). h( ) Paul Dawkins 0

17 ht 8+ e t. ( ) g z z 0 e. ( ) Review : Logarithm Functions Without using a calculator determine the eact value of each of the following.. log 8. log55. log 8. log ln e log 00 Write each of the following in terms of simpler logarithms 7 7. log ( y ) 8. ln ( y ) + z log y z 9. 5 Combine each of the following into a single logarithm with a coefficient of one. 007 Paul Dawkins

18 log + 5log y log z 0.. ln ( t+ 5) ln t ln ( s ). log a 6log b+ Use the change of base formula and a calculator to find the value of each of the following.. log 5. log 5 Review : Eponential and Logarithm Equations For problems 0 find all the solutions to the given equation. If there is no solution to the equation clearly eplain why. 7+. e 7. 0 z z e. 6t t t 0 e. ( ) + + e 5. y y 0 e e e + e ln 7 8. ln ( y ) + ln ( y+ ) 007 Paul Dawkins

19 9. ( w) ( w ) log + log 0. ( z) ( z ) log log 7 0 t w 7 Compound Interest. If we put P dollars into an account that earns interest at a rate of r (written as a decimal as opposed to the standard percent) for t years then, a. if interest is compounded m times per year we will have, tm r A P + m dollars after t years. b. if interest is compounded continuously we will have, rt A Pe dollars after t years.. We have $0,000 to invest for months. How much money will we have if we put the money into an account that has an annual interest rate of 5.5% and interest is compounded (a) quarterly (b) monthly (c) continuously. We are starting with $5000 and we re going to put it into an account that earns an annual interest rate of %. How long should we leave the money in the account in order to double our money if interest is compounded (a) quarterly (b) monthly (c) continuously Eponential Growth/Decay. Many quantities in the world can be modeled (at least for a short time) by the eponential growth/decay equation. kt Q Q0e If k is positive then we will get eponential growth and if k is negative we will get eponential decay. 5. A population of bacteria initially has 50 present and in 5 days there will be 600 bacteria present. (a) Determine the eponential growth equation for this population. (b) How long will it take for the population to grow from its initial population of 50 to 007 Paul Dawkins

20 a population of 000? 6. We initially have 00 grams of a radioactive element and in 50 years there will be 80 grams left. (a) Determine the eponential decay equation for this element. (b) How long will it take for half of the element to decay? (c) How long will it take until there is only gram of the element left? Review : Common Graphs Without using a graphing calculator sketch the graph of each of the following.. y. f ( ). g( ) ( ) sin + 6. f ( ) ( ) h 5. ( ) cos h ln 5 π ( ) ( ) W e + 7. ( ) 8. ( ) ( ) f y y + 9. R( ) 0. g( ) 007 Paul Dawkins

21 h +. ( ) f y y + 8y+. ( ). ( ) ( y ) y y y ( ) ( y ) y ( y ) ( + ) Paul Dawkins 5

22 Limits Introduction Here are a set of practice problems for the Limits chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site. 7. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or, 8. Go to the download page for the site and select the section you d like solutions for and a link will be provided there. 9. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. Tangent Lines and Rates of Change The Limit One-Sided Limits Limit Properties Computing Limits Infinite Limits Limits At Infinity, Part I Limits At Infinity, Part II Continuity The Definition of the Limit Problems for this section have not yet been written. 007 Paul Dawkins 6

23 Rates of Change and Tangent Lines f + and the point P given by answer each of the following questions.. For the function ( ) ( ) (a) For the points Q given by the following values of compute (accurate to at least 8 decimal places) the slope, m PQ, of the secant line through points P and Q. (i) -.5 (ii) -. (iii) -.0 (iv) -.00 (v) (vi) -.5 (vii) -.9 (viii) -.99 (i) () (b) Use the information from (a) to estimate the slope of the tangent line to f ( ) at and write down the equation of the tangent line.. For the function g( ) 8 following questions. + and the point P given by answer each of the (a) For the points Q given by the following values of compute (accurate to at least 8 decimal places) the slope, m PQ, of the secant line through points P and Q. (i).5 (ii). (iii).0 (iv).00 (v).000 (vi).5 (vii).9 (viii).99 (i).999 ().9999 (b) Use the information from (a) to estimate the slope of the tangent line to g( ) at and write down the equation of the tangent line.. For the function W( ) ln ( ) following questions. + and the point P given by answer each of the (a) For the points Q given by the following values of compute (accurate to at least 8 decimal places) the slope, m PQ, of the secant line through points P and Q. (i).5 (ii). (iii).0 (iv).00 (v).000 (vi) 0.5 (vii) 0.9 (viii) 0.99 (i) () (b) Use the information from (a) to estimate the slope of the tangent line to W( ) at and write down the equation of the tangent line. 007 Paul Dawkins 7

24 . The volume of air in a balloon is given by V ( t) questions. 6 t + answer each of the following (a) Compute (accurate to at least 8 decimal places) the average rate of change of the volume of air in the balloon between t 0.5 and the following values of t. (i) (ii) 0.5 (iii) 0.5 (iv) 0.50 (v) (vi) 0 (vii) 0. (viii) 0.9 (i) 0.99 () (b) Use the information from (a) to estimate the instantaneous rate of change of the volume of air in the balloon at t The population (in hundreds) of fish in a pond is given by P( t) t sin ( t 0) each of the following questions. + answer (a) Compute (accurate to at least 8 decimal places) the average rate of change of the population of fish between t 5 and the following values of t. Make sure your calculator is set to radians for the computations. (i) 5.5 (ii) 5. (iii) 5.0 (iv) 5.00 (v) (vi).5 (vii).9 (viii).99 (i).999 ().9999 (b) Use the information from (a) to estimate the instantaneous rate of change of the population of the fish at t The position of an object is given by ( ) questions. s t cos answer each of the following (a) Compute (accurate to at least 8 decimal places) the average velocity of the object between t and the following values of t. Make sure your calculator is set to radians for the computations. (i).5 (ii). (iii).0 (iv).00 (v).000 (vi).5 (vii).9 (viii).99 (i).999 ().9999 (b) Use the information from (a) to estimate the instantaneous velocity of the object at t and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero). 7. The position of an object is given by ( ) ( )( ) s t Note that a negative position here simply means that the position is to the left of the zero position and is perfectly acceptable. 007 Paul Dawkins 8

