) Solve the following inequalities.) ++.) 4 > 3.3) Calculus-Lab { + > + 5 + < 3 +. ) Graph the functions f() = 3, g() = + +, h() = 3 cos( ), r() = 3 +. 3) Find the domain of the following functions 3.) f() = + 3.) f() = 3+ 3.3) f() = 3.4) f() = tan( ) 3.5) f() = 3 3.6) f() = e 3.7) f() = ln(ln ) + 4 3 3.8) f() = arcsin( + 3 ) + ln( ). 4) Decide if the following function are even, odd or neither: f() = + 3, g() = ln sin +, h() =. 5) Given sinh = e e and cosh = e +e, prove the equality cosh sinh =. Calculus-Lab ) Given f() = + 3 +, evaluate:.) lim f().) lim f().3) lim f().4) lim f(). + + ) Evaluate the following limits.) lim (e + +4 + ).) lim.5) lim + e +ln( ).9) lim + ( + ).) lim.3) lim +! sin().3) lim.6) lim 3 + ln( 3) 3.7) lim + e ln( +3 + ) +.) lim 3 + + + 3) Find domain and limits at boundary points for the function 3.) f() = 3.) g() = e + 3.3) h() = arctan() + 4) Draw the graph of a function f() satisfying the following conditions lim f() =, lim f() = +, lim f() =. +.4) lim arctan( +cos ).8) lim arctan( sin )!.) lim + + 3 3 ++ ) Evaluate the following limits.) lim + cos +.) lim + sin()e Calculus-Lab 3 ) Use the known limit lim sin = to evaluate lim cos..3) lim sin.4) lim 3) Discuss the continuity of the following function as a obtains all possible real values { sin, < f() = ; a,. 4) Discuss domain, limits at boundary points and continuity of the following function { arcsin + f() =, > ;,. e +cos. 5) Find domain, limits at boundary points and discuss the continuity of the function f() = +sin. 6) Use the definition of derivative (and the limit evaluated in )) to prove that (cos ) = sin. 7) Find the derivative of the following functions 7.) f() = +3 3 + 7.) f() = e cos 7.3) f() = + 7.4) f() = e sin. 7.5) f() = ln( 3 + ) 7.6) f() = ( + ) ln 7.7) f() = ln ( + ) 7.8) f() = cos.
Calculus-Lab 4 ) Use the bisection method to find a solution of the equation 3 + 4 = with an approimation of two decimal points. ) Use the bisection method to find the intersection point of the graphs of y = e and y = with an approimation of two decimal points. ) Find the tangent line and orthogonal line to the graph of the function f() = ln( +3+)+e + at =. 3) Find the equation of the line passing trough the origin and tangent to the graph of f() = ln. 4) Find the derivative of the following function and compare its domain with the domain of the original function: 4.) f() = ln( +3 3 + ) 4.) f() = e cos ln(+) 4.3) f() = ln 4.4) f() = e + +sin 4.5) f() = ln e ( 3 ) 4.6) f() = ln 4.7) f() = (+) 4.8) f() = ( + ) cos. 5) Use l Hospital s rule the calculate the following limits: 5.) lim + ln() 5.) lim + 5.5) lim e 5.6) lim + e cos + ln (+) 5.3) lim + 5.4) lim + 5.7) lim ( + ) +3 ln( 3 +) 5.8) lim + ( sin ). 6) Determine for what values of a, b IR the following function is continuous and differentiable in all its domain { sin, < f() = ; a + b,. ) Find the derivative of the following function Calculus-Lab 5.) f() = ln( + ).) f() = arctan +. ) Find the Taylor polynomial of degree n at a = for the function.) f() = +.) f() = e. 3) Find the Taylor polynomial of third degree at a = for the function f() = ln. 4) Find maimal intervals where the function is increasing, decreasing, local maimum and minimum for f() = e +3. 5) Find absolute maimum and minimum of f() = 8 + 8 in the closed interval [, 6]. 6) After determining maimal intervals where the function is increasing (decreasing), local maimum, minimum, maimal intervals where the function is concave up (down) and infleion points, graph the function 6.) f() = + 6.) f() = e 6.3) f() = 3 6.4) f() = 3 4 8 3 + 6. Calculus-Lab 6 ) Find maimal intervals where the following function is concave up and concave down, infleion points..) f() = ln.) f() = 3 3 + 3. ) Write down the equation of all asymptotes of the following functions.) f() = +.) f() = e e e.3) f() = ln.4) f() = ln( + ). 3) Find domain, limits at boundary point, asymptotes, maimal intervals where the function is increasing (decreasing), local maimum, minimum, maimal intervals where the function is concave up (down), infleion points, and graph the function 3.) f() = + ln 3.) f() = e 3.3) f() = ln( + ) 3.4) f() = arctan( ).
Calculus-Lab 7 - Evaluate the following integrals ) cos( ) d ) 4) d 3 5) 7) e d 8) ln ) d ) 3) sin d 4) 6) e e d 7) + cos sin d 3) ( e d 6) sin( ) d 9) arctan d ) ln d 5) ( + ln d 8) ) e 3 d d 3 ln() d 3 sin( ) d arcsin d sin cos 3 d Calculus-Lab 8 - Evaluate the following integrals ) ( ) sin( ) d ) 4) sin d 5) 7) ) 3) d 8) 6 e e d ) e d 4) 4e (3 + ) cos( ) d 3) + e ( + e ) d 3 4 + d 6) d 3 d 9) 5 + 6 + ( d ) + + ) + ( )( d 5) + ) + 3 + 8 3 d + 9 (4 sin + 6) cos (sin() )(sin () + 4) d 5(sin() + ) cos (sin() + )(5 cos ()) d Calculus-Lab 9 A) Evaluate the following integrals ) 4) ( + ) cos( + ) d ) e + cos(e + ) d 5) 3 d 3) ( ) 3 4 sin ( ) d 6) + 5 + d ( 5) ln( + ) d 7) 4 + 5 d 8) + 3 3 + + 6 d 9) 3 + 3 + + d ) sin 3 () d ) 5 5 + d ) ( )e d B) The following calculation is wrong. Eplain why. [ = ] = = 3. C) Use a partition of the interval [, ] and the definition of definite integral to approimate the value of sin d.
