1. (10 pts) Sketch the graph of a function, f, with domain (-oo, co) that satisfies all of the following conditions: 1. f has critical numbers at x = 1, x = 2, and 5. lim f(x) =-2. x-+-Do x = 3. 6. lim f (x) = -co. x-1+ 2. f' > 0 on (-co, 1) and (1, 2) 7. f" > 0 on (-co, 1) and (3, 4.5) 3. f' 0 on (-4, 2) U (3, 4) f is concave down on (-4, -2) f"(x) >0 on (-2, 2) U (2, 4) -2 2 3 4 y = f ( x ) 2 -3 -2 2 -1 -2 -3Strategy For Sketching y = f(r) : 1. Identify the domain for y = f(x). the largest set of x-values on which f is defined. 2. Analyze f. a ) zeros ( x, ,0), (X2, 0 ) , ... set f ( x ) = 0 and solve for x. 6 ) y intercept (0, y, ) set x = 0 and plug it into f to find y, c ) asymptotes i ) vertical asymptotes, (eans ) x = Q ., x = az lim f ( x ) = 10 OR lim f (x ) = + 0 a , x=at Review Section 2.2 il ) horizontal asymptotes (eans ) y=b,, y= bz 1 im fix ) = 6, OR lim f (x ) = b, x 700 * 2 - OF Review section 2.5 d ) hole " ex. lim ( x + 1 ) 0 0 )_ lim ( *+ ) = 1+1= 2 X-71 IEx There is a graphical feature in this graph that appears as a hole at ( 1, 2) o e ) symmetry this is discussed f ) periodicity later3. Analyze S'. aj find the critical numbers off coordinates f ' ( c ] =0 or f (c) does not exist and cis indomain b) find intervals of increasing (f'20) and decreasing (fico). * make a number line! intervals consider the sign of fo mark critical #S, points not in our doman. c) identity local extrema Represent as coordinates use First Derivative Test. d) identity absolute extrema. Represent as coordinates use EVT, if possible or use information from ( b ) and ( c ) 4. Analyze J". a ) identify places for potential points of inflection (PPOI ). Find where f" ( x)=0 and f" ( x) does not exist b ) find intervals of concave up (f"20] and concave down (f"co ). make a number line Represent these as intervals. Include values from (a) (PPOI) and x-values not in the domain. c) Identify points of inflection ( POI ) Represent these as coordinates. Look at PPOI from cal and see If concavity changesSymmetry We say that a function, y = /(x) is even, it f(x)=f(-X ) for all r in the domain. symmetric about the y-axis y= COS(x ) Note cos ( ) = cos( -x ) for all x in (- 00, 00 ] see in the graph to the left X cosco] =-1 = cos(-1) 7 careful this COS( T/2 )= 0= cos(-12) is not enough to tell us for all x . COSCX) = 065 (-*) We say that a function, y = /(x) isodd, if fex)2 -f (x ) for all s in the domain. symmetrc about the origin (0,0) Note sinc-x ) = - sin ( x ) 4 = sin (x ) for all x in (-00, 0 0 ] we can see in the graph to the left sin (IE ) = - sm (-2 ) >- sin ( 2 ) = sin(-=) rin (-1 ) = - sin (IT) similarly this is not proof sincx )= We say a function, y = /(x) is periodic, if there exists some number p such that -sink) for all x. f(x) = /(x + p) for all r in the domain. 4 = cos(x ] Note cos ( x ) = f ( x ) is periodic because -zn_ f ( x ) = cos ( x ) = cos ( x+ zm) = flat ZIT ) similarly, sine is periodic with period 217 Also, tangent and cotangent are periodic with period TI some other functions where symmetry may help Include f(x ) = - (odd ) fox ) = x2 ( even) fix ) = *3 (odd ) 4 = f (x ) my = f(x ) y 1 p y = f ( x ) X 3 X 92020 Kelli Karcher, DAkiel Kim