Application of derivatives Questions and Answers

Simplify the following trigonometric expression sect cos O al 1 b cote Od tan 0 d sin 0

oblem 5 For each change equation 1 find the equilibrium solution s if any exist 2 graph the equilibrium solution s along with the graphs of the graphs of y z for the given initial values dy y 3 y 0 1 and y 0 4 da dy b y 2 y 4 y 0 1 y 0 3 and y 0 5 da a

Find the degree measure that corresponds to the given radian measure 5F O a 108 Ob 300 R Od 300 Q108

TABLE OF INDEFINITE INTEGRALS f cf x dx c f f x dx fkdx kx C Xn 1 fxndx C n 1 n 1 fex dx ex C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x tan x dx sec x C S 1 x 1 dx tan x C sinh x dx cosh x C 18 f x g x dx f f x dx f g x dx dx In x C faxdx ax In a cos x dx sin x C fcsc x dx cot x C csc csc x cot x dx C x cot x dx csc x C CSC dx sin x C 1 x cosh x dx sinh x C sin 2x dx

TABLE OF INDEFINITE INTEGRALS fcf x dx cff x dx fkdx dx kx C fxndx C n 1 dx dx In x C Xn 1 n 1 fe dx e C sin x dx cos x C sec x dx tan x C sec x tan x dx sec x C 1 x 1 sinh x dx cosh x C f f x g x dx f f x dx f g x dx dx tan x C ax a dx In a I cos x dx cos x dx sin x C C fcsc x dx cot x C S CSC x cot x dx csc x C x dx f cosh x dx dx sin x C sinh x C 10 v v 2 dv 2

Given the absolute function graphed below find the following 4 a Domain b Range Increasing 6 S Decreasing 4 3 O 7 4 2 a 4 2 c On what interval is the function increasing and on what interval is the function decreasing 4

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or or 81X 3x 2 X

A line passes through the points P 1 7 8 and Q 10 2 2 Find the standard parametric equations for the line written using the base point P 1 7 8 and the components of the vector PQ x 0 y 0 z 0 X Type expressions using t as the variable COLO

4 Evaluate the indefinite integral 4x 3 e z 3x 7 dx a e 4x 3 C b e x 3x 7 C c 4x 3 e x 3x 7 C d e x 3x

15 Find the derivative of the function g x S dz ln a b 1 c ln 24 1 d In 24 1

px 3x 14 Find the derivative of the function g x x 1 t dt a 1 x 3x b 1 t 2t 3 1 c 2 1 2 3x 3 2x 3 d 1 x 3x 2x 3

16 Evaluate sin x dx a 2 b 0 c d 1 2

3 Which substitution will you make to evaluate the indefinite integral 4x 3 sec x 3x 7 dx 4x 4 c u a u x 3x 7 b u 4x 4 c u 4x 3x d u x 3

1 Evaluate the indefinite integral f cos 3x 2 dx a 3 sin 3x 2 C b sin 3x 2 C sin 3x 2 C d cos 3x 2 C e 3 tan 3x 2 C

A particle has a velocity modeled by v t 6 2 ln t 8 measured in cm s over the interval 1 t 6 What is the distance traveled from t 1 to t 6 You may either enter an exact answer or round your answer to the nearest tenth of a centimeter Provide your answer below

Evaluate the indefinite integral given below 6x x 4x x x2 dx

A triangle has an area of 7 in and two of the sides of the triangle have lengths 5 in and 4 in Find the angle included by these two sides 45 573 O 0 795 44 427 no correct answer

Find a function that models the simple harmonic motion having the given properties Assume that the displacement is zero at time t amplitude 2 3 m frequency 0 5 Hz O f t 2 3 sin t O f t 2 3 sin 2t O f t 2 3 sin 2 t O f t 2 3 sin at O f t 2 3 sin 0 5 t

Graph g x x on the Cartesian plane to the right The domain of g x is The range of g x is If any of the above do not exist leave the answer box blank 654 ch 2 10 9 8 7 6 5 4 3 2 21 6 7 8 9 10

Find the terminal point P x y on the unit circle determined by t O 0 87 0 50 O 0 87 0 50 O 0 87 0 50 O 0 99 0 64 5TC 6 Round each value to the nearest hundred

