Calculus Questions

In each triangle below find sin A cos A sin B and cos B 7 8 9 A A 3 1 B C 2 C 1 B 10 A C 4 3 B 2 C B

6 Using rules to combine known integral values Activate Suppose 7 5 4 5 6 f x dx 7 5 6 f x dx 10 10f x 8 dx 7 5 4 5 5 f x dx 8 f x dx 10 9

Problems 1 6 refer to right triangle ABC with C 90 In each case use the given information to find the six trigonometric functions of A 1 b 3 c 5 2 b 5 c 13

6 8 pts Answer the following questions about the rational function f x a Write the equations of any vertical asymptotes of the graph of f f x 3 x 5 x 1 x b This function has a horizontal asymptote Write its equation x Y 3 x 5 x 1 x 2x 8 16x 1x 15x c Find the y intercept of the graph of f Write your answer as an ordered pair If this graph doesn t have a y intercept say so 2X 8 x 6x 12x d Find any x intercepts of the graph of f Write your answer s as an ordered pair s If this graph doesn t 15X have any x intercepts say so 12 12 5

Find the area of the surface generated when the given curve is revolved about the given axis y 22 x for 4 x 15 about the x axis The area of the surface is square units Type an exact answer using as needed

Evaluate the following limit Use l H pital s Rule when it is convenient and applicable y 5y 36 y 4 32 y y lim lim y 5y 36 y 4 32 y y 11 Type an exact answer

4 pts Find the inverse of the function f x f x X 2 X S X 2 X 5 x2 x5 0 4 y

Assume f is a nonnegative function with a continuous first derivative on a b The curve y f x on a b is revolved about the x axis Explain how to find the area of the surface that is gen What is the area of the surface generated when the graph of f on the interval a b is revolved about the x axis O A To find the area of the surface that is generated evaluate the integral S 2x x 1 F x dx B To find the area of the surface that is generated evaluate the integral S OC To find the area of the surface that is generated evaluate the integral S OD To find the area of the surface that is generated evaluate the integral S b x 1 x 1 P x dx b 2x 1 f x dx a Sz f x 1 f x dx a

4 2 pts Which of the rational functions shown below have an oblique asymptote in their graphs x 1 2 3x 1 x 2 x 2x x 5 f x x 5 g x h x 5x 2 X 3 j x 1

8 4 pts The rational function g x asymptote 2x 3x 5 x 1 2x 3x s X 1 has an oblique asymptote Find the equation of its oblique 4X 8X 1x 4x 8x IX

A frustum of a cone is generated by revolving the graph of y 5x on the interval 3 7 about the x axis What is the area of the surface of the frustu The area of the surface is 50x656 square units Type an exact answer using as needed

pts Find the quotient and remainder using polynomial long division 4x 9x 1 x 2

f x x x 6 x 1 3 6 pts Given the rational function 6 a show your algebraic work in finding the x intercepts and y intercept of the graph of f and x 2 x 6 12 X b find the equations of all of the asymptotes vertical horizontal and or oblique of the graph of f f x x f x 6 x 2

2 4 pts 4 pts Sketch the graph of the function f x the asymptotes of the graph using dashed lines x 4 2 3 Clearly show X 4

The function y f x is graphed below What is the average rate of change of the function f x on the interval 7 x 2 6 40 30 39

Determine the point on the graph of the unit circle that 3x 3x 3x andsin 2 2 corresponds to Then findcoS 2 V 3x state which functions are undefined for 2 The point on the unit circle is and C 1 0 Lo 12 Ay 1 0 1 0 0 1 1 0 5

Find the circular function values for the given number s State which functions are undefined for s Do not use calculator S 17x 2 OA cos OB The function is undefined Select the correct choice below and if necessary fill in the answer box to complete your choice 17x 2 Type an integer or a fraction 1 0 2 The function is undefined Ay 2 0 0 1 P 1 0 0 1 2x Q S A

se the identity cos s sin s 1 to find the value of x or y as appropriate Then assuming that s ne given point on the unit circle find the six circular function values for s 21 X 29 in s 29 OS S ans 21 29 x 0 Type an integer or a simplified fraction Type an integer or a simplified fraction 20 29 Type an integer or a simplified fraction Type an integer or a simplified fraction