25 Answer each of the following questions. (a) Compute (accurate to at least 8 decimal places) the average velocity of the object between t 0 and the following values of t. (i) 0.5 (ii) 0. (iii) 0.0 (iv) 0.00 (v) (vi) 9.5 (vii) 9.9 (viii) 9.99 (i) () (b) Use the information from (a) to estimate the instantaneous velocity of the object at t 0 and determine if the object is moving to the right (i.e. the instantaneous velocity is positive), moving to the left (i.e. the instantaneous velocity is negative), or not moving (i.e. the instantaneous velocity is zero). The Limit. For the function f ( ) 8 answer each of the following questions. (a) Evaluate the function the following values of compute (accurate to at least 8 decimal places). (i).5 (ii). (iii).0 (iv).00 (v).000 (vi).5 (vii).9 (viii).99 (i).999 ().9999 (b) Use the information from (a) to estimate the value of 8 lim.. For the function R( t) t + t + answer each of the following questions. (a) Evaluate the function the following values of t compute (accurate to at least 8 decimal places). (i) -0.5 (ii) -0.9 (iii) (iv) (v) (vi) -.5 (vii) -. (viii) -.0 (i) -.00 () (b) Use the information from (a) to estimate the value of lim t t + t +.. For the function g ( θ ) ( θ ) sin 7 answer each of the following questions. θ 007 Paul Dawkins 9

26 (a) Evaluate the function the following values of θ compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations. (i) 0.5 (ii) 0. (iii) 0.0 (iv) 0.00 (v) (vi) -0.5 (vii) -0. (viii) -0.0 (i) () ( ) sin 7θ (b) Use the information from (a) to estimate the value of lim. θ 0 θ. Below is the graph of f ( ). For each of the given points determine the value of f ( a ) and lim f ( ). If any of the quantities do not eist clearly eplain why. a (a) a (b) a (c) a (d) a 5. Below is the graph of f ( ). For each of the given points determine the value of f ( a ) and lim ( ). If any of the quantities do not eist clearly eplain why. f a (a) a 8 (b) a (c) a 6 (d) a Paul Dawkins 0

27 6. Below is the graph of f ( ). For each of the given points determine the value of f ( a ) and lim f a ( ). If any of the quantities do not eist clearly eplain why. (a) a (b) a (c) a (d) a One-Sided Limits. Below is the graph of f ( ). For each of the given points determine the value of f ( a ), lim f ( ), lim f ( ), and lim f ( ) a + a a. If any of the quantities do not eist clearly eplain why. (a) a (b) a (c) a (d) a 007 Paul Dawkins

28 . Below is the graph of f ( ). For each of the given points determine the value of f ( a ), lim a f ( ), lim f ( ), and lim ( ) + a f. If any of the quantities do not eist clearly eplain why. a (a) a (b) a (c) a (d) a 5. Sketch a graph of a function that satisfies each of the following conditions. ( ) ( ) ( ) lim f lim f f +. Sketch a graph of a function that satisfies each of the following conditions. ( ) ( ) ( ) lim f 0 lim f f does not eist + ( ) ( ) lim f f Limit Properties. Given lim f ( ) 9, lim g( ) and h( ) 8 8 lim use the limit properties given in this 8 section to compute each of the following limits. If it is not possible to compute any of the limits clearly eplain why not. (a) lim f ( ) h( ) 8 (b) lim h ( ) 6 8 (c) lim ( ) ( ) ( ) (d) lim ( ) ( ) + ( ) g h f 8 f g h Paul Dawkins

29 . Given lim f ( ), lim g( ) 0 and h( ) lim 7 use the limit properties given in this section to compute each of the following limits. If it is not possible to compute any of the limits clearly eplain why not. (a) (c) ( ) ( ) ( ) ( ) f h lim g f f ( ) ( ) + ( ) lim + h( ) g h (b) lim f ( ) g( ) h( ) (d) lim h( ) h( ) + 7 f ( ). Given lim f ( ) 6, lim g( ) and h( ) 0 0 lim use the limit properties given in this 0 section to compute each of the following limits. If it is not possible to compute any of the limits clearly eplain why not. (a) lim ( ) + ( ) (b) lim g ( ) h ( ) f h 0 0 ( ) (c) lim + g( ) 0 (d) f lim 0 h( ) g( ) For each of the following limits use the limit properties given in this section to compute the limit. At each step clearly indicate the property being used. If it is not possible to compute any of the limits clearly eplain why not.. lim ( 6t+ t ) t 5. lim( + 7 6) 6 6. w 8w w lim w lim lim z + 6 z Paul Dawkins

30 9. lim ( + ) 0 Computing Limits For problems 9 evaluate the limit, if it eists.. lim( 8 + ) 6+ t. lim t t lim z lim z 8 y y 8 z z y y lim y lim h 0 ( h) h 7. lim z z z 8. lim lim Given the function f ( ) 7 < + Evaluate the following limits, if they eist. 007 Paul Dawkins

31 (a) lim f ( ) 6 (b) lim f ( ). Given ( ) h z 6z z 9z z > Evaluate the following limits, if they eist. (a) lim h( z) z 7 (b) lim h( z) z For problems & evaluate the limit, if it eists.. lim( ) 5. t + lim t t +. Given that 7 f ( ) + for all determine the value of lim f ( ). 5. Use the Squeeze Theorem to determine the value of lim 0 π sin. Infinite Limits For problems 6 evaluate the indicated limits, if they eist. 9. For f ( ) ( ) 5 (a) lim f ( ) evaluate, (b) lim f ( ) + (c) lim ( ) f. For ht ( ) (a) lim ht ( ) t 6 t evaluate, 6 + t (b) lim ht ( ) + t 6 (c) lim ht ( ) t 6. For g( z) z + ( z + ) evaluate, 007 Paul Dawkins 5

