Application of derivatives Questions and Answers

9 A particle moves along the x axis with its position on the x axis at time t given by p t 2t 0 t 6 At what time is the velocity equal to 3 6 t

11 An open box with a square base is to have a volume of 4000 in Determine the length of the side of the square base that would result in the least amount of material needed to construct the box

4 Use the 8 definition technique to prove lim x 5 1 1 x 1

In class 30 points SHOW Mathematics is not about numbers equations computations or algorithms it is about understanding 3 5 pts 6 4 177 Determine the length of y 4 William Paul Thurston American mathematician from x 1 to x 2

Pure mathematics is in its way the poetry of logical ideas Albert Einstein SHOW WORK ON EACH PROBLEM FOR ANY PARTIAL CREDIT 10 pts each 10 All the edges of a cube are expanding at a rate of 4 in per second How fast is the volume changing when each edge is 10 inches long

6 Determine the equation of the tangent line to the curve f x x x 3x at x 1

An airplane is flying in the heading of 20 with an air speed of 300 mph Its ground speed and true ourse are 350 mph and 30 respectively Approximate the speed and direction of the wind 75 mph N163 E V Zocos 300 ZOSIN300 Vaz 30

S 36 2 dz Evaluate the definite integral

A company predicts that if it sells its toys at a price of x dollars their income y also x 3000x 3000 Based in dollars will be given by the formula y on the graph of this function at what price should they sell their toys in order to maximize income Round your answer to the nearest cent

Find the vertex focus and directrix of the parabola Then sketch the parabola y 10x vertex focus directrix O S No Gelton 0 0 5 30 Graph L Hon

Your friend says that the helium used to inflate balloons is a product of radioactive decay Another friend says no way With whom do you agree

The vector function r t is the position of a particle in space at time t Determine the graph of the position function r t 3t cos t i 2t sin t j 2tk

2 5 pts Openstax 6 2 102 Graph the equations and shade the area between the curves Determine the area rotated around the y axis x 9 y x e y 0 y 3 6 4 2 6 4 2 2 6 2 4 20 6

Integration Appl Name SHOW WORK FOR ANY CREDIT We are what we repeatedly do Excellence then is not an act but a habit Aristotle TAKE HOME PORTION USE ONLY OUR NOTES AS RESOURCES 70 points 12 5 pts Openstax 6 9 410 Calculate sinh x d dx

Test the series for convergence or divergence 1 1 In 4 In 5 1 converges O diverges 1 In 6 1 In 7 1 In 8

A ladder 10 ft long rests against a vertical wall If the bottom of the ladder slides away from the wall at a rate of 1 3 ft s how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall That is find the angle s rate of change when the bottom of the ladder is 8 ft from the wall rad s

1 f x 8 6 x 6x 8 4x 8x 32 42 8 6 2 y X

5 f x 2x 6x 4 3 x 3x 2x 8 6 4 2 8 6 2 2 6 y 8 X

4 x 8 6 4 4 x x 6 2 5 2 2 2 4 Ay 8 6 2 X

6 f x 48 6 x 2x 8x 3 X x x 2x 3 X 4 2 2 2 4 6 Ay 8 2 X

4 f x 8 6 x x 3 x x 2x 3 X 4 2 12 4 2 No 2 4 y 2 6 8 x

Select the answer for each question below 20 Select the vertex form of f x 3x 12x 8 10 A f x 3 x 2 4 D f x 3 x 2 4 B f x 3 x 1 2 f x 3 x 2 E 21 If tan 120 5x cot 6 x then x 52 A 0 D 25 B E 10 30 C f x 3 x 4x 8 C 20

24 Megan started her own retail company and just released three new clothing items She is selling a t shirt for 65 10 a pair of shorts for 15 and a hat for 10 She sold a total of 100 items and made 1 125 after the first week of the release If she sold as many hats as the other two items combined how many of each type of clothing did Megan sell

Which inequality does the shaded region represent y 2x 3 y 2x 3 y 1 2 x 3 y 1 2 x 3

ress your answer in terms of t where x x 2y z 33 2x 3y 4z 70 3x 6y 3z 98

8 2 5 a Notice that f 5 0 65 What does this tell us about the numerator and the denominator of f When z 5 the value of is times as large as the value of 5 Hint In this case the first answer should be an expression in terms of r and the second answer should be a number Suppose f x b Use function notation to represent the answer to the question When a 10 1 2 8 is how many times as large as 2 52 Preview

