Eecrices Mathematics FRDIS Vytvořeno s podporou projektu Průřezová inovace studijních programů Lesnické a dřevařské fakulty MENDELU v Brně (LDF) s ohledem na disciplíny společného základu http://akademie. ldf.mendelu.cz/cz (reg. č. CZ..7/../) za přispění finančních prostředků EU a státního rozpočtu České republiky. Simona Fišnarová MENDELU
Derivatives Find the derivative of the functions:. y = + ln sin tg. y = + ln cos. y = + arctg e 4. y = ln + sin 5. y = ln + (sin + 4)5 y = e ( + ) + sin(ln ) y = e + ln(cos ) y = sin sin + e y = e ( + ) + cos(ln ). y = cos + ln(sin ). y = + + sin(e ). y = ln + sin (e ). y = sin + ln(cos ) 4. y = + + cos(e ) 5. y = sin ln + e y = e + ( + ln ) 4 y = e ( + ) + ln y = + ( ) y = ln +
Investigation of functions. Polynomials Investigate the following polynomials: (a) Find the domain, find the parts above/below -ais. (b) Monotonicity. (c) Concavity. (d) Sketch the graph.. y = 6 6.. y = 5 5 4.. y =. 4. y = 4 +. 5. y = 4 4. y = +. y = 4 + 4. y = 6 + y = 9 +.. Rational functions Investigate the following functions: (a) Find the domain, find the parts above/below -ais. (b) Monotonicity. (c) Concavity. (d) Asymptotes. (e) Sketch the graph.. y = Hint: y = 4 ( ), y =. y = + Hint: y = + ( + ), y =. y = ( ) 8 ( ). ( + ). Hint: y = ( ), y = + 8 ( ) 4. 4. y = ( + ) Hint: y = 5. y = ( + ). Hint: y = y = +. Hint: y = ( + ), y = ( + ) 4. ( + ), y = 4 ( + ) 4. ( + ), y = 6 ( + ).
y = ( ). Hint: y = ( ), y = + 4 ( ) 4. y = ( ). Hint: y = ( ), y = 8 ( ). 4 y = + +. Hint: y = + ( + ), y =. y =. ( + ). Hint: y = 4, y = 5. 4
Integrals. Basic integrals. ( + ) d 5. d d.. d d 4 d + d 4. d d. + + d. Basic substitutions. e d. cos d. e d 4. sin 5 d 5. e 4 d. cos d sin d ( 5) 5 d d ( + 5) d... 4. + d tg d cotg d + 5 d. Substitutions. sin d. sin( + ) d. e d 4. cos d 5. e + d. sin cos d cos sin d sin cos 4 d sin sin cos d + 5 d... 4. cos sin d cos + 7 + d d + d 5
.4 By parts. cos d 5. ln d arctg d. e d sin d. arctg d. ln d ln d. e d 4. e d cos d. sin d.5 Definite integrals. ( + + ) d 4. ( + ) d π cos d. ( + ) d 5. ( + ) d e + d. ( + ) d + + d + d 6
4 Vectors, matrices, determinants 4. Operations with matrices. Let A =, B =. Calculate (A I) T B, where I is the identity matri.. Let A =, B =. Calculate (A T + I) B, where I is the identity matri.. Let A =, B =, Calculate (A B), where I is the identity matri. 4. Let Calculate A. 5 A =. 5. Let A =. Calculate (A T I)A, where I is the identity matri. 4. Determiannts, inverse matri, linear dependence/independence of vectors. Let (a) Evaluate the determinant of A. A = (b) Using the value of det A answer the following questions: (i) Are the rows of A linearly dependent or independent? (ii) Is rank (A) >, rank (A) < or rank (A) =? (iii) Does the inverse matri A eist? If A eists, find it. 7
. Let A =. 4 (a) Evaluate the determinant of A. (b) Using the value of det A answer the following questions: (i) Are the rows of A linearly dependent or independent? (ii) Is rank (A) >, rank (A) < or rank (A) =? (iii) Does the inverse matri A eist? If A eists, find it.. Let 4. (a) Evaluate the determinant of A. (b) Using the value of det A answer the following questions: (i) Are the rows of A linearly dependent or independent? (ii) Is rank (A) >, rank (A) < or rank (A) =? (iii) Does the inverse matri A eist? If A eists, find it. 4. Let A =. (a) Evaluate the determinant of A. (b) Using the value of det A answer the following questions: (i) Are the rows of A linearly dependent or independent? (ii) Is rank (A) >, rank (A) < or rank (A) =? (iii) Does the inverse matri A eist? If A eists, find it. 5. Let A =. 5 (a) Evaluate the determinant of A. (b) Using the value of det A answer the following questions: (i) Are the rows of A linearly dependent or independent? (ii) Is rank (A) >, rank (A) < or rank (A) =? (iii) Does the inverse matri A eist? If A eists, find it. Evaluate the determinants:, 5, 5, 5. Are the following vectors liearly dependent or independent? 8
(a) a = (,,, ), b = (,,, ), c = (,,, ), d = (,,, ) (b) a = (,,, ), b = (,,, ), c = (,,, ), d = (,,, ) (c) a = (,,, ), b = (,,, ), c = (,,, ), d = (,,, ) 9
5 Systems of linear equations Solve the following systems using the Gauss method. (a) Find the rank of the coeffcient and of the augmented matri and determine how many solutions the system has. (b) Find the solution of the system (if eists any).. 8 + 6 + 4 = 9 + + 5 4 = + + 4 = 8 + 7 4 = 4. + + 4 4 = 4 + 4 = + + 4 = 5 + + 4 4 = 4.. + + 4 = + + 4 = + + 4 = + 4 =. + 5 + 4 = + 4 4 = + + 4 = 6 + 4 + 6 4 =.. + 4 = + + 5 4 = + + 4 4 = + 4 =. + + 4 = + + + 4 = 6 4 + 5 + 4 = 6 + 4 6 + 4 = 4.. + + 4 = + + 4 4 = 4 + 4 = 4 7 6 + 5 8 4 = 4. + + 4 = + 4 = + + 4 = + + =. 5.. + + 4 = + + 5 4 = + 4 = 8 + + 8 4 =. + + 4 = + + 4 = + + + 4 = + 4 =.. + + 4 = + 5 + 4 = + 4 =9 + 4 + 9 4 =. + + 5 4 = + + 4 = + 4 7 4 = + 4 =.
Solve the following systems using (a) the Cramer rule, (b) the inverse of the coefficient matri... + y = 4 + 7y = + y = + 5y =