32 (a) lim g( z) z (b) lim g( z) + z (c) lim g( z) z g + 7. For ( ) (a) lim g( ) evaluate, (b) lim g( ) + (c) lim g( ) 5. For h( ) ln ( ) (a) lim h( ) 0 evaluate, 6. For R( y) tan ( y) (a) lim R( y) y π evaluate, (b) lim h( ) + 0 (b) lim R( y) y π + (c) lim h( ) 0 (c) lim R( y) y π For problems 7 & 8 find all the vertical asymptotes of the given function. 7. f ( ) 8. g( ) 7 ( 0 ) 8 ( + 5)( 9) Limits At Infinity, Part I f evaluate each of the following limits.. For ( ) 7 (a) lim f ( ). For ( ) (b) lim ( ) f ht t+ t t evaluate each of the following limits. (a) lim ht ( ) t (b) lim ht ( ) t For problems 0 answer each of the following questions. (a) Evaluate lim f ( ) (b) Evaluate lim ( ) f.. (c) Write down the equation(s) of any horizontal asymptotes for the function. 007 Paul Dawkins 6

33 . f ( ). f ( ) 5. f ( ) f ( ) 7. f ( ) 8. f ( ) f ( ) + 0. f ( ) Limits At Infinity, Part II For problems 6 evaluate (a) lim f ( ) and (b) lim ( ) f.. f ( ). f ( ) f e e e e 0e 6 7. ( ) 007 Paul Dawkins 7

34 f e 8e e 5 0. ( ) 5. f ( ) 6. f ( ) 8 e e 9e 7e 8 7 e e e e + 6e + e 0 For problems 7 evaluate the given limit. 7. lim ln ( 9t t ) t 8. lim ln z z + z lim ln lim tan ( ).. + 7t lim tan t t lim tan w w 9w w w Continuity. The graph of f ( ) is given below. Based on this graph determine where the function is discontinuous. 007 Paul Dawkins 8

35 . The graph of f ( ) is given below. Based on this graph determine where the function is discontinuous. For problems 7 using only Properties 9 from the Limit Properties section, one-sided limit properties (if needed) and the definition of continuity determine if the given function is continuous or discontinuous at the indicated points (a), (b) 0, (c)?. f ( ) 6 g z z z 0 (a) z, (b) z 0, (c) z 5?. ( ) 5. g( ) < Paul Dawkins 9

36 (a), (b) 6? 6. ht ( ) t t < t + 6 t (a) t, (b) t 0? < g 6< < > (a) 6, (b)? 7. ( ) For problems 8 determine where the given function is discontinuous. 8. f ( ) R t t 9. ( ) 0. h( z) t 9t cos y + 7 e. ( ). g( ) tan ( ) ( z) For problems 5 use the Intermediate Value Theorem to show that the given equation has at least one solution in the indicated interval. Note that you are NOT asked to find the solution only show that at least one must eist in the indicated interval , on [ ]. w ln ( 5w+ ) 0 on [ 0, ] 5. 0 t t t + 0 e e on [, ] 007 Paul Dawkins 0

37 The Definition of the Limit Problems for this section have not yet been written. Derivatives Introduction Here are a set of practice problems for the Derivatives chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site. 0. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or,. Go to the download page for the site and select the section you d like solutions for and a link will be provided there.. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. The Definition of the Derivative Interpretation of the Derivative Differentiation Formulas Product and Quotient Rule Derivatives of Trig Functions Derivatives of Eponential and Logarithm Functions Derivatives of Inverse Trig Functions Derivatives of Hyperbolic Functions Chain Rule 007 Paul Dawkins

38 Implicit Differentiation Related Rates Higher Order Derivatives Logarithmic Differentiation The Definition of the Derivative Use the definition of the derivative to find the derivative of the following functions.. f ( ) 6. V ( t) g. ( ) Q t 0 + 5t t. ( ) W z z 9z 5. ( ) f 6. ( ) g + 7. ( ) 8. R( z) 9. V ( t) 5 z t + t + 0. Z( t) t. f ( ) 9 t 007 Paul Dawkins

39 Interpretations of the Derivative For problems and use the graph of the function, f ( ), estimate the value of f ( a) given values of a.. (a) a (b) a for the. (a) a (b) a For problems and sketch the graph of a function that satisfies the given conditions.. f ( ), f ( ), f ( ) 5, f ( ). f ( ) 5, f ( ), f ( ), f ( ) 0, f ( ), f ( ) 007 Paul Dawkins

40 For problems 5 and 6 the graph of a function, f ( ), is given. Use this to sketch the graph of the. derivative, f ( ) W z z 9z. (a) Is the function increasing or decreasing at z? (b) Is the function increasing or decreasing at z? (c) Does the function ever stop changing? If yes, at what value(s) of z does the function stop changing? 7. Answer the following questions about the function ( ) 8. What is the equation of the tangent line to f ( ) at Paul Dawkins

41 t +. t + (a) Determine the velocity of the object at any time t. (b) Does the object ever stop moving? If yes, at what time(s) does the object stop moving? 9. The position of an object at any time t is given by s( t) 0. What is the equation of the tangent line to f ( ) 5 at?. Determine where, if anywhere, the function ( ). Determine if the function Z( t) t (a) t 5 (b) t 0 (c) t 00 g + stops changing. increasing or decreasing at the given points.. Suppose that the volume of water in a tank for 0 t 6is given by Q( t) 0 + 5t t. (a) Is the volume of water increasing or decreasing at t 0? (b) Is the volume of water increasing or decreasing at t 6? (c) Does the volume of water ever stop changing? If yes, at what times(s) does the volume stop changing? Differentiation Formulas For problems find the derivative of the given function. f ( ). y t 0t + t g z z z + 9z 7 7. ( ) h y y 9y + 8y +. ( ) 5. y Paul Dawkins 5

42 f ( ) 5 f t 8 t 6t + t 7. ( ) 5 R z 6 z + 8z z 8. ( ) 0 9. z ( 9) 0. g( y) ( y )( y+ y ). h( ) f ( y) 5 y 5y y + y. Determine where, if anywhere, the function ( ) f is not changing.. Determine where, if anywhere, the function y z z z is not changing. 5. Find the tangent line to g ( ) 6 at. f at Find the tangent line to ( ) s t t 0t + 6t 9. (a) Determine the velocity of the object at any time t. (b) Does the object ever stop changing? (c) When is the object moving to the right and when is the object moving to the left? 7. The position of an object at any time t is given by ( ) 8. Determine where the function ( ) 5 9. Determine where the function ( ) ( )( ) h z 6 + 0z 5z z is increasing and decreasing. R + is increasing and decreasing. 007 Paul Dawkins 6

