King Fahd University of Petroleum and Minerals Prep-Year Math Program

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1 King Fahd University of Petroleum and Minerals Prep-Year Math Program Math 00 Class Test II Textbook Sections: 6. to 7.5 Term 17 Time Allowed: 90 Minutes Student s Name: ID #:. Section:. Serial Number:. Provide neat and complete solutions. Show all necessary steps for full credit and write the answer in simplest form. No Calculators, Cameras, or Mobiles are allowed during this exam. Question Points Student s Score Total

2 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page 1 of 5 Q1. (5 points): Given y 1+ cos x, 0 x. (a): Graph the function over the given interval. (b): Find the intervals where the function is increasing over the given interval. Find the intervals where the function is decreasing over the given interval. Solution (a): 0 x + x x 6 (b): 5,,, ,,, 6 6 Q. (5 points): Given y sec x, 0 x. (a): Graph the function over the given interval. (b): Determine the equations of vertical asymptotes over the given interval. Find the intervals where the function is decreasing over the given interval. Solution (a): 0 x + x x 6 y cos x (b): x 0, x, x, x 5,,,,, 6 6 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page 1 of 5

3 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5 Q. (5 points): Given the function y tan x where < x < (a) Graph the function over the given interval (b) Find the x-intercepts over the given interval (c) Find the equation(s) of vertical asymptote over the given interval. Solution (a): < x + < < x + < < x < 0 < x < y x (b): x, x, x x 0, x Q. (5 points): If sin(7 ) t, then sin6 + sin07 Solution: sin6 + sin07 sin(6 70 ) + sin(60 07 ) sin1 sin(5 ) sin(10 1 ) sin(5 ) + sin7 cos(90 5 ) t cos(7) t t 1 sin 7 1 t [ ] Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5

4 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5 Q5. (5 points): Given: f ( x) cosx + sinx. 11 (a): Sketch the graph of f over the interval (b): Find the interval where the function is decreasing over the interval Find the interval where the graph is above the x-axis over the interval (d): Find the interval where the graph is below the x-axis over the interval 1 1 Solution (a): ( ) f ( x) cosx + sin x 1 + sin( x + α) sin( x + α) 1 sinα α f ( x) sin x 6 cosα x + x x (b): Decreasing on, 6 The graph is above the x-axis 5, 1 1 Q6. (5 points) (7.1 Textbook Exercises ): Prove the following identities (Show all steps) (a): ln tan x sin x ln sin x + ln sec x (b): ln tan x + ln cot x 0 e e e e x + lnsinx x (d): e e sin x ln x (e): xe x sin x tan x sec x cos x Solution: sin x (a): LHS ln tan x sin x ln t an x sin x ln tan x + ln sin x ln + ln sin x cos x ln sinx ln cos x + ln sin x ln sin x + ln cos x ln sin x + ln sec x RHS (b): LHS ln tan x + ln cot x ln tan x cot x ln1 0 RHS sin tan 1 cos sec 1 1 cos sec 1 cos sec cos sec x x x x x + x x + x x x LHS e e e e e e e e RHS x+ lnsinx x lnsinx x lnsinx x x (d): e e e e e e sin x e sin x RHS lnx ln x (e): xe x e x x x RHS Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5

5 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5 Q7. (5 points): Whenever possible, find the value of each of the following: (a): cot ( ) (b): sin ( sin ) cos cos 5 (d): sec( tan ) (e): cos tan 1 5 cot ( ) tan ( ) 6 Solution (a): 1 1 (b): 1 ( ) sin 1 ( 0) sin sin cos cos undefined because D cos [ 1,1] (d): Let θ tan where < θ < then tanθ, θ QI tanθ x 1, y and r 5 1 sec( tan 1 ) se 5 r cθ 5 x 1 cos tan cos tan cos undefined because [ 1,1] D (e): ( ) 6 6 Q. (5 points) (7.1 Textbook Exercise 10): If sin + cos A + B sin cos then find A and B. (Show your work) 6 6 Solution: sin + cos ( sin ) + ( cos ) + Another Method: 6 6 sin + cos sin + cos ( sin + cos )( sin sin cos + cos ) ( 1) ( sin + cos + sin cos sin cos sin cos ) ( sin cos ) + sin cos sin cos 1 sin cos ( ) ( ) ( 1 cos ) ( cos ) + A B sin cos then A 1 and B s + cos cos + 6 ( 1 co ) co 1 cos + cos 1 cos sin 1 sin cos 1 cos (1 cos ) ( s ) cos Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page of 5

6 Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page 5 of 5 Another Method: ( ) ( cos ) 6 6 sin + cos sin + ( sin + cos )( sin sin cos + cos ) sin sin cos + cos sin sin sin cos + cos cos sin ( 1 cos ) sin cos + cos (1 sin ) sin sin cos sin cos + cos sin cos sin + cos sin cos 1 sin cos Q9. (5 points): (7. Textbook Exercise 70): Refer to the figure. Find the exact value of γ?. Solution: Let A and B be the two angles shown in the diagram. Then A 90 α and B α γ α + γ α 90 5 γ α + tanα + tan tan γ tan( α + χ) 1 tanα tan tan γ γ tan OR γ α + cos + cos Q10. (5 points): Solve sin θcosθ sin θ cosθ + 6 0, where 0 θ < 10 Solution: sin θcosθ sin θ cosθ + 0 sin θ ( cosθ ) (cosθ ) 0 ( cosθ )( sin θ ) 0 cosθ 0, sin θ 0 cosθ, sin θ θ 0 + k60, θ 0 + k60, θ 5 + k60, θ 15 + k60 where k is an integer. θ 10 + k10, θ k10, θ 15 + k10, θ 5 + k10 k 0 θ 10, 110, 15, 5 k 1 θ 10, 0 [0, 10 ), 15, 165 SS 10,15,5,110,10,15,165 { } Math 00 Test II, (Textbook: 6. and 7.5), Instructor: Sayed Omar, Term 17, Page 5 of 5