Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
lim x0 x+1 x 2 +1lim x0 2x x 2 +4lim x4 x 2 4x x4lim x3 x 2 9 x 2 3xlim x0 ( e x +1 )14AYUlim x0 cosx1 x , x in radianslim x0 tanx x , x in radiansIn Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x2 f( x )In Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x4 f( x )In Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x2 f( x )In Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x2 f( x )In Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x2 f( x )In Problems 17-22, use the graph shown to determine if the limit exists. If it does, find its value. lim x2 f( x )In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x4 f( x ),f( x )=3x+1In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x1 f( x ),f( x )=2x1In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x2 f( x ),f( x )=1 x 2In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x1 f( x ),f( x )= x 3 1In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x3 f( x ),f( x )=| 2x |In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x4 f( x ),f( x )=3 xIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x/2 f( x ),f( x )=sinxIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x f( x ),f( x )=cosxIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )= e xIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x1 f( x ),f( x )=lnxIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x1 f( x ),f( x )= 1 xIn Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x2 f( x ),f( x )= 1 x 2In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ x 2 ifx0 2xifx0In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ x1ifx0 3x1ifx0In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x1 f( x ),f( x )={ 3xifx1 x+1ifx1In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x2 f( x ),f( x )={ x 2 ifx2 2x1ifx2In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ xifx0 1ifx=0 3xifx0In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ 1ifx0 1ifx0In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ sinxifx0 x 2 ifx0In Problems 23-42, graph each function. Use the graph to find the indicated limit, if it exists. lim x0 f( x ),f( x )={ e x ifx0 1xifx0In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x1 x 3 x 2 +x1 x 4 x 3 +2x2In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x1 x 3 + x 2 +3x+3 x 4 + x 3 +2x+2In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x2 x 3 2 x 2 +4x8 x 2 +x6In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x1 x 3 x 2 +3x3 x 2 +3x4In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x1 x 3 +2 x 2 +x x 4 + x 3 +2x+2In Problems 43-48, use a graphing utility to find the indicated limit rounded to two decimal places. lim x3 x 3 3 x 2 +4x12 x 4 3 x 3 +x3The limit of the product of two functions equals the __________ of their limits.lim xc b= _____lim xc x= a. x b. c c. cx d. x cTrue or False The limit of a polynomial function as x approaches 5 equals the value of the polynomial at 5.True or False The limit of a rational function at 5 equals the value of the rational function at 5.True or false The limit of a quotient equals the quotient of the limits.In Problems 7- 42, find each limit algebraically. lim x1 5In Problems 7- 42, find each limit algebraically. lim x1 ( 3 )In Problems 7-42, find each limit algebraically. lim x4 xIn Problems 7-42, find each limit algebraically. lim x3 xIn Problems 7-42, find each limit algebraically. lim x2 ( 5x )In Problems 7-42, find each limit algebraically. lim x4 ( 3x )In Problems 7-42, find each limit algebraically. lim x2 ( 5 x 4 )In Problems 7-42, find each limit algebraically. lim x3 ( 2 x 3 )In Problems 7-42, find each limit algebraically. lim x2 ( 3x+2 )In Problems 7-42, find each limit algebraically. lim x3 ( 25x )In Problems 7-42, find each limit algebraically. lim x1 ( 3 x 2 5x )In Problems 7-42, find each limit algebraically. lim x2 ( 8 x 2 4 )In Problems 7-42, find each limit algebraically. lim x1 ( 5 x 4 3 x 2 +6x9 )In Problems 7-42, find each limit algebraically. lim x1 ( 8 x 5 7 x 3 +8 x 2 +x4 )In Problems 7-42, find each limit algebraically. lim x1 ( x 2 +1 ) 3In Problems 7-42, find each limit algebraically. lim x2 ( 3x4 ) 2In Problems 7-42, find each limit algebraically. lim x1 5x+4In Problems 7-42, find each limit algebraically. lim x0 12xIn Problems 7-42, find each limit algebraically. lim x0 x 2 4 x 2 +4In Problems 7-42, find each limit algebraically. lim x2 3x+4 x 2 +xIn Problems 7-42, find each limit algebraically. lim x2 ( 3x2 ) 5/2In