43 0. Determine where, if anywhere, the tangent line to ( ) y +. f 5 + is parallel to the line Product and Quotient Rule For problems 6 use the Product Rule or the Quotient Rule to find the derivative of the given function.. f ( t) ( t t)( t 8t + ). y ( + )( ). h( z) ( + z+ z )( 5z+ 8z z ). g( ) 5. R( w) f 6 w+ w w ( ) 7. If f ( ) 8, f ( ), g ( ) 7 and g ( ) determine the value of ( ) ( ) 8. If f( ) g( ), g ( 7), g ( 7) 9 determine the value of ( 7) 9. Find the equation of the tangent line to f ( ) ( )( ) f. + at 9. fg. 0. Determine where f ( ). Determine where V ( t) ( t )( 5t ) is increasing and decreasing is increasing and decreasing. 007 Paul Dawkins 7

44 Derivatives of Trig Functions For problems evaluate the given limit.. ( z) sin 0 lim z 0 z. ( α ) ( α ) sin lim α 0 sin 5. ( ) cos lim 0 For problems 0 differentiate the given function.. f ( ) ( ) ( ) cos 6sec + 5. g( z) 0 tan ( z) cot ( z) 6. f ( w) tan ( w) sec( w) 7. ht ( ) t t sin ( t) 8. y 6 + csc( ) 9. R( t) 0. Z( v) sin v+ tan + csc ( t) cos( t) ( v) ( v) + at π.. Find the tangent line to f ( ) tan ( ) 9cos( ). The position of an object is given by s( t) 7 cos( t) object is not moving.. Where in the range [,7] is the function ( ) cos( ) decreasing. + determine all the points where the f is increasing and 007 Paul Dawkins 8

45 Derivatives of Eponential and Logarithm Functions For problems 6 differentiate the given function.. ( ) 8 f e. g( t) log ( t) ln ( t). R( w) w log ( w) 5 z. y z e ln ( z) y h y e 5. ( ) 6. f ( t) y + 5t ln ( t) f + e at Find the tangent line to ( ) 7 8. Find the tangent line to f ( ) ( ) ( ) ln log at. t 9. Determine if V ( t ) e t is increasing or decreasing at the following points. (a) t (b) t 0 (c) t 0 0. Determine if G( z) ( z 6) ln ( z) is increasing or decreasing at the following points. (a) z (b) z 5 (c) z 0 Derivatives of Inverse Trig Functions For each of the following problems differentiate the given function.. T( z) cos( z) + 6cos ( z) 007 Paul Dawkins 9

46 . g( t) csc ( t) cot ( t). 6 y 5 sec ( ). ( ) ( ) f w sin w + w tan ( w) 5. h( ) ( ) sin + Derivatives of Hyperbolic Functions For each of the following problems differentiate the given function.. f ( ) sinh ( ) + cosh ( ) sech ( ). R( t) tan ( t) + t csch ( t). g( z) z + tanh ( z) Chain Rule For problems 6 differentiate the given function. f ( ) ( ). ( ) ( ) g t t t+. y 8z. R( w) csc( 7w) ( ) 5. G( ) sin + tan ( ) 007 Paul Dawkins 0

47 6. hu ( ) tan ( + 0u) 7. f ( t) 8. g( ) H z 5 + e e 7 t+ t ( ) cos z 6 9. ( ) 0. u( t) tan ( t ). F( y) ln ( 5y + y ) ( ). V ( ) ln sin ( ) cot ( ). h( z) sin ( z 6 ) + sin 6 ( z). S( w) 7 w+ e w 5. g( z) z 7 sin ( z + 6) 6. ( ) ( ) ( ) ( ) 0 f ln sin ht t 5t t 7. ( ) 6 8. q( t) t ln ( t 5 ) 9. g( w) cos( w) sec( w) 0. ( t) sin y + t. K( ) + e + tan ( ) 007 Paul Dawkins

48 . f ( ) cos( e ). z 5+ tan ( ) ( ) 6t. f ( t) e + sin ( t) ( ) 0 5. g( ) ln ( + ) tan ( 6 ) 6. h( z) tan ( z + ) ( ) 7. ( ) + sin ( ) f f 6e at. 8. Find the tangent line to ( ) 9. Determine where ( ) ( ) V z z z 8 is increasing and decreasing. 0. The position of an object is given by s( t) sin ( t) t interval [ 0, ] the object is moving to the right and moving to the left. At 5 t. Determine where ( ). Determine where in the interval [, 0] increasing and decreasing. +. Determine where in the te is increasing and decreasing. the function f ( ) ln ( 0 00) + + is Implicit Differentiation For problems do each of the following. (a) Find y by solving the equation for y and differentiating directly. (b) Find y by implicit differentiation. (c) Check that the derivatives in (a) and (b) are the same.. y 007 Paul Dawkins

49 . + y. + y For problems 9 find y by implicit differentiation.. y + y 6 7 y + sin y 5. ( ) 6. e sin ( y) 7. y + y ( ) y cos + y + e tan y + y 9. ( ) For problems 0 & find the equation of the tangent line at the given point y at (, ). y y+ 0,. e at ( ) For problems & assume that ( t), y y( t) and z z( t) equation with respect to t. and differentiate the given. y + z. cos( y) sin ( y + z) Related Rates. In the following assume that and y are both functions of t. Given, y and determine y for the following equation. 007 Paul Dawkins

50 y 6y + e. In the following assume that, y and z are all functions of t. Given, y, z, 9 and y determine z for the following equation. y + 5z yz + ( ). For a certain rectangle the length of one side is always three times the length of the other side. (a) If the shorter side is decreasing at a rate of inches/minute at what rate is the longer side decreasing? (b) At what rate is the enclosed area decreasing when the shorter side is 6 inches long and is decreasing at a rate of inches/minute?. A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m /sec at what rate is the radius decreasing when the area of the sheet is m? 5. A person is standing 50 feet away from a model rocket that is fired straight up into the air at a rate of 5 ft/sec. At what rate is the distance between the person and the rocket increasing (a) 0 seconds after liftoff? (b) minute after liftoff? 6. A plane is 750 meters in the air flying parallel to the ground at a speed of 00 m/s and is initially.5 kilometers away from a radar station. At what rate is the distance between the plane and the radar station changing (a) initially and (b) 0 seconds after it passes over the radar station? See the (probably bad) sketch below to help visualize the problem. 7. Two people are at an elevator. At the same time one person starts to walk away from the elevator at a rate of ft/sec and the other person starts going up in the elevator at a rate of 7 ft/sec. What rate is the distance between the two people changing 5 seconds later? 8. Two people on bikes are at the same place. One of the bikers starts riding directly north at a rate of 8 m/sec. Five seconds after the first biker started riding north the second starts to ride directly east at a rate of 5 m/sec. At what rate is the distance between the two riders increasing 0 seconds after the second person started riding? 007 Paul Dawkins

