Differential Calculus

Transcription

1 Differential Calculus. Compute the derivatives of the following functions a() = b() = c() = 3 + d() = sin cos e() = sin f() = log g() = tan h() = 3 6e 5 4 i() = + tan 3 j() = e k() = arctan l() = log (log ) m() = sin n() = o() = sin log (sin ) p() = q() = arctan ( + e ) r() = log ( + arctan ) s() = log ( + + ) t() = arctan + arctan u() = arctan log ( + ) v() = log + w() = arcsin + y() = log e + z() = log 3 log e + α() = log + 3 log β() = log e + 3e γ() = + log ( ) δ() = log ε() = 4 + cos + cos 6 + e η() = arctan log (arccos ) ϑ() = [ + log ( sin )] e sin λ() = sin cos ϕ() = ( 9)e ψ() = e.. Find the equation of the tangent line to the graph of the following functions at the given point 0 : a) f() = 4 3, ( 0 = ) b) f() = +, ( 0 = ) c) f() =, ( 0 = ) d) f() = e, ( 0 = 0).

2 3. Discuss the differentiability of f() = Determine the monotonicity intervals of the following functions: a) f() = b) f() = ( ) c) f() = e d) f() = e) f() = e. 5. Find local maima and minima of the following functions a) f() = b) f() = c) f() = 3 d) f() = + e) f() = log. 6. Find local maimum and minimum points of f() = sin on [0, π]. 7. Find (local and global) maimum and minimum points of f() = 3 + on [, 3]. 8. Compute the number of real roots of the equation = Let f() = ( )e + arctan(log ) +. Prove that f is invertible on its domain and find R (f). 0. Compute f and f of the following functions (a) f() = (d) f() = (b) f() = arctan (c) f() = cos(sin ) ( + ) (e) f() = + sin (f) f() = arcsin.. Find (if possible) the asymptotes of the following functions a) f() = 3 + b) f() = 3 c) f() = 5 + d) f() = e e e) f() = Find the asymptote as + of the function f() = 3 + sin Compute the following limits using de l Hopital s Theorem: a) lim 0 ( + ) 4 tan d) lim 0 cos g) lim 0 e e sin l) lim 0 e tan e. b) lim sin e) lim 0 ( cos ) cos sin + cos h) lim π sin cos c) lim 0 sin 3 + sin 3 sin ( sin ) f) lim π cos sin i) lim 0 + sin

3 4. For which values of α R is the function ( ) + α f() = arcsin + defined on R? { , if < 0 5. Let f() = ln ( + + ) for which values of k is the function f continuous on R? + k, if 0; Which is the highest order derivative of f we can compute on R? 6. Which of the following functions satisfy the assumptions of Rolle s Theorem on the interval [, ]?, if [, 0] a) f() = b) f() = c) f() = 0, if (0, ] ( ), if (, ] 7. Find the range of the function, if [, 0] d) f() = 0, if (0, ], if (, ] ), if [, e) f() = 0, if =, if (, ]. f() = arctan + arctan, Using Rolle s Theorem show that the derivative of f() = zeros in (0, ). 9. How many solutions has the equation arctan(a) =, where a > 0? { sin π, if > 0 0, if = 0 0. Let f() = + cos. Verify that f is invertible and compute (f ) (). has an infinite number of. Prove that cos, R. 3

4 Solutions. a () = b () = + 4 ( + ) c () = 3 3 ( + ) d () = cos + sin e () = sin + cos f () = g () = ( + tan ) = h () = e 54 i () = + cos e q () = + ( + e ) r () = s () = t () = 0 arctan ( + )( + arctan ) + u () = arctan v () = ( + ) w () = ( + ) 3 4 y () = e e + e (e + ) j () = 3 e tan 3 ( + tan 3) k () = arctan + l () = log + m () = sin cos log n () = (log + ) ( o () = sin cos log + sin ) z () = log 7 log + 5 (log ) α () = 6 log + β () = e + 3e e + 3e γ () = ( ) + log ( ) δ () = sin cos ( + cos ) ( ) cos p () = log sin log sin ε () = 3 e log + sin 6 + e [ ] log (arccos ) η arctan () = arctan + arccos [ ] cos ϑ () = e sin + cos ( + log ( sin )) sin λ () = ( + ) cos + ( ) sin sin cos { e ( + + 9), if > 0 ϕ () = e ( + 9), if < 0 4

5 ( e ) se 0, ψ () = ( ). + e se < 0, +. Since the equation of the tangent line is y = f( 0 ) + f ( 0 )( 0 ) we have: a) y = + 8; b) y = + ; c) y = (horizontal tangent line); d) y =. 3. Since f() =, f the derivative of f eists whenever R \ {}. 4. a) Increasing when > 5/6, decreasing when < 5/6; b) increasing when < /3 and when >, decreasing when /3 < < ; c) Increasing for every ; d) Increasing when 0 < <, decreasing when > ; e) Increasing when > 0, decreasing when < a) Maimum at =, minimum at = ; b) maimum at = ; c) f does not have any maimum or minimum; d) minimum at = 3 ; e) maimum at =. 6. Maimum at = π and minimum at = 0, π. 7. Absolute minimum at = 3, where f ( ) 3 = 3 4. Absolute maimum at = 3, where f(3) = 5. = and = are local maima (f() = 3 and f( ) = 3). = 3 is a local minimum and f ( ) 3 = Eactly one real solution. 9. f is strictly increasing on (0, + ), hence invertible. R (f) = ( π, + ). 0. (a) f () =, f 6 () = ; ( ) 3 ( ) 5 (b) f () = ( + ) arctan, f arctan + () = ( + ) arctan 3 ; (c) f () = sin (sin ) cos, f () = cos (sin ) cos + sin (sin ) sin ; (d) f () = f () = ( + ) [ ( log + ) ], + ( { + ) [ ( log + ) ] } + ( + ) ; (e) f () = ( + cos ), f () = ( 3 + cos ) 4 sin ; (f) f () =, if > 0, if < 0, f () = ( ) 3 ( ) 3, if > 0, if < 0.. a) = vert. as., y = 3 hor. as. ; b) no as. ; c) = vert. as., y = obl. as. ; d) = 0 vert. as., y = 0 hor. as. at ; y = hor. as. at + ; e) no as... y =. 3. a) 4; b) 0; c) 3; d) lim 0 ± f() = ± ; e) ; f) 0; g) ; h) ; i) lim 0 ± f() = ± ; l) α = 0. 5

6 5. f is continuous if k = 0; f eists on R, f () = if < 0 and f () = /( + ) if 0. Computing the limits of the difference quotient of f as 0 + and 0, we obtain f +(0) = and f (0) = 6, hence f has only the first derivative on R. 6. a) No (f( ) f()); b) No (f is not differentiable); c) Yes; d) No (f is not differentiable); e) No (f is not continuous). 7. R (f) = { π/, π/}. 8. f(/n) = 0 for every n, hence the assumptions of the Rolle s Theorem are satisfied in [/(n + ), /n] for every n, that is in infinitely many intervals. 9. If 0 < a < there eists a unique solution. If a, 3 solutions eist. 0. Since f () = sin for every R, f is strictly increasing and then invertible. We look for 0 such that f( 0 ) =. We have 0 = 0, hence g () =.. Let f() = cos + /, we have f () = sin + and f () = cos +. Since f () 0 for every R, f is increasing. We remark that f (0) = 0, hence f () 0 when 0 and f () 0 when 0. Hence f decreases on (, 0], increases on [0, + ) and f() f(0) = 0. 6