Determine the values of the trigonometric functions of a where sin a 25 can a CSC sec a 7 24 25 7 Type a simplified fraction Type a simplified fraction Type a simplified fraction and cc

3 Find the trigonometric function values for 6x using the graph of the unit circle A tan 60 Simplify your answer B The function is undefined OA cot 6x Simplify your answer OB The function is undefined 4 1 0 2 0 1 0 0 1 1 0 Select the correct choice below and if necessary fill in the answer box to complete your choice et A O 5

3 T Let f x cos x The Mean Value Theorem guarantees the existence of a value c 1 c 3 What is this value A 0 765 B 1 817 C 2 270 D 2 817

Graph the following function COS X 4 16 Use the graphing tool to graph the function Click to enlarge graph For any answer boxes shown with the grapher type an exact answer Type the word pi to insert the symbol as needed 81 6x 4x Ay 4 0 8 10 6 2x 9 4 0 2 0 2 6 08 H 2x 4x 6x 2 it ct

K Find the a amplitude b period c phase shift if any d vertical translation if any and e range of the following function Then graph the function over at least one period y cos 1 2 T a The amplitude is

The function graphed to the right is of the form y a sec bx c or y a csc bx c for some a 0 b 0 Determine the equation of the function Ay 10 18 6 14

The function graphed is of the form y a sin bx or y a cos bx where b 0 Determine the equation of the graph y Type an expression using x as the variable 0

The graph shows an idealized description of the temperature in F for approximately the last 150 thousand years of a location at the same latitude Complete parts a c a ring the nignest and lowest temperatures recordea The highest temperature is 60 F The lowest temperature is 20 F b Use these two numbers to find the amplitude The amplitude is 20 c Find the period of the function The period is yr 4 C Average Annual Temperature Idealized 120 000 90 000 60 000 Years ago 70 60 50 40 F 30 20 30 000 Q 10

The function graphed to the right is of the form y a sec bx c or y a csc bx c for some a 0 b 0 Determine the equation of the function C 10 Ay 8 6 4 112 77 LS Lat 817 10

nd the a period b phase shift if any and c range of the following function 1 y 2 csc 3x 1 a The period of y 2 csc 3x is Type an exact answer using as needed Use integers or fractions for any numbers in the

Find the a period b phase shift if any and c range of the following function 1 X 3 5 y y 3 cot I 5 a The period is Type an exact answer in terms of

The function graphed to the right is of the form y a tan bx or y a cot bx for some a 0 b 0 Determine the equation of the function Half and quarter points are identified by dots An equation of the function shown is y 10 8 6 4 6 Ay 10

In calculus it is often desired to replace radial expressions with trigonometric expressions The next group of exercises illustrates this Simplify the given expression as much as possible by making the given substitution Assume 0 0 90 65 Vx 9 x 3 tan 0 66 V 16 x 4 tan 0 67 36 x x 6 sin 0 69 Vx 25 x 5 sec 0 5 71 V4x 25 x tan 0 73 V16 9x x 4 3 sin 8 68 V100x x 10 sin 70 Vx 49 x 7 sec 0 72 9x 1 x tan 8 74 V10 x x 10 sin 8

In calculus it is often desired to replace radial expressions with trigonometric expressions The next group of exercises illustrates this Simplify the given expression as much as possible by making the given substitution Assume 0 0 90 65 Vx 9 x 3 tan 0 66 Vx 16 x 4 tan 0