51 9. A light is mounted on a wall 5 meters above the ground. A meter tall person is initially 0 meters from the wall and is moving towards the wall at a rate of 0.5 m/sec. After seconds of moving is the tip of the shadow moving (a) towards or away from the person and (b) towards or away from the wall? 0. A tank of water in the shape of a cone is being filled with water at a rate of m /sec. The base radius of the tank is 6 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing with the radius of the top of the water is 0 meters?. The angle of elevation is the angle formed by a horizontal line and a line joining the observer s eye to an object above the horizontal line. A person is 500 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 5 ft/sec. At what rate is the angle of elevation, θ, changing when the hot air balloon is 00 feet above the ground. See the (probably bad) sketch below to help visualize the angle of elevation if you are having trouble seeing it. Higher Order Derivatives For problems 5 determine the fourth derivative of the given function. ht t 6t + 8t t+ 8. ( ) 7 V +. ( ) f 8. ( ) 5 w. f ( w) 7sin ( ) + cos( w) 5z 5. y e + 8ln ( z ) For problems 6 9 determine the second derivative of the given function. 007 Paul Dawkins 5

52 6. g( ) sin ( 9) 7. z ln ( 7 ) 8. Q( v) ( 6+ v v ) 9. H( t) cos ( 7t) For problems 0 & determine the second derivative of the given function y y. 6y y Logarithmic Differentiation For problems use logarithmic differentiation to find the first derivative of the given function. f +. ( ) ( ) sin y ( z+ z ) ( 6 z ). ht ( ) + ( t) 5t 8 9cos t + 0t For problems & 5 find the first derivative of the given function. g w w 7 w. ( ) ( ) 8 5. f ( ) ( e ) ( ) sin 007 Paul Dawkins 6

53 Applications of Derivatives Introduction Here are a set of practice problems for the Applications of Derivatives chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site.. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or,. Go to the download page for the site and select the section you d like solutions for and a link will be provided there. 5. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. Rates of Change Critical Points Minimum and Maimum Values Finding Absolute Etrema The Shape of a Graph, Part I The Shape of a Graph, Part II The Mean Value Theorem Optimization Problems More Optimization Problems L Hospital s Rule and Indeterminate Forms Linear Approimations Differentials Newton s Method Business Applications 007 Paul Dawkins 7

54 Rates of Change As noted in the tet for this section the purpose of this section is only to remind you of certain types of applications that were discussed in the previous chapter. As such there aren t any problems written for this section. Instead here is a list of links (note that these will only be active links in the web version and not the pdf version) to problems from the relevant sections from the previous chapter. Each of the following sections has a selection of increasing/decreasing problems towards the bottom of the problem set. Differentiation Formulas Product & Quotient Rules Derivatives of Trig Functions Derivatives of Eponential and Logarithm Functions Chain Rule Related Rates problems are in the Related Rates section. Critical Points Determine the critical points of each of the following functions. f ( ) R t + 80t + 5t t. ( ) 5 g w w 7w w. ( ) g + 8. ( ) 6 5 h z z z + 9z+ 5. ( ) 6. ( ) ( ) ( ) Q 8 9 f z 7. ( ) R 8. ( ) z + z + z Paul Dawkins 8

55 r y 5 y 6 y 9. ( ) 0. ( ) ( ) ht 5 t t 8t+ 7 s z z z. ( ) cos( ). ( ) sin ( y ) y 9 f y +. V ( t) ( t) sin + f 5e 9. ( ) 5. g( w) e w w 7w 6. R( ) ln ( + + ) 7. At ( ) t 7 ln ( 8t+ ) Minimum and Maimum Values. Below is the graph of some function, f ( ). Identify all of the relative etrema and absolute etrema of the function. 007 Paul Dawkins 9

56 . Below is the graph of some function, f ( ). Identify all of the relative etrema and absolute etrema of the function.. Sketch the graph of g( ) and identify all the relative etrema and absolute etrema of the function on each of the following intervals. (a) (, ) (b) [, ] (c) [, ] (d) [,5 ] (e) (, 5] h + and identify all the relative etrema and absolute etrema of the function on each of the following intervals.. Sketch the graph of ( ) ( ) (a) (, ) (b) [ 5.5, ] (c) [, ) (d) [, ] 5. Sketch the graph of some function on the interval [, 6 ] that has an absolute maimum at 6 and an absolute minimum at. 6. Sketch the graph of some function on the interval [,] that has an absolute maimum at and an absolute minimum at. 007 Paul Dawkins 50

57 7. Sketch the graph of some function that meets the following conditions : (a) The function is continuous. (b) Has two relative minimums. (b) One of relative minimums is also an absolute minimum and the other relative minimum is not an absolute minimum. (c) Has one relative maimum. (d) Has no absolute maimum. 007 Paul Dawkins 5

58 Finding Absolute Etrema For each of the following problems determine the absolute etrema of the given function on the specified interval.. f ( ) on [ 8, ]. f ( ) on [, ] 5. R( t) + 80t + 5t t on [.5, ] 5. R( t) + 80t + 5t t on [ 0,7 ] 5. h( z) z z + 9z+ on [,] 6. g( ) on [, 5 ] 7. ( ) ( ) ( ) on [,] Q h w w w+ on, 8. ( ) ( ) 5 z + 9. f ( z) z + z+ 8 on [ 0,0] 0. At ( ) t ( 0 t) on [, 0.5 ]. ( ) sin ( y ) y 9 f y + on [ 0,5]. g( w) w w 7w 5 e on, + + on [, ]. R( ) ln ( ) 007 Paul Dawkins 5

59 The Shape of a Graph, Part I For problems & the graph of a function is given. Determine the open intervals on which the function increases and decreases.... Below is the graph of the derivative of a function. From this graph determine the open intervals in which the function increases and decreases. 007 Paul Dawkins 5

