Calculus Problem Sheet Prof Paul Sutcliffe. 2. State the domain and range of each of the following functions

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1 f() f() f() Calculus Problem Sheet Prof Paul Sutcliffe. By applying the vertical line test, or otherwise, determine whether each of the following equations gives a function y() (a) + (y ) = (b) y = + (c) + y = (d) + y = (e) y = 4 Applying the vertical line test yields gives the following answers to whether the equation determines a function: (a) no, (b) yes, (c) no, (d) yes, (e) no.. State the domain and range of each of the following functions (a) f() = 7 (b) f() = 5 (c) f() = 3 (d) f() = /( + 4) (e) f() = /( 6) (f) f() = e /( 4) (a) Dom f = R, Ran f = [ 7, ) (b) Dom f = [, ), Ran f = (, 5] (c) Dom f = (, 3] [ 3, ), Ran f = [, ) (d) Dom f = R, Ran f = [, ) (e) Dom f = R\{±4}, Ran f = R (f) Dom f = R\{±}, Ran f = (, e /4 ] (, ) 3. Let f() = + a +, where a R. Derive the allowed values of a such that R, f() = f(). f = f iff f so the required condition is equivalent to R, f(). By completing the square f() = ( + a) + 4 a hence a 4 giving a [, ]. 4. Graph the following functions (a) f() = + 3 (b) f() = e (+) (c) f() = 4

2 9 h() g() f() For [, 4π], on the same drawing, graph the following three functions h() = sin, g() = + sin, f() = /( + sin ). 6. For [ π, π], on the same drawing, graph the following three functions h() = cos, g() = cos(), f() = / cos(). 4 3 h() g() f() For [, ], on the same drawing, graph the following three functions h() = e, g() = e, f() = /(e ) 5 4 h() g() f() Are the following functions even, odd or neither? Justify your answers. (a) f() = ( )( ) (b) f() = n k= k+ (c) f() = sin( ) (d) f() = ( +) cos (a) f() = 3 +, so f( ) = Since f() f( ) and f() f( ) this function is neither even nor odd. (b) f( ) = n k= ( )k+ k+ = n k= k+ = f() hence this function is odd. (c) f( ) = sin(( ) ) = sin( ) = f() hence this function is even. (d) is odd, but both + and cos are even, hence f() is the product of one odd function and two even functions and is therefore an odd function.

3 9. Are the following functions even, odd or neither? Justify your answers. (a) f() = e (b) g() = tan (c) h() = e log (d) k() = log (e) p() = ( 3 + )/( 3 ) (f) q() = sin (4) (g) r() = 4 sin (a) f( ) = e. Since f() f( ) and f() f( ) this function is neither even nor odd. (b) g( ) = sin( )/ cos( ) = sin / cos = tan = g() hence this function is odd. (c) h( ) = e log( ) = e log = h() hence this function is odd. (d) k( ) = log = log = k() hence this function is even. (e) 3 + is an odd function and so is 3 hence p() is the ratio of two odd functions and is therefore even. (f) q( ) = sin ( 4) = (sin( 4)) = ( sin(4)) = sin (4) hence this function is even. (g) r( ) = ( ) 4 sin( ) = + 4 sin. Since r() r( ) and r() r( ) this function is neither even nor odd.. If f : R R is an even function and g : R R is an odd function then determine whether the following functions are even, odd or neither? Justify your answers. { f() if > (a) f () = f() if < (b) f () = f() + f() (c) f 3 () = (g f)() (d) f 4 () = (f g)() (e) f 5 () = (g g)() (a) On R\{} { f( ) if > f ( ) = f( ) if < is odd. = { f() if < f() if > = f () hence this function (b) f ( ) = f( ) + f( ) = f() + f() = f () hence this function is even. (c) f 3 ( ) = (g f)( ) = g(f( )) = g(f()) = (g f)() = f 3 () hence this function is even. (d) f 4 ( ) = (f g)( ) = f(g( )) = f( g()) = f(g()) = (f g)() = f 4 () hence this function is even. (e) f 5 ( ) = (g g)( ) = g(g( )) = g( g()) = g(g()) = (g g)() = f 5 () hence this function is odd.