Problems 7-42, find each limit algebraically. lim x1 ( 2x+1 ) 5/3In Problems 7-42, find each limit algebraically. lim x2 x 2 4 x 2 2xIn Problems 7-42, find each limit algebraically. lim x1 x 2 +x x 2 1In Problems 7-42, find each limit algebraically. lim x3 x 2 x12 x 2 9In Problems 7-42, find each limit algebraically. lim x3 x 2 +x6 x 2 +2x3In Problems 7-42, find each limit algebraically. lim x1 x 3 1 x1In Problems 7-42, find each limit algebraically. lim x1 x 4 1 x1In Problems 7-42, find each limit algebraically. lim x1 ( x+1 ) 2 x 2 1In Problems 7-42, find each limit algebraically. lim x2 x 3 8 x 2 4In Problems 7-42, find each limit algebraically. lim x1 x 3 x 2 +x1 x 4 x 3 +2x2In Problems 7-42, find each limit algebraically. lim x1 x 3 + x 2 +3x+3 x 4 + x 3 +2x+2In Problems 7-42, find each limit algebraically. lim x2 x 3 2 x 2 +4x8 x 2 +x6In Problems 7-42, find each limit algebraically. lim x1 x 3 x 2 +3x3 x 2 +3x4In Problems 7-42, find each limit algebraically. lim x1 x 3 +2 x 2 +x x 4 + x 3 +2x+2In Problems 7-42, find each limit algebraically. lim x3 x 3 3 x 2 +4x12 x 4 3 x 3 +x3In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=2 ; f( x )=5x3In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=2 ; f( x )=43xIn Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=3 ; f( x )= x 2In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=3 ; f( x )= x 3In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=1 ; f( x )= x 2 +2xIn Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=1 ; f( x )=2 x 2 3xIn Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=0 ; f( x )=3 x 3 2 x 2 +4In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=0 ; f( x )=4 x 3 5x+8In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=0 ; f( x )=4 x 3 5x+8In Problems 43-52, find the limit as x approaches c of the average rate of change of each function from c to x . c=1 ; f( x )= 1 x 2In problems 53-56, use the properties of limits and the facts that lim x0 sinx x =1 lim x0 cosx1 x =0 lim x0 sinx=0 lim x0 cosx=1 where x is in radians, to find each limit. lim x0 tanx xIn problems 53-56, use the properties of limits and the facts that lim x0 sinx x =1 lim x0 cosx1 x =0 lim x0 sinx=0 lim x0 cosx=1 where x is in radians, to find each limit. lim x0 sin( 2x ) x [Hint: Use a Double-angle Formula.]In problems 53-56, use the properties of limits and the facts that lim x0 sinx x =1 lim x0 cosx1 x =0 lim x0 sinx=0 lim x0 cosx=1 where x is in radians, to find each limit. lim x0 3sin+cosx1 4xIn problems 53-56, use the properties of limits and the facts that lim x0 sinx x =1 lim x0 cosx1 x =0 lim x0 sinx=0 lim x0 cosx=1 where x is in radians, to find each limit. lim x0 sin 2 x+sinx( cosx1 ) x 2For the function f( x )={ x 2 ifx0 x+1if0x2 5xif2x5 , find f( 0 ) and f( 0 ) . (pp. 100-101)What are the domain and range of f( x )=lnx ?True or False The exponential function f( x )= e x is increasing on the interval ( , ) . (pp. 282-287)Name the trigonometric functions that have asymptotes. (pp. 423-429)True or False Some rational functions have holes in their graph. (pp. 237-238)True or False Every polynomial function has a graph that can be traced without lifting pencil from paper. (pp. 182)In Problems 7-42, find each limit algebraically. If we approach c from only one side, then we have a(n) _______ limit.In Problems 7-42, find each limit algebraically. The notation ____________ is used to describe the fact that as x gets closer to c but remains greater than c , the value of f(x) gets closer to R .In Problems 7-42, find each limit algebraically. If lim xc f( x )=f( c ) , then f is ________ at ________________.In Problems 7-42, find each limit algebraically. True or False For any function f , lim xc f( x )= lim x c + f( x ) .In Problems 7-42, find each limit algebraically. If f is continuous at c , then lim x c + f( x )= ______. (a) lim x c + f( x )= (b) lim x c + f( x )= (c) f( c ) (d) All of theseIn Problems 7-42, find each limit algebraically. True or False Every polynomial function is continuous at every real number.In Problems 7-42, find each limit algebraically. What is the domain of f ?In Problems 7-42, find each limit algebraically. What is the range of f ?In Problems 7-42, find each limit algebraically. Find the x-intercept( s ) , if any, of f .In Problems 7-42, find each limit algebraically. lim x3 ( 25x )In Problems 7-42, find each limit algebraically. lim x1 ( 3 x 2 5x )In Problems 7-42, find each limit algebraically. lim x2 ( 8 x 2 4 )In Problems 7-42, find each