Application of derivatives Questions and Answers

Determine the derivative of the function f x 225 3e6e

2 Given f x x 12 and g x x 3 5 points svin sit worb A Find fog 5 points Smotenont Find gof 5 points

Given f x 8 3x6 1 3 Find lim e

What are the minimum and maximum values of x for which x x 6x 0 O minimum 3 maximum 0 O minimum 0 maximum 2 O minimum 3 no maximum minimum 2 no maximum

Evaluate the integral by using a substitution prior to integration by parts Se 5m 5m 9 5m 9 dm

Sketch the triangle 28 32 2C 48 b 45 CH A C 45 329 B to 18 A 45 48 32 B C C 45 A 32 B 18 Solve the triangle using the Law of Sines Round side lengths to the nearest Integer a 45 32 48 A B

Evaluate the integral 3x secx 0x 2 dx 2 3x see x dx

Solve the triangle using the Law of Sines Assume c 60 ZA 48 and 28 15 Round lengths to two decimal places x x X ZC U B

Evaluate the integral using integration by parts fxsin x dx x sin x dx

What is the solution of x x 9 x 1 X 3 2 1 0 3 TT 3 3 2 31 L 0 0 2 1 1 2 2 2 3 3 43 3 4 4

he given complex number a state the real part B state the imaginary or nonreal complex hat is the real part of 2 979 PUT part and 6 identify the num

Simplify the power of i 1 22 C 1 i 22 Simplify your answer Type your answer in the form a

Find the complex conjugate of the given complex number 4 2i The complex conjugate is Type an exact answer using radicals and as needed

Find the product and write the result in standard form i 6 4i 6 4i www i 6 4i 6 4i Type an integer or a simplified fraction Type your answer in the form a bi

Simplify the following expression to i 1 i or 1 35 35

5 4i 2 7i 5 4i 2 7i Simplify your answer Type your answer in the form a bi Do not fa

2 2 2 2 Simplify your answer Type your answer in the form a bi Type an exact answer using radicals as

Simplify the following expression to i 1 i or 1 i77 177 0

uply 2 6i 2 6i Simplify your answer Type your answer in the form a bi

implify 15 5 3 15 5 3

the following complex numbers Write the difference in star 5 9i 10i 8 9i 10i 8 Type your answer in the form a bi

8 5i 5 5i 8 5i 5 5i Simplify your answer Type your answer in the form a bi

14 14 Simplify your answer

Perform the indicated multiplication 4i 2 8i 4i 2 8i Simplify your answer Type your answer in the form a ba

EX 9 9 Simplify your answer Type your answer in the form a b

1 Determining values of 6 from a graph The function in the fig ure satisfies lim f x 6 Determine the largest value of 8 0 x 3 satisfying each statement a If 0 x 3 8 then f x 6 3 b If 0 x 3 8 then f x 6 1 YA 9 6 1 0 1 y f x 3 REGA 6 x

Consider the function f x 5 f 7 f 4 7 4 51 on the interval 4 7 Find the average rate of change of the function on this interval i e By the Mean Value Theorem we know there exists a c in the open interval 4 7 such that f c is equal to this average rate of change For this problem there is only one e that works Find this value of c

a Evaluate the integral 32 S x 4 Your answer should be in the form k where k is an integer What is the value of k darctan a Hint dz dx a1 a2 a3 1 224 1 k 4 b Now lets evaluate the same integral using power series First find the power series for the function f x 324 Then integrate it from 0 to 2 and call it S S should be an infinite series 4 What are the first few terms of S ao a4 c The answers to part a and b are equal why Hence if you divide your infinite series from b by k the answer to a you have found an estimate for the value of 7 in terms of an infinite series Approximate the value of TT by the first 5 terms d What is the upper bound for your error of your estimate for if you use the first 7 terms Use the alternating series estimation

over each of the indicated intervals a Interval 10 0 1 Absolute maximum 2 Absolute minimum b Interval 7 2 1 Absolute maximum 2 Absolute minimum c Interval 10 2 1 Absolute maximum 2 Absolute minimum f x x 12x 27x 6

The critical numbers of the function are t and t2 with tit f t 2t 3 5 3

me function f x 7 4 e 21 has one critical number Find the critical number

After a drug is taken orally the amount of the drug in the bloodstream after t hours is f t 118 e 0 4t et units Complete the following a Graph f t f t and f t in the window 0 12 by 20 75 b How many units of the drug are in the bloodstream after 7 hours c At what rate is the level of drug in the bloodstream increasing after 1 hour d While the level is decreasing when is the level of drug in the bloodstream 10 units e What is the greatest level of drug in the bloodstream and when is this level reached f When is the level of drug in the bloodstream decreasing the fastest a Choose the correct graph below OA B b After 7 hours there are 7 units of the drug in the bloodstream Round to the nearest integer as needed c After 1 hour the level of drug in the bloodstream is increasing at a rate of Round to the nearest integer as needed units per hour c Q O D

