FINAL EXAM REVIEW PACKET

Transcription

1 FINAL EXAM REVIEW PACKET Chapter 4.8, 5, 6.3, 9.1, 9., 9.3, 10.6 Chapter 1 Derivatives: power rule, trig, word problems 1) A ship leaves a port at 1:00 PM traveling at 13 knots directly north. Another ship leaves the same port at :00 PM traveling due east at 15 knots. At 9:00 PM, how far apart are the ships? Along what heading will the northern ship have to travel to reach the eastern ship which is now stopped? ) An airplane leaves an airport and travels due east for 55 miles. It then heads due south for 330 miles. From its current position, along what heading (bearing) should it travel to reach the airport and how far is it? 3) Simplify (1 tan y)(cos y 1) 3 csc x(cos x tan x sin x) sin (5 x) cos (5 x) c) d) cos x cos x 4 4 tan 5x tan y e) 1 tan 5xtan y f) sin x tan x cos xsec x g) tan x sin xcos x sin x h) (sec x1)(1 cos x) sin x i) 1 1 sec x tan x sec x tan x 1 sin (sin x), x j) k) cot A(sec A cos A) 4) Solve for

2 l) cos 3sin 3 0 m) 8tan x 5 n) 4sin(3 x) 1 0 o) sec( x 35 ) 4 p) sin x cos x 5) Solve for 0 x : 3sin x sin x1 0 3 csc x 7 c) cos x cos x 6) Find the solutions of each equation in the indicated interval. (cos x 3)(tan x 3) 0, x tan xsin x tan x 0, x 0 c) 3sin x 3cos x 0,90 x 450 d) sin x cos x sec x, 90 x 90 7) Evaluate without a calculator. c) 3 sin( x) cos x sec csc tan cot tan ( 3) d) cos(165 ) e) f) g) sin cos sin cos cos sin tan 4 where 1 tan 3 1 3

3 8) Given that 3 sin and cos, where, find each of the following. 5 cos sin c) sin( ) d) cos( ) e) 1 tan f) sin g) cos( ) h) tan( ) 9) Suppose 3 3 cos x,csc y,0 x, and y, find sin( x y) ) If A is acute and 4 cos A, then find sin 4A. (exact value) 5 11) Verify each identity csc sin sin cos sin cot tan sec c) csc cot csc cot 1 d) csc 4 csc cot 4 cot sin x e) csc x cot x 1 cos x f) 8csc 3cot 3 5csc g) 1 1 cot 1 sec 1 sec h) cos cos i) sin( ) cot cot sinsin j) cos( ) cos( ) cos cos

4 1 sin sin cos k) l) m) 1 tan cos 1 tan sec 1 cos 1) Consider the sequence 3, 6, 11, 18, 7, State whether the sequence is arithmetic, geometric, or neither. Find a 6. c) Find a formula for a n. d) Find a ) Find a n for the sequence -, 6, -18, 54, 14) Express as an infinite series and as a rational number. 15) Find S 6 as an exact value for the series: ) Find an explicit formula for the sequence that is made up of all digit numbers that are divisible by 3. 17) Find the sum of the first 50 odd integers. 18) Find the sum of all numbers less than 1000 that are divisible by 3. 19) Find the sum: 6 1 i + 1 i= 0) Find the third partial sum and the sum of the series: i i 1) Use sigma notation to write the sum: (5) 3(6) 3(7) 3( n)

5 ) Simplify: 3 n! 3n! n! n1! n n 1! 1! 3) Eliminate the parameter and write the corresponding rectangular equation. x 3t y 3t x cos y sin 4) Find a set of parametric equations for the rectangular equation using t x and t 4 x x4y 5 5) Sketch the curve given by the parametric equations x t 3 and y t, 1 t 3

6 6) Evaluate each limit or state that it does not exist. 3 x x lim x0 x x lim3x 5 c) d) e) f) g) h) i) x 3 x 5x lim x x x 1 x 4 lim x4 x x 1 lim 1 x1 x 1 x 4 lim x 0 x 10 lim x3 x 3 x 5 lim x5 x 5 3x lim x x 9 j) lim x 4 4x 3 5x x 4 x 6 k) lim x x

7 7)

8 8) George Bush goes skydiving. He jumps from a plane at an altitude of 0,000 feet. Find his velocity after 6 seconds. If George is supposed to release the chute 10 seconds before would be impact when should he release the chute? c) How fast is he falling when the chute is released? 9) A marshmallow was thrown upward from the top of a 300 foot building with an initial velocity of 30 ft/ sec. Find the velocity of the marshmallow at seconds. Lindsay Lohan lives on the 4 th floor of the building (60 feet above ground) and just as she was sticking her head out to wave to Kim Kardashian, the marshmallow hit her. How long was the marshmallow in motion before it hit Lindsay? c) What was the velocity of the marshmallow when it hit Lindsay? 30) Use the definition of the derivative to find the derivative of: f ( x) 3x 4x 7 1 f( x) x 5 31) Find the derivative: f(x) = 3x 1 x+5 y = 6x(cos x)

9 c) g(x) = x 4 d) s(t) = t 5 6t 3 + 5t 10 3) Find the equation of the tangent line (in slope-intercept form) to the curve 3 f ( x) x 4x 5x 9 when x 1. 33) Find the area between the curve 4 left endpoint rectangles 4 right endpoint rectangles c) 4 midpoint rectangles d) the limit process f ( x) x x 11 and the x-axis on the interval [-1,3] using: 34) Use the formal definition of continuity to determine if f( x ) is continuous. x 4 x f x x x sin x x 0 ( ) Sketch a curve the satisfies the following:

10 36. Let v = 3,7 and w = 8,7 Find: q) v r) v w s) 3v + 7w 37. Find the component form and magnitude of a vector that has initial point (3, -8) and terminal point (-,6). 38. Are the vectors equivalent? How do you know?