Differential Equaitons Equations

Transcription

1 Welcome to Multivariable Calculus / Dierential Equaitons Equations The Attached Packet is or all students who are planning to take Multibariable Multivariable Calculus/ Dierential Equations in the all. The irst quiz will include the materials covered in this packet. I there are any topics that you do not understand, you are epected to get help on your own. I look orward to meeting each one o you. Have a great summer! Mr. Choi Have a Great Summer!

2 Calculus Summer Review Packet I. Trigonometry Review o. Find the eact radian measure o 0.. Find the eact degree measure o π. 6. Find the values o the trigonometric unctions i θ is an acute angle and tanθ.. I θ is in standard position and Q(, ) is on the terminal side oθ, ind the values o the trigonometric unctions.. Find the eact value: (a) π sin (b) π tan 6 (c) sin π (d) π sec 6 6. I ( ) cos, show that ( ) cos sin h cos h sin h h h h 7. Veriy the identities: (a) ( t)( t) sin tan (b) csc θ cot θ tan θ (c) csc cot csc cot 8. Find all solutions o the equations: (a) cos θ 0 (b) sinθ 0 9. Find the solution o the equations or 0 θ < π. (a) sin θ sin θ (b) (c) sin θ sinθ 0 tanθ sec θ 0

3 II. Limits Find the it, i it eists. 0. ( ). ( )( ) ( )( ) Reer to the graph to ind each it: (a) (b) (c) (d) 0 (e) 0 () 0.

4 Find each it, i it eists.. (a) <,, (b) (c). (a),, > (b) (c). <,,, > (a) (b) (c) Find the it, i it eists: A unction satisies the given conditions. Sketch a possible graph o, assuming that does not cross a horizontal asymptote. 8.,,, ) and 9.,,,, ( III. DEFINITION OF THE DERIVATIVE : 0. Use the deinition o the derivative to ind (a) (b)

5 IV. TECHNIQUES OF DIFFERENTIATION AND TANGENT LINE PROBLEMS. Find the derivative using the power rule, the product rule or the quotient rule.. 8. ( 7)( ). 6. ( ) Find the -coordinate o a point on the graph o y at which the tangent line is: (a) horizontal (b) parallel to the line y 8 8. Find the equation o the line tangent to the curve at the point (, 7 ) y P. g,, g and 9. I and are unctions such that epression: g evaluate the ( ) () ) () (c) ( (a) g (b) ( g ) (d) g ( g ) () (e) () 0. A weather balloon is released and rises vertically such that its distance s( t) above the ground during the irst 0 seconds o light is given by s() t 6 t t where s( t ) is in eet and t is in seconds. (a) Find the velocity o the balloon at t, t, t 8 (b) Find the velocity o the balloon at the instant the balloon is 0 eet above the ground Use the power rule, the product rule, the quotient rule and/or the rules or derivatives o trigonometric unctions to ind the derivatives:. ( ) 7tan. sin. cot tan sin tan cot.. ( ) ( csc )

6 6. I cos sin ; 0 < π ; ind the points where the tangent line is horizontal. Use the chain rule to ind the derivative: 7. ( 8) 8. sin ( ) ( ) ( ). ( 6 7) ( 8 9). sin cos V. IMPLICIT DIFFERENTIATION. 8 y 0. y y 0. y secy 6. sin ( y) 7. Find the slope o the tangent line to the graph o y y 7,. at the point 8. I y ; P, (a) Find the equation o the tangent line and the normal line to the graph o the equation at the point P. (b) Find the -coordinate o the graph at which the tangent line is horizontal. 9. Assuming that the equation y determines a unction such that ; ind y d y d. and 60. Suppose and g are unctions such that ; ; ; g ; g g. Find the value o each o the ollowing at. (a) ( g ) (b) ( ) (c) ( ) g g (d) ( g ) (e) g () g

7 Problems 6-6 should be completed without the use o a calculator. 6. (987 AB) Let sin. a. What is the domain o?. b. Find c. What is the domain o.. d. Write an equation or the line tangent to the graph at (988 AB) Let be the unction given by a. Find the domain o. b. Describe the symmetry, i any, o the graph o.. c. Find d. Find the slope o the line normal to the graph o at (989 AB-) Let be the unction given by a. Find the zeros o.. b. Write an equation o the line tangent to the graph o at 6. (989 AB-) Let be the unction given by. a. Find the domain o. b. Write an equation or each vertical asymptote to the graph o. c. Write an equation or each horizontal asymptote to the graph o.. d. Find 6. (99 AB, BC) Consider the curve deined by the equation y cos y or 0 y π a. Find dy in terms o y. d b. Write an equation or each vertical tangent to the curve. d y c. Find in terms o y. d

8 . 7π 6 SUMMER REVIEW PACKET ANSWERS 6. LDNE 7. LDNE. 0 o. sinθ cosθ tanθ. sinθ cosθ tanθ cscθ secθ cotθ cscθ secθ cotθ 8. (a) (b) (c) LDNE (d) (e) () LDNE 0 9. (a) (b) (c) (d) (e) () 0. (a) (b) (c) LDNE (d) (e) (). (a) (b) (c) (d) (e) (). (a) 0 (b) (c) LDNE. (a) (b) (c). (a) (b) (c). (a) (b) (c) (d). 8. (a) 8. (b) 9. (a) 9. (b) 9. (c) 0. π π nπ ; nπ π 8nπ 7π 8nπ ; π, π, π 6 6 π π, π π 0, π,, 6. LDNE (a) 0 (b). 0. LDNE

9 ( ) 0 ( ) ( ) 0 ( ) 6 ( 6 9) ( ) 7. (a) or (b) 0 or. cos sin sin cos 8. y 8 9. (a) (b) (c) (d) (e) () 9 0. (a) v() s () t v s v s t t (b) /sec 0 /sec /sec t.708; v.708. t sec 7sec.. cos sin. csc cot sec tan.. csc cot csc or ( sin tan sin sec tan cos csc csc cot csc cot π 7π 6., or, 7. ( 8) 8. 8( ) cos( ) ). ( 6 7) ( 8 9)( 68 8).. dy 8 d y dy 0 y d 8y dy secy. d sec y tan y y (a) 9. dy y d cos y y ( y) cos dy cos or d y y or y ( ) dy d Tangent Line : y Normal Line : y (b) 0 dy d y y 9 ; d d 6y 60. (a) (b) 7 (c)

10 (d) 0 9 (e) () 9 7 D set o all real numbers OR 6. a. { } cos sin b. π c. D set o all real π n, n any integer y OR y d. ( 0) 6. a. D :,, 0 b. Graph is symmetric with respect to the y-ais. c. 6 d. Slope o the normal ( ) a. Zeros:,, - b. y ( ) OR y 8 6. a. D : < or > b. Vertical asymptotes:, c. Horizontal asymptotes: y, y d. 6. a. dy π,0 y π and y d sin y π d y cos y d sin y b. Equation o vertical tangent: c.