Find the area of a triangle with sides of length 9 and 13 and included angle 56 48 499 32 713 86 730 Ono correct answer

A 680 ft rope anchors a hot air balloon as shown in the figure 680 ft 0 a Express the angle 0 as a function of the height h of the balloon 0 b Find the angle 0 if the balloon is 510 ft high Round your answer to one decimal place 0 O

A 40 ft pole casts a shadow as shown in the figure 40 ft a Express the angle of elevation of the sun as a function of the length s of the shadow 0 b Find the angle of elevation of the sun when the shadow is 25 ft long Round your answer to one decima 0

A 84 ft tree casts a shadow that is 110 ft long What is the angle of elevation of the sun Round your answer to one decimal place

A sample of 12 grams of radioactive material is placed in a vault Let P t be the amount remaining after t years and let P t satisfy the differential equation P t 0 027 Answer parts a through g c What is the decay constant 0 027 d How much of the material will remain after 5 years 10 5 gram s Type an integer or decimal rounded to one decimal place as needed e How fast is the sample disintegrating when just 1 gram remains Use the differential equation Choose the correct process to find how fast the samp disintegrating when just 1 gram remains OA Solve P 1 0 027P t for P t OB Evaluate P t 0 027P 1 OC Solve 1 0 027P t for P t OD Evaluate P t 0 027 1 D

The price of a particular crop per bushel at time t in months is approximated by f t 8 0 002t 0 02e t What is the percentage rate of change of f t at t 0 t 1 t 2 HELLER The price of the crop is expected to change at a relative rate of 0 22 per month when t 0 Do not round until the final answer Then round to two decimal places as needed The price of the crop is expected to change at a relative rate of per month when t 1 Do not round until the final answer Then round to two decimal places as needed

20 Points t a t 1 t 2 t 3 V The displacement in meters of a particle moving in a straight line is given by the equation of motion s 5 t2 where t is measured in seconds Find the velocity of the particle at times t a t 1 t 2 and t 3 V V DETAILS V m s SCALCET8 2 7 015 m s m s m s MY NOTES PRACTICE ANC

49 60 Evaluate each limit 4x 10 49 lim x2 V 2x 51 lim X T cos x 3 cos x 2 cos x 1 53 lim 7 x 3 55 lim 57 lim sin x 1 2 Vsin x 1 2 cos x 1 x 0 sin x 1 59 lim 2 0 1 50 lim x 4 x 9 x 1 52 lim X 3 2 54 lim 56 lim 0 0 X 1 5 21 V 5 1 2 58 lim x 0 60 lim sin x 6 sin x 5 sin x 1 1 2 sin 0 sin 0 1 cos x sin x In x 5 ln x 6 In x 2

Rewrite the expression cos sin x as as an algebraic expression in x an

Find the angle in degrees rounded to one decimal 39 36

Find the angle in degrees rounded to one decimal O 8 10 6

An entrepreneur invests in a new play The cost includes an overhead of 31 200 plus production costs of 2100 per performance A sold out performance brings in 2750 Assume every performance is sold out and let x represent the number of sold out performances R x 2750x Type an expression using x as the variable c Determine the break even point 48 132000 Type an ordered pair Do not use commas in large numbers Describe what the break even point means Select the correct choice below and fill in the answer box to complete your choice O A For OB For O C For sold out performances both cost and revenue are sold out performances the gain is sold out performances the loss is

Problem 3 Let f x 7e a What are the domain and range of f b Find any asymptotes on the graph f c Find any x or y intercepts on the graph of f d Evaluate the average rate of change of f over the intervals 2 1 an

Sketch a triangle that has acute angle 8 13 csc 8 13 5 10 13 2 22 2 22 A A A A 5 10 13 13 2 22 2 22 O cos 8 O Find the other five trigonometric ratios of 8 sin 8 tan 8 9 O

Find the side labeled x X 30 19 X

Find the area of a triangle with the given description Round your answer to one decimal place a triangle with sides of length 5 and 6 and included angle 74

Find the area of the triangle whose sides have the given lengths Round your answer to one decimal place a 6 b 8 C 10 A

Find the indicated side x Use either the Law of Sines or the Law of Cosines as appropriate Assume LA 34 and ZC 86 Round your answer to two decimal pl X C B