Show that each of the following statements is an identity by transforming the left side of each one into the right side 43 cos tan 0 sin 0 45 sin 8 sec 0 cot 0 1 47 csc 8 49 cot 8 sec 8 sec 8 csc 8 51 sin tan 0 cos 0 sec 0 tan 8 53 tan 0 cot 0 sec 0 csc 0 cos 0 sin 0 55 csc sin 0 57 1 sin 0 cos 0 44 sin cot cos 0 46 cos csc tan 0 1 sec 0 48 50 tan 8 csc 8 csc 0 cot 0 sec 0 52 cos cot sin 0 csc 0 54 tan 0 1 sec 0 56 sec cos 0 si sin 0 cos 8 58 1 cos 0 sin 0

41 sin 8 c

41 sin 0 cc

Show that each of the following statements is an identity by transforming the left side of each one into the right side 43 cos tan 0 sin 0 45 sin sec 0 cot 0 1 47 49 csc 0 cot 0 sec 8 csc sec 0 tan 8 44 sin cot cos 0 46 cos csc tan 0 1 48 50 sec 8 tan 8 csc 8 sec 8 csc 8 cot 0

Factor 33 2

33 25 35 1 x 37 sin 0

Multiply 21 sin 4 sin 0 3 23 2 cos 0 3 4 cos 0 5 25 1 sin 1 sin 0 22 cos 2 cos 0 5 24 3 sin 0 2 5 sin 0 4 26 1 cos 0 1 cos 0

13 In year 2015 the population of a town was an annual 12000 and it was growing at rate of 3 2 Use this information for following Questions a write an exponential function of form pct town ab P Population of the and Number of years to b use your function from part a find algebraically the year the population reach 20000

Multiply 21 sin 4 sin 0 3 23 2 cos 0 3 4 cos 0 5 25 1 sin 0 1 sin 0 27 1 tan 0 1 tan 8 29 sin cos 0 22 cos 2 cos 0 5 0 24 3 sin 0 2 5 sin 0 4 26 1 cos 0 1 cos 0 28 1 cot 0 1 cot 8 30 cos sin 8

C 0 045 The function A 72e modies The decay of a sample of radio active material as a function of time In years In this modle A is the amount of radio active material remaining in grams at time t a How much radio active material Is preasent after 8 years round to tenth necrest The b Algebraically determine half life of this material show steps round to nearest tenth

Multiply 21 sin 4 sin 0 3

Add and subtract as indicated Then simplify your answers if possible answers in terms of sin 0 and or cos 0 sin 0 1 13 cose sin 0 15 1 sin 8 17 sin 0 1 cos e 1 cos 8 14 16 cos 8 sin 8 1 cos e 18 cos 0 sin 8 cos e 1 sin 0 1 sin 0

12 find all of the asymptotes vertical Horizontal and or oblique of the ration function f x 3x x x 4x 3

6 Find a curve that passes through the point 1 2 and has an arc length on the interval 2 6 given by 2 What is one such curve y 0 1 400x 6

The profile of the cables on a suspension bridge may be modeled by a parabola The central span of the bridge is 1280 m long and 160 m high The parabola y 0 00039x gives a good fit to the shape of the cables where x 640 and x and y are measured in meters Approximate the length of the cables that stretch between the tops of the two towers The length of the cables is approximately Round to the nearest whole number meters 160 m 1280 m

Find the arc length of the curve below on the given interval x 3 y 1 4x on 1 2 The length of the curve is Type an exact answer using radicals as needed

Write each of the following in terms of sin 0 and cos 0 and then simplify if possible Leave all answers in terms of sines and cosines 1 csc 0 cot 8 2 sec 0 cot 0 3 csc tan 0 5 sec 8 csc 8 4 sec tan csc 0 6 csc 8 sec 8

a Write and simplify the integral that gives the arc length of the following curve on the given interval b If necessary use technology to evaluate or approximate the integral 5 y for 5 x 15 a Which of the following is a simplified form of the integral OA OB O C D 15 51 15 615 15 S 51 15 5 5 dx dx 25 25 dx dx b The arc length of the curve is approximately