Application of derivatives Questions and Answers

Find the length of the following two dimensional curve r t 10 cos t 10 sin t for Osts The length of the curve is Type an exact answer using as needed

For the following trajectory find the speed associated with the trajectory and then find the length of the trajectory on the given interval r t 4t t3 6t for Ost 4

3 y 3 x 2x 4x 1 Avg Ava rate Avg rate between x 1 and x 41 A la

Consider the following trajectory of a moving object Find the tangential and normal components of the acceleration r t 11 cost 11 sint 24t a N T Type exact answers using radicals as needed

If f x f x f 5 4 sin x 2 cos x then

Use the alternative curvature formula x K r t 4 21 t 0 axv v to find the curvature of the following parameterized curve

43 Draw a diagram and find the remaining part of the right triangle If b 48 A 36 a OA 51 B 35 C 66 D 29

Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve Verify that T N 1 and T N 0 r t T 2 31 6

31 sin 8 48 cos 20 What is the second step to finde A 8 48 28 90 B 28 0 48 90 C 8 48 20 180 O D 28 8 48 180

35 Draw and label a right triangle using the given sides then solve for the unknown side and write the trigonometric functions of angles A and B if a 5 and c 7 cos A OA OB 75 C 7 2 6 C 2 6 7 5

Find the unit tangent vector T and the curvature x for the following parameterized curve r t 7 cost 7 sint 7t Choose the correct answer for the unit tangent vector of r t 14 4 OA T t B T t OC T t K OD T t 14 4 1 VIA 1 4 sint 14 14 sin t 4 14 14 4 sin t sint cost 14 4 cost 14 2 cost 4 4 cost 2 4 2 4 2 4

32 sin 8 48 cos 28 8 A 132 OB 14 O C 44 O D 42

26 sec 246 OA 1 095 OB 2 459 O C 1 095 D 2 459

Graph the following function Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima f x 4x Use the graphing tool to graph the function Click to enlarge graph Determine on which interval s f x is increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f is increasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB There is no interval on which the function f x is increasing Determine on which interval s f x is decreasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f is decreasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB There is no interval on which the function f x is decreasing Determine the relative maxima or minima Select the correct choice below and if necessary fill in the answer box to complete your choice A The function has a relative minimum and the minimum value is OB The function has no relative minimum at x

x cos y x y dx x dy 0 FINAL ANSWER y arctan ln x c1 x anx X

y x 2 dy 1 12 In 12 2 1 12 10 x 1 21 In x 1 In y arcsec c X c1 X Show the whole solution of the given problem and regarding with the final answer thats the one with below the equation which is 1 In and so on

If f x 5x sin x cos x find f x f 3

If g x sin x then g x cos x g 2 0 42

Find the x values of all points where the function has any relative extrema Find the value s of any relative extrer f x x6 ex 8 Select the correct choice below and if necessary fill in any answer boxes within your choice OA There are no relative minima The function has a relative maximum of at x Use a comma to separate answers as needed Type exact answers in terms of e at x OB There are no relative maxima The function has a relative minimum of Use a comma to separate answers as needed Type exact answers in terms of e OC The function has a relative maximum of at x and a relative minimum of Use a comma to separate answers as needed Type exact answers in terms of e There are no relative extrema at x

Find the point if it exists at which the following plane and line intersect 4x 8y 6z 19 and x 10 7t y 9 9t z 3 4t Select the correct choice below and if necessary fill in any answer boxes to complete your choice A The point at which the line intersects the plane is Simplify your answers Type an ordered triple OB The line and the plane do not intersect

15 Find the angle e to the nearest minute sin 8 0 1855 A 9 32 B 9 38 OC 9 39 D 9 31

Given the following number as an approximate measurement 49 5 feet What is the range of values for the actual value OA 49 0 to 50 0 OB 49 45 to 49 55 OC 49 49 to 49 51 D 49 4 to 49 6

3 Perform the indicated operation and give each result with the proper number of significant digits 4 7 2 65 A 2 05 B 2 1 C 2 15 D 2 0