4 . Write each of the following functions as the sum of an even function f even () and an odd function f odd (). (a) f() = (b) f() = e 3 (c) f() = log + (d) f() = 3 /( ) (a) By inspection f even () = 4 and f odd () = 3 5. (b) f even () = (f() + f( )) = (e3 + e 3 ) and f odd () = (f() f( )) = (e3 e 3 ) (c) f even () = (f()+f( )) = (log + +log ) and f odd() = (f() f( )) = (log + log ) (d) Since f( ) = 3 /( ) = f() is an odd function f even () = and f odd () = f() = 3 /( ). In the following cases write a formula for the functions f g and g f and find the domain and range of each of them. (a) f() = + and g() = /. (b) f() = and g() =. (a) (f g)() = f(g()) = f(/) = Ran (f g) = [, )\{ }. +. Dom (f g) = (, ] (, ) (g f)() = g(f()) = g( + ) = / + Dom (g f) = (, ), Ran (g f) = (, ). (b) (f g)() = f(g()) = f( ) = ( ) = + Dom (f g) = [, ), Ran (f g) = [, ). (g f)() = g(f()) = g( ) = = Dom (g f) = R, Ran (g f) = (, ]. 3. Given f() = and g() = /( + ), find (a) (f g)( ) (b) (f f)() (c) (g f)() (d) (g g)() (a) (f g)( ) = f(/3) = /3 (b) (f f)() = f() = (c) (g f)() = g( ) = / (d) (g g)() = g(/3) = 3/4

5 4. Given u() = 3, v() = 4 and f() = /, find (a) (u (v f))() (b) (v (u f))() (c) (f (v u))() (d) (v (f u))() (a) (u (v f))() = u(v(/)) = u(/ 4 ) = / 4 3 (b) (v (u f))() = v(u(/)) = v(/ 3) = (/ 3) 4 (c) (f (v u))() = f(v( 3)) = f(( 3) 4 ) = /( 3) 4 (d) (v (f u))() = v(f( 3)) = v(/( 3)) = /( 3) 4 5. For each f() given below, find the inverse function f () and identify its domain and range. (a) f() = 5 (b) f() = 3 + (c) f() = /, > (d) f() = 4, (e) f() = 7 (f) f() = / 3, Write y = f () and use f(y) =. (a) f(y) = y 5 = hence y = 5 = f (). Dom f = Ran f = R and Ran f = Dom f = R. (b) f(y) = y 3 + = hence y = ( ) 3 = f (). Dom f = Ran f = R and Ran f = Dom f = R. (c) f(y) = /y = hence y = / = f (). Dom f = Ran f = (, ) and Ran f = Dom f = (, ). (d) f(y) = y 4 = hence y = 4 = f (). Dom f = Ran f = [, ) and Ran f = Dom f = [, ). (e) f(y) = y 7 = hence y = + 7 = f (). Dom f = Ran f = R and Ran f = Dom f = R. (f) f(y) = /y 3 = hence y = 3 = f (). Dom f = Ran f = R\{} and Ran f = Dom f = R\{}. 6. Which of the following functions are injective? Find the inverses of those which are and specify the domain of the inverse. (a) f() = ( + 3) 3 on R (b) f() = ( ) on [, ] (c) f() = ( ) on [, ] (d) f() = ( )/( + ) on R\{ } (e) f() = + on [, ] (f) f() = + on [, ]

6 (a). It is injective. Apply horizontal line test or f( ) = f( ) iff ( + 3 ) 3 = ( + 3 ) 3 iff ( + 3 ) = ( + 3 ) iff =. Write y = f () and use f(y) =. So f(y) = (+3y) 3 = hence y = 3 ( 3 ) = f (). Dom f = Ran f = R. (b). It is injective. Apply horizontal line test. Write y = f () and use f(y) =. So f(y) = (y ) = hence y = + = f (). Dom f = Ran f = [, ]. (c). It is not injective. Apply horizontal line test or eg. f() = = f(). (d). It is injective. Apply horizontal line test. Write y = f () and use f(y) =. So f(y) = (y )/(y + ) = hence y = ( + )/( ) = f (). Dom f = Ran f = R\{}. (e) It is injective. Apply horizontal line test or f() = + = ( + ) so [, ] is only on one side of the turning point = of the quadratic. Write y = f () and use f(y) =. So f(y) = (y + ) = hence y = + = f (). Dom f = Ran f = [, ]. (f) It is not injective. Apply horizontal line test or eg. f( ) = = f(). 7. Complete the following tables. g() f() (f g)() g() f() (f g)() / + g() f() (f g)() g() f() (f g)() / For, define the following si functions f () =, f () =, f 3() =, f 4 () =, f 5() =, f 6() =. These have the property that the composition of any two of these functions is again one of these functions. Complete the following table f f f 3 f 4 f 5 f 6 f f f 4 f 3 f 4 f 5 f 6