60 . This problem is about some function. All we know about the function is that it eists everywhere and we also know the information given below about the derivative of the function. Answer each of the following questions about this function. (a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maimums, relative minimums or neither. f ( 5) 0 f ( ) 0 f ( ) 0 f ( 8) 0 ( ) < 0 on ( 5, ), (, ), ( 8, ) ( ) > 0 on (, 5 ), (,8) f f For problems 5 answer each of the following. (a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maimums, relative minimums or neither. f ( ) ht t 5t t 6. ( ) 5 7. y + 0 p cos + on, 8. ( ) ( ) 9. R( z) 5z sin ( z ) on [ 0,7] ht t t 7 0. ( ) 007 Paul Dawkins 5

61 f w we. ( ) w. g( ) ln ( + ). For some function, ( ) f, it is known that there is a relative maimum at. Answer each of the following questions about this function. (a) What is the simplest form for the derivative of this function? Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative that you can come up with. (b) Using your answer from (a) determine the most general form of the function. (c) Given that f ( ) find a function that will have a relative maimum at. Note : You should be able to use your answer from (b) to determine an answer to this part.. Given that f ( ) and g( ) are increasing functions. If we define h( ) f ( ) + g( ) show that h( ) is an increasing function. 5. Given that f ( ) is an increasing function and define h( ) f ( ). Will h ( ) be an increasing function? If yes, prove that h( ) is an increasing function. If not, can you determine any other conditions needed on the function f ( ) that will guarantee that h( ) will also increase? The Shape of a Graph, Part II. The graph of a function is given below. Determine the open intervals on which the function is concave up and concave down. 007 Paul Dawkins 55

62 . Below is the graph the nd derivative of a function. From this graph determine the open intervals in which the function is concave up and concave down. For problems 8 answer each of the following. (a) Determine a list of possible inflection points for the function. (b) Determine the open intervals on which the function is concave up and concave down. (c) Determine the inflection points of the function. f + 6. ( ) g z z z + 8z+. ( ) ht t + t + 6t 6t+ 5. ( ) 6. h( w) 8 5w+ w cos( w) on [, ] R z z z+ 7. ( ) ( ) 8. h( ) e For problems 9 answer each of the following. (a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maimums, relative minimums or neither. (d) Determine the open intervals on which the function is concave up and concave down. 007 Paul Dawkins 56

63 (e) Determine the inflection points of the function. (f) Use the information from steps (a) (e) to sketch the graph of the function. g t t 5t ( ) 5 f ( ) h z z z z. ( ). Q( t) t 8sin ( t ) on [ 7,]. f ( ) ( ) w. P( w) we on [, ] 5. Determine the minimum degree of a polynomial that has eactly one inflection point. 6. Suppose that we know that f ( ) is a polynomial with critical points, and 6. If we also know that the nd derivative is f ( ) +. If possible, classify each of the critical points as relative minimums, relative maimums. If it is not possible to classify the critical points clearly eplain why they cannot be classified. The Mean Value Theorem For problems & determine all the number(s) c which satisfy the conclusion of Rolle s Theorem for the given function and interval.. f ( ) 8 on [, ]. g( t) t t t on [,] For problems & determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval.. h( z) z 8z + 7z on [,5 ] 007 Paul Dawkins 57

64 t. At ( ) 8t+ e on [,] 5. Suppose we know that f ( ) is continuous and differentiable on the interval [ 7,0] f ( 7) and that f ( ). What is the largest possible value for f ( 0)? f has eactly one real root. 6. Show that ( ), that Optimization. Find two positive numbers whose sum is 00 and whose product is a maimum.. Find two positive numbers whose product is 750 and for which the sum of one and 0 times the other is a minimum.. Let and y be two positive numbers such that y 50 + and ( )( y ) + + is a maimum.. We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $0/ft, the cost of the bottom is $/ft and the cost of the top is $7/ft. If we have $700 determine the dimensions of the field that will maimize the enclosed area. 5. We have 5 m of material to build a bo with a square base and no top. Determine the dimensions of the bo that will maimize the enclosed volume. 6. We want to build a bo whose base length is 6 times the base width and the bo will enclose 0 in. The cost of the material of the sides is $/in and the cost of the top and bottom is $5/in. Determine the dimensions of the bo that will minimize the cost. 7. We want to construct a cylindrical can with a bottom but no top that will have a volume of 0 cm. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. 8. We have a piece of cardboard that is 50 cm by 0 cm and we are going to cut out the corners and fold up the sides to form a bo. Determine the height of the bo that will give a maimum volume. More Optimization Problems 007 Paul Dawkins 58

65 . We want to construct a window whose middle is a rectangle and the top and bottom of the window are semi-circles. If we have 50 meters of framing material what are the dimensions of the window that will let in the most light?. Determine the area of the largest rectangle that can be inscribed in a circle of radius.. Find the point(s) on y that are closest to (,0).. An 80 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into a rectangle with one side times the length of the other side. Determine where, if anywhere, the wire should be cut to maimize the area enclosed by the two figures. 5. A line through the point (,5 ) forms a right triangle with the -ais and y-ais in the st quadrant. Determine the equation of the line that will minimize the area of this triangle. 6. A piece of pipe is being carried down a hallway that is 8 feet wide. At the end of the hallway there is a right-angled turn and the hallway narrows down to feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 7. Two 0 meter tall poles are 0 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used? Indeterminate Forms and L Hospital s Rule Use L Hospital s Rule to evaluate each of the following limits..... lim lim w sin w + 6 ( π w) ( t) ln lim t t z 0 6 ( ) z ( z+ ) sin z + 7z z lim 007 Paul Dawkins 59

66 5. lim e z z + e lim z z e lim t ln t + t z 8. lim w ln ( w ) + w 0 π 9. lim ( ) tan ( ) + 0. lim cos( y) + y 0 y. lim e + Linear Approimations For problems & find a linear approimation to the function at the given point. f e at 5. ( ) 0. ( ) ht t 6t + t 7 at t g z. Find the linear approimation to ( ) z at z. Use the linear approimation to approimate the value of and 0. Compare the approimated values to the eact values.. Find the linear approimation to f ( t) cos( t) approimate the value of cos( ) and ( ) values. at t. Use the linear approimation to cos 9. Compare the approimated values to the eact 5. Without using any kind of computational aid use a linear approimation to estimate the value 0. of e. 007 Paul Dawkins 60