5 Round off 4 465 to one significant digit A 0 OB 5 OC 4 OD 3

3 Round off 4 465 to three significant digits A 4 45 B 4 50 C 4 46 D 4 47

for the angle whose terminal side passes through the point p 5 4

stion factors determine the amount of simple interest ean all that apply Number of bank accounts Time Interest rate

Find parametric equations that describe the circular path of the following object Assume x y denotes the position of the object relative to the origin at the center of the circle Use the units of time given in the description A go cart moves clockwise with constant speed around a circular track of radius 200 m completing one lap in 1 8 min Assume the center of the track is at the origin and that the go cart starts at 200 0 What are the parametric equations x t and y t Type exact answers using as needed

2 1 f x x 5x 7 2 f x 24 3 f x x 11x 6 4 f x x 3x 5 f x x 3x 1 Note 1 2 x x 41

Consider the following parametric equations x t 2 y 4 t 0sts 16 a Eliminate the parameter to obtain an equation in x and y b Describe the curve and indicate the positive orientation

A regular Pentagon with sides that are 20m long is shown What is the approximate Area of the Pentagon

e interest is compounded more than once per year which rate is higher the ose the correct answer below A The stated rate is higher B The effective rate and the stated rate are the same C There is not enough information to decide D The effective rate is higher

the present value the amount that should be invested now to accumulate the followi Dounded as indicated 9000 at 9 compounded quarterly for 9 years present value is ot round until the final answer Then round to the nearest cent as needed

Question 13 5 2 3 ect all that apply at factors determine the amount of interest eamed on a fixed prin ount A Interest rate B Number of compounding periods C Frequency of periodic payments D None of the above EO

passes through 2 2 perpendicular to y 5x 8 CHALLENGE Find the value of n so that the line perpendicular to the line with equation 2y 4 6x 8 passes through the points at n 4 and 2 8

ptions Let x x 1 2 A 2 27 A 2 3 27 7 2 7 7 2 a sinh 2 4 12 f x x sin y f y dy Then a b is equal to 0 cos x xe IR be

ODD function Question 3 25 points A function f x is an EVEN function If f 4 2 which other point lies on the graph of g x 4 2 O 4 2 0 4 2 2 4

The definition of the derivative of a function is f x h f x f x lim h 0 h Write but do not attempt to compute the following derivatives I If f x 2 sin x f x lim h 0 II If f x ex tan x f x lim h 0 X

What is the area of the largest equilateral triangle that can be cut from a circular piece of material 50 centimeters in diameter

2 12 13 15 17 and E 13 Insert or to make the A E 3 Corto make the statement true E

Find the minimum value of the function f x y x y xy subject to the constraint 2x y 1 0

9 Find a general solution to the following difference equation Yn 2 2Yn 1 Yn 1 3 Answer y C C n n n

Consider the following function Function Factors f x 2 x 3 x 2 6 x 3 quad 2 x 1 x sqrt 3 a Verify the given factors of f x b Find the remaining factor s of f x Enter your answers as a comma s c Use your results to write the complete factorization of f x f x d List all real zeros of f Enter your answers as a comma separated list e Confirm your results by using a graphing utility to graph the function

0 You have 208 feet of fence to make a rectangular garden What are the dimensions of the garden that will enclose the largest area Type a response

For each function in Problems 25 30 find the relative maxima relative minima and horizontal points of inflection then sketch the graph Check your graph with a graphing utility 25 y x x x 1

35 Find the numbers at which the function f x x 2 if x 0 et 2 x if 0 x 1 if x 1 is discontinuous At which of these points is f continuous from the right from the left or neither Sketch the graph of f

he separable IVP x t dx dt 4 with initial condition z 0 6 has solution X

du e u 3t u 0 13 dt has solution u t

On the interval 0 0 this separable IVP dy 9 11x dx xy has solution y x y 1 3

Find an equation of the tangent line at x 0 y x cos x Use symbolic notation and fractions where needed equation