limit algebraically. lim x1 ( 5 x 4 3 x 2 +6x9 )In Problems 7-42, find each limit algebraically. lim x1 ( 8 x 5 7 x 3 +8 x 2 +x4 )Find lim x 4 f( x ) .Find lim x 4 + f( x ) .Find lim x 2 f( x ) .Find lim x 2 + f( x ) .Does lim x4 f( x ) exist? If it does, what is it?Does lim x0 f( x ) exist? If it does, what is it?Is f continuous at 4 ?Is f continuous at 6 ?Is f continuous at 0?Is f continuous at 2?Is f continuous at 4?Is f continuous at 5?lim x 1 + ( 2x+3 )lim x 2 ( 42x )lim x 1 ( 2 x 3 +5x )lim x 2 + ( 3 x 2 8 )lim x/ 2 + sinxlim x ( 3cosx )lim x 2 + x 2 4 x2lim x 1 x 3 x x1lim x 1 x 2 1 x 3 +1lim x 0 + x 3 x 2 x 4 + x 2lim x 2 + x 2 +x2 x 2 +2xlim x 4 x 2 +x12 x 2 +4xf( x )= x 3 3 x 2 +2x6c=2f( x )=3 x 2 6x+5c=3f( x )= x 2 +5 x6 c=3f( x )= x 3 8 x 2 +4 c=2f( x )= x+3 x3 c=3f( x )= x6 x+6 c=6f( x )= x 3 +3x x 2 3x c=0f( x )= x 2 6x x 2 +6x c=0f( x )={ x 3 +3x x 2 3x ifx0 1ifx=0 c=0f( x )={ x 2 6x x 2 +6x ifx0 2ifx=0 c=0f( x )={ x 3 +3x x 2 3x ifx0 1ifx=0 c=0f( x )={ x 2 6x x 2 +6x ifx0 1ifx=0 c=0f( x )={ x 3 1 x 2 1 ifx1 2ifx=1 3 x+1 ifx1 c=1f( x )={ x 2 2x x2 ifx2 2ifx=2 x4 x1 ifx2 c=2f( x )={ 2 e x ifx0 2ifx=0 x 3 +2 x 2 x 2 ifx0 c=0f( x )={ 3cosxifx0 3ifx=0 x 3 +3 x 2 x 2 ifx0 c=0f( x )=2x+3f( x )=43xf( x )=3 x 2 +xf( x )=3 x 3 +7f( x )=4sinxf( x )=2cosxf( x )=2tanxf( x )=4cscxf( x )= 2x+5 x 2 4f( x )= x 2 4 x 2 9f( x )= x3 Inxf( x )= lnx x3R( x )= x1 x 2 1 , c=1 and c=1R( x )= 3x+6 x 2 4 , c=2 and c=2R( x )= x 2 +x x 2 1 , c=1 and c=1R( x )= x 2 +4x x 2 16 , c=4 and c=4R( x )= x 3 x 2 +x1 x 4 x 3 +2x2R( x )= x 3 + x 2 +3x+3 x 4 + x 3 +2x+2R( x )= x 3 2 x 2 +4x8 x 2 +x6R( x )= x 3 x 2 +3x3 x 2 +3x4R( x )= x 3 +2 x 2 +x x 4 + x 3 +2x+2R( x )= x 3 3 x 2 +4x12 x 4 3 x 3 +x3R( x )= x 3 x 2 +x1 x 4 x 3 +2x2 Graph R(x) .R( x )= x 3 + x 2 +3x+3 x 4 + x 3 +2x+2 Graph R( x ) .R(x)= ( x 3 2 x 2 +4x8) ( x 2 +x6) Graph R( x ) .86AYU87AYU88AYU89AYU90AYUFind an equation of the line with slope 5 containing the point ( 2,4 ).( p.33 )2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYUf( x )=cosx at 033AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYU42AYU43AYU44AYU45AYU46AYU47AYUInstantaneous Velocity of a Ball In physics it is shown that the height s of a ball thrown straight down with an initial velocity of 48ft/sec from a rooftop 160 feet high is s=s( t )=16 t 2 48t+160 where t is the elapsed time that the ball is in the air. (a) When does the ball strike the ground? That is, how long is the ball in the air? (b) What is the average velocity of the ball from t=0 to t=1 ? (c) What is the instantaneous velocity of the ball at time t ? (d) What is the instantaneous velocity of the ball at t=1 ? (e) What is the instantaneous velocity of the ball when it strikes the ground?Instantaneous Velocity on the Moon Neil Armstrong throws a ball down into a crater on the moon. The height s (in feet) of the ball from the bottom of the crater after t seconds is given in the following table: (a) Find the average velocity from t=1 to t=4 seconds. (b) Find the average velocity from t=1 to t=3 seconds. (c) Find the average velocity from t=1 to t=2 seconds. (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous velocity at t=1 second.Instantaneous Rate of Change The following data represent the total revenue R (in dollars) received from selling x bicycles at Tunney’s Bicycle Shop. (a) Find the average rate of change in revenue from x=25 to x=150 bicycles. (b) Find the average rate of change in revenue from x=25 to x=102 bicycles. per bicycle (c) Find the average rate of change in revenue from x=25 to x=60 bicycles. per bicycle (d) Using a graphing utility, find the quadratic function of best fit. (e) Using the function found in part (d), determine the instantaneous rate of change of revenue at x=25 bicycles.In Problems 29-32, find the first five terms in the sequence of partial sums for each series. k=1 k 22AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYURepeat Problem 9 for f( x )=4x .11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYUConsider the function f( x )= 1 x 2 whose domain is the interval [ 1,1 ] . (a) Graph f . (b) Approximate the area under the graph of f from 1 to 1 by dividing [ 1,1 ] into five subintervals, each of equal length. (c) Approximate the area under the graph of f from 1 to 1 by dividing [ 1,1 ] into five subintervals, each of equal length. (d) Express the area as an integral. (e) Evaluate the integral using a graphing utility. (f) What is the actual area?1AYU2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYU32AYU