67 Differentials For problems compute the differential of the given function.. f ( ) sec( ). w e +. h( z) ln( z) sin( z). Compute dy and y for 5. Compute dy and y for y e as changes from to.0. 5 y 7 + as changes from 6 to The sides of a cube are found to be 6 feet in length with a possible error of no more than.5 inches. What is the maimum possible error in the volume of the cube if we use this value of the length of the side to compute the volume? Newton s Method For problems & use Newton s Method to determine for the given function and given value of 0. f 7 + 8, 0 5. ( ) f cos, 0. ( ) ( ) For problems & use Newton s Method to find the root of the given equation, accurate to si decimal places, that lies in the given interval in [,6 ] + 5e in [, ] 007 Paul Dawkins 6

68 For problems 5 & 6 use Newton s Method to find all the roots of the given equation accurate to si decimal places sin ( ) Business Applications. A company can produce a maimum of 500 widgets in a year. If they sell widgets during the year then their profit, in dollars, is given by, ( ) 0, 000, , P How many widgets should they try to sell in order to maimize their profit?. A management company is going to build a new apartment comple. They know that if the comple contains apartments the maintenance costs for the building, landscaping etc. will be, C ( ) The land they have purchased can hold a comple of at most 500 apartments. How many apartments should the comple have in order to minimize the maintenance costs?. The production costs, in dollars, per day of producing widgets is given by, ( ) C What is the marginal cost when 75 and 00? What do your answers tell you about the production costs?. The production costs, in dollars, per month of producing widgets is given by, ( ) C What is the marginal cost when 00 and 500? What do your answers tell you about the production costs? 5. The production costs, in dollars, per week of producing widgets is given by, 007 Paul Dawkins 6

69 ( ) C and the demand function for the widgets is given by, ( ) p What is the marginal cost, marginal revenue and marginal profit when 00 and 00? What do these numbers tell you about the cost, revenue and profit? Integrals Introduction Here are a set of practice problems for the Integrals chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site. 6. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or, 7. Go to the download page for the site and select the section you d like solutions for and a link will be provided there. 8. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. Indefinite Integrals Computing Indefinite Integrals Substitution Rule for Indefinite Integrals More Substitution Rule Area Problem 007 Paul Dawkins 6

70 Definition of the Definite Integral Computing Definite Integrals Substitution Rule for Definite Integrals Indefinite Integrals. Evaluate each of the following indefinite integrals. (a) (b) d 5 6 d Evaluate each of the following indefinite integrals. (a) (b) (c) d d d 9 + For problems 5 evaluate the indefinite integral t t t dt w w w dw 6 z + z z dz Determine f ( ) given that f ( ) h t t t + t + t. 7. Determine ht ( ) given that ( ) Computing Indefinite Integrals For problems evaluate the given integral d 007 Paul Dawkins 6

71 z z z dz t t t dt 5 w + 0w 8dw 5. dy 6. w + 0 w dw d + d y y y dy 0. ( t )( + ) t dt. z z dz z. 8 5 z 6z + z dz z. d 6. sin ( ) + 0csc ( ) d 5. cos( ) sec( ) tan ( ) w w w dw 6. + ( ) ( ) + ( ) csc θ sin θ csc θ d θ 007 Paul Dawkins 65

72 7. 8. z + 5 dz e 6 z t t e t e dt 9. 6 dw w w d 6cos. ( ) z + z dz. Determine f ( ) given that f ( ) and ( ) f 7. 7 z. Determine g( z ) given that g ( z) z + e and ( ) z g 5 e.. Determine ht ( ) given that h ( t) t 8t+, h ( ) 9 and ( ) h. Substitution Rule for Indefinite Integrals For problems 6 evaluate the given integral.. ( 8 )( ) d. ( + ) 7 t t dt. ( )( 6 + 7) 0. ( ) w w w dw 5 z z 8z dz sin ( + 6 ) d 007 Paul Dawkins 66

73 6. sec( z) tan ( ) z dz 7. ( 5 5 ) cos( 6 + ) t t t t dt y 7y y y e dy 8. ( 7 ) 9. w + dw w + 6w ( cos )( sin ( ) ) 5 0. ( ) z t t t t dt z z. e cos ( e + ln ) e. sec ( v) + tan( v) dv z dz 0sin cos cos 5 d. ( ) ( ) ( ). ( ) cot ( ) ( ) csc csc 6 5. dy 7 + y d 6. 9w dw 7. Evaluate each of the following integrals. (a) d + 9 (b) ( + 9 ) (c) d + 9 d 007 Paul Dawkins 67

74 More Substitution Rule Evaluate each of the following integrals. 5+ 9t t dt. ( ) cos + 8 e d. ( ). 7w ( 8e ) 7w 7w 6e e + 8e 7w cos( + ) 5. e dw d z ( z) ( z) sin cos 8 dz 6. 0e + e + 7w 6 8w 8w w dw + 7t 9t 5t + dt 7. ( ) 8. 6 csc d ( + )( + ) + sin( + 8 ) y y y y dy 0. sec ( ) tan ( ) tan ( ). 8 w dw w + 9 t t t dt d 007 Paul Dawkins 68

75 7. ( 8+ ) 8 z z dz Area Problem For problems estimate the area of the region between the function and the -ais on the given interval using n 6 and using, (a) the right end points of the subintervals for the height of the rectangles, (b) the left end points of the subintervals for the height of the rectangles and, (c) the midpoints of the subintervals for the height of the rectangles.. f ( ) + on [, ]. g( ) + on [, ]. h( ) cos( ) on [ 0, ] 5. Estimate the net area between f ( ) 8 and the -ais on [, ] using n 8 and the midpoints of the subintervals for the height of the rectangles. Without looking at a graph of the function on the interval does it appear that more of the area is above or below the -ais? The Definition of the Definite Integral For problems & use the definition of the definite integral to evaluate the integral. Use the * right end point of each interval for i.. + d. ( ) 0 6 d. Evaluate : cos ( e + ) + d 007 Paul Dawkins 69