Calculus-Lab - Evaluate the following improper integrals ) ( + )( + ) d ) d 3) + + e 3 d 4) d 5) 3 d 6) d 7) e d 8) + 5 d 9) 3 d ) 3) 6) 4 e e + d ) d 4) ( + ) d 7) + e + d ) d 5) + e d 8) + sin d e e + 3 d (sin )e d Calculus-Lab Discuss the convergence of the following series and, if possible, find its sum. ) k= 4) k= 7) k= ) k= k+ 3 k ) k=3 k+ k+ 5) k+ k 3 +k+3 k= 8) k= k! ) k ( 3) k+ 3k 4 3) k= 3 k k+ 6) k= 3k+ k 9) ( k k= k= k+ )k ) Calculus-Lab - Discuss the converge and absolute convergence of the following series k 3 k (k+)! ( 3 a ) a IR ( k k= k+ )k ) k= 4) k= ( ) k k+ ) k= ( ) k k 5) k= ( ) k+ k +4 sin k k +4 3) ( ) k k k+ k= 6) k= ( ) k (+ln k) a IR - Use the known criterions to discuss for what values of IR the following series is convergent ) k= ( ) k (+ ) k ) ( + k= )k 3) k= (k+5) 4 k! k Calculus-Lab 3 ) Find domain of f(, y) = ln ++y +y. ) Find the domain of f(, y, z) = y+z.
3) Graph f(, y) = + y, g(, y) = 4 + 9y, discuss level curves. 4) Approach zero along different paths to prove that DNE, lim (,y) (,) 4 4 +y lim 4 y (,y) (,) 4 +y lim (,y) (,) +y DNE, DNE. 5) Given f(, y, z) = y z 3 ln( + y + 3z), find D(f), f, f y, f z. 6) Given f(, y, z) = e y + 4 y 4 z 3, verify that f y = f y, f z = f z, f zy = f yz. Find f yz. 4) Is f(, y) = y a solution of Laplace equation? 7) Find the linearization of f(, y, z) = e + cos(y + z) at (, 4, 4 ). 8) Find tangent plane to z = ln( + y) at (, 3, ). 9) z = + y, = te t, y = e t, find dz dt. ) z = sin cos y, = (s t), y = s t, find z s ) u = y + yz + z, = st, y = e st, z = t, find u s ) Verify that y + 3 y = 3 + 8 defines y as a function of around (, 4), find dy. 3) Find f for f(, y) = ln( + y ), (, ). z t. u t. 4) Find f for f(, y, z) = e +y cos z + (y + ) sin, (, /). 5) Find D u f for f(, y, z) = 3e cos(yz), (,, ), v =<,, >. 6) Find D u f for f(, y, z) = + y 3z, (,, ), v =<,, >. 7) Find the maimal and minimal rate of change of f(, y) = e y + 3y at (, ) in the direction in which they occur. 8) Find the maimal and minimal rate of change of f(, y, z) = y + y z at (4,, ) in the direction in which they occur. 9) Find local maimum, minimum and saddle points for f(, y) = 6 3 + 3y + 6y. ) Find local maimum, minimum and saddle points for f(, y) = 4y 4 y 4. ) Find two numbers a b such that b a (6 ) d has largest value. Find a geometrical interpretation of the problem. ) Find absolute etrema of f(, y) = + y + y 6 + on the rectangle 5, 3 y. 3) The temperature of a heated plate is given by T (, y) = 4 4y + y. A bug walks on the plate along a circle centered at (, ) with radius 5. Find the coordinates of the hottest and coldest points reached by the bug and the temperature there. 4) Find the points on y = 54 nearest to the origin. Calculus-Lab 4 d ) Evaluate ) Evaluate ln 8 ln y D sin e +y ddy. da, where A is the triangle with vertices (, ), (, ), (, ). 3) Change order of integration to evaluate ln 3 ln 3 y/ e ddy. 4) Evaluate R e +y ddy, where R is the half disk with center (, ) and radius lying above the -ais by changing the integral into polar coordinates. 5) Evaluate with the use of a double integral the area of a disk of radius one. 6) Find the volume of the solid bounded by z =, and the paraboloid z = y. 7) Knowing that the average value of a function f over a region R is by definition Average(f(, y)) = f(, y) da Area(R) R
find the average value of f(, y) = cos(y) over the rectangle R = [, ] [, ]. 8) Use a substitution to evaluate ( + y) 3 y da, where R is the closed region bounded by y =, y =, + y =, + y =. 9) Use a substitution to evaluate R + y(y ) dyd. Calculus-Lab 5 ) Find the area of the region bounded by the curves y =, y =, y =. ) Find the volume of the solid of rotation obtained rotating the region under the graph of y = arcsin, with [, ], around the -ais. 3) Use an integral to find the area of the surface of the cone with base radius r and hight h. 4) Using 4 metres of fence we need to surround three sides of a rectangular region of the garden with the fourth side formed by the house wall. What are the measures of the rectangle s sides that make the area of the surrounded garden maimal? 5) An isosceles trapezoid has three consecutive sides equal to cm, find the size of the fourth side that makes maimal the area of the trapezoid. 6) Consider the finite region R bounded by the curves y = sin( ), y =, =. Find the volume of the solid obtained by revolving the region R about the y-ais.