Use the Law of Sines to find x Let c 21 LA 58 25 24 O 46 23 A C C X 70 B

What is the side labeled x equal to if y 22 22 O 11 O 44 Y 30 X

Find the reference angle for the angle measuring O O 4 T 11 4T 15 11 4 4 T 11 15T 11

D C 23 S x 1 A B AH 24 S x 2 1 A B 6 D 2

Divide Write your answer in fraction form 16 x 6x 10x 10x 16 x 1 11 A x 5x 5x 5 x 1 B r 5x2 5x 5 6 C r 5x2 5x 5 D x 3x 4x 6 x 1 6 x 1 6 x 1 17 4x 25x 28x 6 x 5 A 4x 5x 3 B 4x 5x 3 C 4x 5x 3 D 4x 5x 3 9 x 5 2 x 5 2 x 5 9 x 5

State if the given functions are inverses 10 x 2 5 g x s x x 2 A No B Yes 6 S x x 2 8 x A No 2 3 B Yes

Which of the following sets are subspaces of R OA 7x 4y 5x 9y 6x 9y x y arbitrary numbers OB x x 2 x 8 x arbitrary number c x y z 2x 8y 3z 0 D x y z x y z OE x y z x y z 7 A

Do 4 The natatorium pool is 25 0 meters by 25 0 yards Assuming the pool is uniformly 396 cm deep what is the volume of the pool in liters 2 26x10 liters 2 0 LICE

Do 3 Determine the number of grams in the following A 15 85 moles of Helium 63 45gte 142atte 1 th

B 71 60 g Si 2 549 mores Si 28 085951 71 600 Spo mole si

1 point Find the equation for the line tangent to the parametric curve Y X y at the points where t 4 and t 4 For t 4 the tangent line in form y mx b is y For t 4 the tangent line is t 16t 16t 14

1 point dy dx parameter Find and d at the given point without eliminating the dx X dy dx t 6 y t t t 2 d y dx

21 Find the exact values of the three trigonometric functions sine cosine tangent of the real number t a b a D 5 13

1 point Find the exact length of the curve given by 1 t 3 Length X 2t y 0 t 1

Find all angles a in radians that satisfy the equation Your answers should be formulas that look like angle k period where k 2 1 0 1 2 ranges over the integers 0 angle period and period is the smallest positive period that makes the formula work Note decimal answers are not allowed so for example if appears in your answer call it pi not 3 14159 The solutions of the equation are C or 2 sin x tan x sin x 0 A x

Find all solutions of the equation tan5 x 9 tan x 0 The answer is Ak where k is any integer the constant A

13 Simplify the expression algebraically sin 7x 7y sin 7x 7y O sin 7x cos 7y O sin 7x 7y O2sin 7x cos 7y O2 sin 7x O sin 7x 7y

7 Complete the table 0 sin 0 0 7 0 8 0 9 1 1 1

Find all solutions of the equation cos x sin x 2 cos x 0 The answer is A Bkn where k is any integer and 0 A T A 9 2 B

Find all solutions of the equation sec 4x 2 0 The answer is A and B where k is any integer and 0 A B 2 A B 9

Find the all solutions of this equation in the interval 0 2 X FI Give your answers in radians Do not use decimal approximations Use pi to represent Use commas to separate your answers tan x 10

If sin x A B cos 2x C cos 4x then A B C 4

If cos x 4 5 csc x 0 then sin 2x cos 2x tan 2x A G C

Use half angle formulas to fill in the blanks in the identity below sin 2x 4 cos x cos

Simplify the expression as much as possible sin 2x cos x help formulas

If tan 0 7 8 and cos 0 0 then sin 20 cos 20 tan 20 G 9 Fl

Find the exact value 8 2 cos 7 17 sin

Evaluate the following expressions Your answer must be an angle in radians and in the interval 2 2 Note that is already provided in the answer so you just have to write the appropriate multiple E g if the answer is 2 you should write 1 2 Do not use decimal numbers The answer should be a fraction or an integer a tan tan 7 6 b tan tan 57 4 c tan tan 7 4 G TT T

Evaluate the following expressions 2 sin sin 1 a b cos cos 0 1 c tan tan 1 D A CI

Find the domain and the range of g x sin 3x 1 A sas y Domain Range G F