7 f f f 3 f 4 f 5 f 6 f f f f 3 f 4 f 5 f 6 f f f f 4 f 3 f 6 f 5 f 3 f 3 f 5 f f 6 f f 4 f 4 f 4 f 6 f f 5 f f 3 f 5 f 5 f 3 f 6 f f 4 f f 6 f 6 f 4 f 5 f f 3 f 9. State any vertical and horizontal asymptotes of the following functions (a) f() = /( 9) (b) f() = 3/( + ) (c) f() = 3 /( + ) (d) f() = ( )(7 4 +9) (a) Vertical asymptotes = ±3, horizontal asymptote y =. (b) Horizontal asymptote y =. (c) No asymptotes. (d) Vertical asymptote =, horizontal asymptote y = 3/7.. Write each of the following rational functions as the sum of a polynomial and a proper rational function (a) , (b) , (c) , (d) (a) = (b) = (c) = (d) = Evaluate each of the following epressions (without using a calculator) (a) log e (b) log 3 9 (c) log 4 6 (d) log (e) log 5 65 (f) log n, (g) n Z log(ne) m log n+log e m, n, m > (a). (b). (c). (d) 3. (e) 4. (f) n. log(ne) log n+ (g) m log n+log e = m m log n+m = m.. Demonstrate graphically that, R. Prove this inequality (the graphical demonstration should provide a hint).

8 y Figure : The red curve is the graph of. This is obtained by shifting the graph of vertically down by unit and then reflecting in the -ais any portion of the curve below this ais. The green curve is the graph of. This is obtained by shifting the graph of to the right by unit. The inequality corresponds to the fact that the red curve is never above the green curve. Define f() =. We need to show that f(), R. The graph suggests we consider three regions. If then f() = =. If then f() = ( ) = + ( + ) =. If then f() = ( ) = + ( ) =. Together these three regions cover R hence we have shown that f(), R. 3. Consider the given graph of the function f(). Are the following statements true or false? (a) lim f() eists, (b) lim f() =, (c) lim f() = (d) lim f() =, (e) lim f() =, (f) lim a f() eists a (, ). (a) true, (b) true, (c) false, (d) false, (e) false, (f) true.

9 y 4. Consider the given graph of the function f(). Are the following statements true or false? (a) lim f() does not eist, (b) lim f() =, (c) lim f() does not eist, (d) lim a f() eists a (, ) (e) lim a f() eists a (, 3). (a) false, (b) true, (c) true, (d) true, (e) true. 5. If f() > a and lim a f() = L, can we conclude that L >? Justify your answer. No. An eample is provided by f() = with a = so that L = which is not positive. 6. Justify whether the following statement is true or false. If lim a f() eists then so does lim a f(). False. An eample is provided by f() =, with a =. Here lim f() eists (and is equal to ) but f() is not a real function. 7. Calculate the following limits (a) lim ( ), (b) lim 3, (c) lim π/ sin, (d) lim π cos π. (a), (b), (c) π/, (d) π. 8. Calculate the following limits (a) lim +, (b) lim 4 3, (e) lim 4 +3, (f) lim 4. (c) lim , (d) lim 9 3 9, (a)lim + (+)( ) + = lim ( ) = lim = 3. (b) lim 4 3 = lim ( +)(+)( ) ( )( ++) (c) lim = lim (d) lim (e) lim ( )( ++4) (+)( )( +4) = lim = lim 9 ( 3)( +3) ( 9)( +3) +3 = lim ( )( +3+) = lim ( +)(+) ++ = 4/ (+)( +4) = 3/8. 9 = lim 9 ( 9)( +3) = lim 9 +3+) ( +3 )( +3+) = lim ( )( +3 4 = 4. (f) lim 4 4 = lim 4 (4 )(+ ) ( )(+ ) = lim 4(( + )) = = /6.