76 For problems & 5 determine the value of the given integral given that ( ) 6 ( ) g d f ( ) d 5. ( ) ( ) 6g 0 f d Determine the value of f ( ) d 9 given that f ( ) d and ( ) 0 7. Determine the value of f ( ) d 0 0 given that f ( ) d, ( ) f ( ) d. 0 5 f d 7 and 5 6 f d 8. f d 9 and For problems 8 & 9 sketch the graph of the integrand and use the area interpretation of the definite integral to determine the value of the integral d 5 0 d For problems 0 differentiate each of the following integrals with respect to. 0. ( + ) 9cos t 6 t dt. ( ) sin 6 7 t + dt. t e dt t Computing Definite Integrals. Evaluate each of the following integrals. 007 Paul Dawkins 70

77 cos d a. ( ) 5 b. cos( ) c. cos( ) d 5 d 5 Evaluate each of the following integrals, if possible. If it is not possible clearly eplain why it is not possible to evaluate the integral d z z dz 5 w w + wdw 5. 8 t t dt z + 7z z 6 + d dz 8. ( ) d 9. y 6y dy y π 0. 7sin( t ) cos( ) 0 π. ( ) ( ) 0 t dt sec z tan z dz 007 Paul Dawkins 7

78 π. sec ( ) 8csc( ) cot ( ) π 6 w w w dw. 0 e + d e y + dy y t t > t t 5. f ( t ) dt where f ( t) 0 6. g ( z ) dz where g( z) 6 z z > z e z d 0 w + dw Substitution Rule for Definite Integrals Evaluate each of the following integrals, if possible. If it is not possible clearly eplain why it is not possible to evaluate the integral ( + )( + ) 0 d 0 π. 0 8cos ( t) ( t) 9 5sin dt 0. sin ( z) cos ( ) π z dz. w e w dw 007 Paul Dawkins 7

79 y + dy 5 y 6. + e d π 6sin 7 cos w dw 7. ( w ) ( ) π d t + t + dt ( 6t ) 0. ( ) + sin ( π ) + cos( π ) z z z dz Applications of Integrals Introduction Here are a set of practice problems for the Integrals chapter of my Calculus I notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. Solutions can be found in a number of places on the site. 9. If you d like a pdf document containing the solutions go to the note page for the section you d like solutions for and select the download solutions link from there. Or, 0. Go to the download page for the site and select the section you d like solutions for and a link will be provided there.. If you d like to view the solutions on the web or solutions to an individual problem you can go to the problem set web page, select the problem you want the solution for. At this point I do not provide pdf versions of individual solutions, but for a particular problem you can select Printable View from the Solution Pane Options to get a printable version. 007 Paul Dawkins 7

80 Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have practice problems written for them. Average Function Value Area Between Two Curves Volumes of Solids of Revolution / Method of Rings Volumes of Solids of Revolution / Method of Cylinders More Volume Problems Work Average Function Value For problems & determine f avg for the function on the given interval.. f ( ) 8 + 5e on [ 0, ] on π, π. f ( ) cos( ) sin ( ). Find avg f c for f for f ( ) + 5 on [,] and determine the value(s) of c in [,] f. which ( ) avg Area Between Curves. Determine the area below ( ). Determine the area to the left of ( ) f + and above the -ais. g y y and to the right of. For problems determine the area of the region bounded by the given set of curves.. y +, y sin ( ), and. 8 y, y and 007 Paul Dawkins 7

81 5. + y, y, y and y 6. y y 6 and y+ 7. y +, y e, and the y-ais 8. y +, y 6, and y, y ( ) + y +, 5 +,,, y and y. e, + y e, y and y y Volumes of Solids of Revolution / Method of Rings For problems 8 use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given ais.. Rotate the region bounded by y, y and the y-ais about the y-ais.. Rotate the region bounded by y 7,, and the -ais about the -ais.. Rotate the region bounded by and 5 about the y-ais. y y. Rotate the region bounded by y and y about the -ais. 5. Rotate the region bounded by about the line y. y 6e and y 6+ between 0 and 6. Rotate the region bounded by the line y 8. +, y 0 6 y 0 + 6, and 5 about 7. Rotate the region bounded by y and 6 y about the line. 8. Rotate the region bounded by y +, and y about the line. 007 Paul Dawkins 75

82 Volumes of Solids of Revolution / Method of Cylinders For problems 8 use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given ais.. Rotate the region bounded by ( ) y, the -ais and the y-ais about the -ais.. Rotate the region bounded by y,, and the -ais about the y-ais.. Rotate the region bounded by y and that 0.. Rotate the region bounded by y and that 0. y y about the y-ais. For this problem assume about the -ais. For this problem assume 5. Rotate the region bounded by y +, y and about the line y Rotate the region bounded by y and 6 y about the line y Rotate the region bounded by 8. Rotate the region bounded by. y and e y +, + 6 about the line 8. y y 5, and 6 about the line More Volume Problems. Find the volume of a pyramid of height h whose base is an equilateral triangle of length L.. Find the volume of the solid whose base is a disk of radius r and whose cross-sections are squares. See figure below to see a sketch of the cross-sections. 007 Paul Dawkins 76

83 . Find the volume of the solid whose base is the region bounded by y and y and whose cross-sections are isosceles triangles with the base perpendicular to the y-ais and the angle between the base and the two sides of equal length is π. See figure below to see a sketch of the cross-sections.. Find the volume of a wedge cut out of a cylinder whose base is the region bounded by y, and the -ais. The angle between the top and bottom of the wedge is π. See the figure below for a sketch of the cylinder and the wedge (the positive -ais and positive y-ais are shown in the sketch they are just in a different orientation). 007 Paul Dawkins 77

84 Work. A force of F( ) ( ) cos +, is in meters, acts on an object. What is the work required to move the object from to 7?. A spring has a natural length of 8 inches and a force of 0 lbs is required to stretch and hold the spring to a length of inches. What is the work required to stretch the spring from a length of inches to a length of 6 inches?. A cable that weighs ½ kg/meter is lifting a load of 50 kg that is initially at the bottom of a 50 meter shaft. How much work is required to lift the load ¼ of the way up the shaft?. A tank of water is 5 feet long and has a cross section in the shape of an equilateral triangle with sides feet long (point of the triangle points directly down). The tank is filled with water to a depth of 9 inches. Determine the amount of work needed to pump all of the water to the top of the tank. Assume that the density of water is 6 lb/ft. 007 Paul Dawkins 78