Classify the equilibrium is x 0 0 for the first order system x 1 d dt x2 1 2 2 38 1 x2 1

1 Use the graph of the function f shown below to estimate the following limits a 1 b C lim f x X lim f x 1 1 lim f x 1 1

6 Find a the average rate of change of the function over the interval 3 3 01 and b the instantaneous rate of change at x 3 F x x 2x 3

11 Find the volume of the solid A x 2 y x 0 y 9 about the y axis 9 25 3 2 55 1 7 1 47 9 36 2 92 1 B y x y 0 x 1 about x 2 y 429 9 S272 5 X 2

Find the consumers surplus at a price level of p 120 for the price demand equation below p D x 200 0 04x What is the consumer surplus

Find the total income produced by a continuous income stream in the first 5 years if the rate of flow is given by the following function where t is time in years f t 4000 How much was earned over 5 years

The life expectancy in years of a certain brand of clock radio is a continuous random variable with the probability density function below f x 2 x 2 ifx20 otherwise A Find the probability that a randomly selected clock lasts at most 6 years B Find the probability that a randomly selected clock radio lasts from 6 to 9 years C Graph y f x for 0 9 and show the shaded region for part A A What is the probability that a clock will last no more than 6 years Type a decimal rounded to three decimal places as needed

ea calculator to find the LRAM area approximation the area under the graph f x x 8 from 0 to x 2 with 20 approximating rectangles area is

Find the average rate of change of f x 3cos x over the interval Select one Ob It 3 O It 9 ft It 2it 3

The following table lists the population statistics for a certain city Year Population 1975 695 854 1985 720 928 1995 742 547 2005 750 370 What is the average rate of change in the population with respect to time between 1975 and 2005 ne average rate of change is Frite your answer as a decimal

Find the indefinite integral 16 tan 16 tan x dx 16 tan x dx

Find the indefinite integral Sx 8 cos xdx 7 8 X cos xdx

Use the given information to solve the triangle shown in the figure to the right A 50 b 8 2 B a C O B Round to the nearest degree as needed Type an integer or decimal rounded to one decimal place as needed Type an integer or decimal rounded to one decimal place as needed A b C

Recall that radians corresponds to 180 Mentally convert the given radian measure to degree measure T 6 H rad

Find the trigonometric ratio by referring to the figure cos a cos 10 8 B

Convert the radian measure to degrees 134450 THE STATE ASS 18 yea GERMAKNALA VESTEERI MARKALA NIE SOMER ARTESARFEDTEG SIRAVERSTERN DENGEMBENEX O DIKETA ERYKABIN 11T 9 LUATENUPER BRUSINAARING LA DESER 11T BERAR FER WARS 15164 ERRONK MI radians 2

Discuss the validity of each statement If the statement is always true explain why If it is not always true give a counterexample If the area under the graph of f on a b is equal to both the left sum L and the right sum R for some positive integer n then f is constant on a b Select the correct answer below A BRE The statement is always true because if a function is either increasing or decreasing over an interval the left sum will not equal the right sum The statement is false The area under the graph of f x x 1 on 0 2 satisfies the given conditions but is OB not constant Oc The statement is always true because the only time the left sum equals the right sum is for constant functions The statement is false Any linear function on a closed and bounded interval satisfies the given conditions but is not necessarily constant D

Question 8 Give decreasing interval for f x in interval notatio f x x 1 5 x 6x O 2 1 O none of these answers O 1 2 O 1 2 Question 9

Question 11 Find equation of Horizontal Asymtotes p x 3x 1 x x 2 Oy 3 O y 3 34 O y 0 Othere is no Horizontal Asymtotes

find the coordinate of the focus for this parabola x 6y O 0 3 O 0 3 O none of these answers

O none of these answers O 1 2 O 1 2 Question 9 Find domain of the g x g x 2x 6 O none of these answers O any number less than or equal to 2 O any number greater than or equal to 1 3 O any number greater or equal to 2

Onone of these answers O 3 0 Question 6 Solve this equation note base of the log is 3 Log3 2x 1 log3 x 4 2 05 07 O 3 O none of these answers Question 7

Which one of the graphs given below is for y log x 2 O c Od Ob y X 2 X 2 7x XV y X 2 X 2 O 74 74

y log x 2 What is equation of the asymptote for above function Oy 2 a horizontal Asymptotes Ox 2 a vertical Asymptotes O y 2 a horizontal Asymptotes Onone of these Question 4