10 9. Calculate the limit as of the following (a) cos, (b) cos, (c) tan, (d) cos, (e) cos 4. (a) lim cos = lim ( cos ) = lim u ( cos u) {( cos ( cos )(+cos ) (b) lim = lim = lim (+cos ) u =. sin { tan sin (c) lim = lim cos = lim sin cos cos = lim ) +cos } = /. ( sin cos )( (d) lim cos = lim (+cos ) ( cos )(+cos ) = lim () (+cos ) 4 sin = /. (e) lim cos 4 = lim (+cos 4) ( cos 4)(+cos 4) = lim (4) (+cos 4) 6 sin 4 = /8. cos ) } =. 3. Does lim sin(+ ) eist? If so, find it. For >, sin(+ ) = sin. Hence lim sin(+ ) sin sin + = lim + = lim + For <, sin(+ ) =. Hence lim sin(+ ) =. The left-sided and right-sided limits eist but are not equal, hence the limit does not eist. 3. Calculate lim π/ {( π/) tan }. Set u = π/ then lim π/ ( π/) tan = lim u u tan(u+π/) = lim u u sin(u+π/) cos(u+π/) = lim u u cos u sin u =. 3. In each case either evaluate the limit or state that no limit eists (a) lim , (b) lim 3 + 3, (c) lim 3 ( + ) 3, (d) lim 3 ( + ) ( 3), (e) lim h /h +/h, (a) no limit eists, (b) lim (f) lim h +/h +/h. = lim 3 (+4)( 3) 3 = 7. (c) lim 3 ( + ) 3 = lim 3 (+4) ( 3) 3 = lim 3 ( + 4) ( 3) =. (d) ( + ) ( 3) (e) lim h /h +/h = (+4)( 3) ( 3) = +4 3 = lim h h h + =. (f) lim h +/h +/h = lim h h +h h + =. hence no limit eists. 33. Calculate the limit as of the following (a) 6+7, (b) ( +) ( ) (+) 3 (+) 3, (c) +sin. (a) lim 6+7 = lim 6+ 7 = 3. ( (b) lim +) ( ) 4 = lim (+) 3 (+) = lim 4 = 4/ (c) First note that sin. sin As lim = then by the pinching theorem lim =. Thus lim +sin = lim + sin =. =.

11 34. Calculate the following limits (a) lim sin, (b) lim cos(/) +(/), (c) lim ( ) /, (d) lim (3 + ) cos(/), (e) lim {( 3 cos(/))( + sin(/))}. Set u = / in each case (a) lim sin = lim u sin u u =. cos(/) (b) lim +(/) = lim u cos u +u =. (c) lim ( ) / = lim u u u = lim u ep(log u u ) = lim u ep(u log u) = e =. (d) lim (3 + ) cos(/) = lim u (3 + u) cos u = 3. (e) lim ( 3 cos(/))( + sin(/)) = lim u (3u cos u)( + sin u) =. 35. For each of the following statements, either give a proof that it is true or a counter eample to show that it is false: (a) If g() > > and lim (f() g()) = then lim (f()/g()) =. (b) If g() > > and lim (f()/g()) = then lim (f() g()) =. (a) False. A counter eample is provided by f() = / and g() = /. (b) False. A counter eample is provided by f() = and g() = In each case either evaluate the limit or state that no limit eists (a) lim u 5 u 5 u, (b) lim y (y 8) /3, (c) lim ( )( cos 3), (d) lim t 5 t 5 t 5, (e) lim , (f) lim , (g) lim t 5t 3 +8t 3t 6t 4, (h) lim 3 tan(( 3)) 3, (i) lim , (j) lim + 4, (k) lim t t +t t, (l) lim t t3 + t +. (a) 5/, (b), (c), (d) /, (e) 3/, (f) 3/5, (g) 8/3, (h), (i) /, (j) /, (k) 3/, (l) no limit eists.