Application of derivatives Questions and Answers

Find an equation of the largest sphere with center 4 2 7 that is contained in the first octant 2 8x 4y 12z 65 0 x

Name Moksha scontinuity holes vertical asymptotes x int erce range of each Then sketch the graph right 2 reflect over x strech 3 f x 3 x 2

log O 0 log 4 x log 4 log 4 log x 1 log log 4 log x

t 3x 2y 24 If y equals 8 and the total points t scored was 54 What is x

Solve the set of lincar equations with the LU Factorization Method Triangulation Method 2x x2 5x 2x X3 1 2x3 2 x3 5 3x3 x Note Show what you have done in your calculations at every step in detail Otherwise even if your calculation result is correct it will not be graded

7 10 Find all solutions in 0 2 Suppose sin a sin 2x 7 10 where 0 a

Question Function g is shown on the graph T 5 g x 2 0 5 4 2 Drag the pieces of function g to the correct locations in the description Not all pieces will be used For help this worked example X 24 3 0 52 x 3 3 x 2 x 2 0 52 4x 5 054 5 3 0 5 4

Calculate the number of possibilities How many hockey teams of 6 players can be formed from 14 players without regard to the position played

Calculate the number of possibilities Electing four candidates to a municipal planning board from a field of seven candidates

Oliver s grandparents set up an annuity for him He will be paid 350 at the end of each month for the next 5 years The first payment will be made 1 month from now How much do Oliver s grandparents need to deposit in the annuity Assume the money earns 5 7 compounded monthly for the toolbar press ALT F10 PC or ALT FN F10 Mac

Lisa has decided to invest 3000 at the end of each six months for the next five years in a savings account that pays 8 per annum compounded semi annually Determine the amount of the annuity For the toolbar press ALT F10 PC or ALT FN F10 Mac

2 12pts Sketch the graph of the following function by first graphing the appropriate Basic function Then show each following transformation on a separate graph one step at a time Label the function at each step Show and label any intercepts and asymptotes in each graph a f x ln x 4 2

graph of y f x shown as a red dashed curve drag the green movable points to draw the graph of y f x Notice that you can control the positioning of the reflective function with the coordinate labeled Drag Function and contr the width of the reflection with the coordinate labeled Control Width Provide your answer below 10 Drag Function 1 5 1 10 5 0 Control Width 5 5 10

1 In the function f x 3 4 the function shifts down 4 units and to the left 3 units If the horizontal shift is f x 3 why does it move towards the negative direction left 2 Given the function f 1 4 x would the transformation be horizontal or vertical and would it be a stretch or a compression

Find and classify the critical points of z x 2x y 8y Local maximums Local minimums DNE Saddle points 0 0 2 0 0 8 2 8 OF o Invalid not

The graph of y f x is shown below dashed curve Manipulate the green draggable points to obtain the graph of y f x 3 2 solid curve Provide your answer below 10 5 10 0 5 5 10

For the function f x 2x e determine intervals of increase and decrease local maximum and minimum values intervals of concavity and inflection points Do x show how you found your answers Write your solution on paper and upload a PDF or image of your solution

Question Identify the local maximum and local minimum of the function shown in the graph below 3 2 2 1 14 0 1 2 3 2 3 N 5 61

Identify the local maximum and local minimum of the function shown in the graph below Provide your answer below 8 7 6 5 3 2 4 5

Given the graph of the function f x shown below find the average rate of change of the function f x on the interval 100 250 300 250 200 150 100 50 0 10 50 f x 100 150 200 250 300

The graph of y f x is shown in blue Draw the graph of y 3f x shown in red by determining how a stretch or compression will transform the location of the indicated blue points Drag the movable red points to the desired coordinates Provide your answer below 164 15 14 13 12 11 10 9 8 1 8 7 9 3 2 5 4 0607 191 2 3 5 6 A 8 8 4 9 10 11 12 13 14 15 16

Given the graph of y f x below solve for f x 3 If you have multiple answers enter your answer in the form x a b 5 2 1 22 3 4 5 6

x a b graph of y f x below solve for f x 2 If you have multiple answers enter your answer in the form 6 5 4 3 4

B Most cars decrease in value after you buy it while other cars if deemed Classic increase in value after a certain time A new Porsche that costs 195 000 decreases by about 20 per year while a 1970 Classic Mustang presently worth 11 500 increases by about 7 per year a How long with it take for the value of the Mustang to be equal to the value of the Porsche round to one decimal b What will the value of both cars be

2 Determine whether each characteristic is True or x 2 4y 2 4x 8u 8 0

Suppose f x x 2 235 x 5 453 x 1 218 Determine all critical numbers of f x and the nature of each critical number i e local maximum local minimum or neither Explain how you found your answer

3 An arrow is shot straight up into the air The function H t 16t2 90t 6 models the height in feet of an arrow after t seconds Round answers to two decimal places as needed All answers must include appropriate units of measure Draw a GOOD graph of this function on the grid below Use it to help you answer the following questions a How long does it take for the arrow to reach its maximum height and what is the maximum height Write your answer in a complete sentence b How long does it take for the arrow to hit the ground Write your answer in a complete sentence c Identify the vertical intercept Write it as an ordered pair and interpret its meaning in a complete sentence d Determine the practical domain and range Write your answers using proper notation e Use your graphing calculator to solve the equation H t 80 Write a complete sentence explaining the meaning of your answer in the context of the problem

For the function f x 4 ln 1 X 1 Find all critical numbers of f 2 Find intervals of increase and decrease of f 3 Find all local maximum and minimum values of f

Find the sum of each series 1 1 2 1 4 1 128

Find the sum of the following series Question 6 12 18 to 23 terms 40 25 10 to 15 terms 36 31 26 to 18 terms 18 29 40 106 0 1 0 8 0 3 0 2 2 7 I Answer

tion x x 1 3 Harry and Ron are arguing over a concept of asymptotes Ron says that a function can only have either an HA or an SA but not both while Harry says that a function can have both an SA and an HA Who is right Explain c

2 What relationship determines Reynolds number in hydrodynamics P gV mgh 4 R pul 77 17 n R 2 R P gh 17 3 R 5 R R P gl 5 V

5 Let the function differentiated on the interval xe a b monotonically lecrease Then on this interval its first derivative is 1 y x 0 2 y x 0 3 y x 0 4 any

COMPARE Exponential vs Linear Now that you ve explored exponential growth consider how it compares to 1 Complete the table with a description of how each function is changing in the pattern and in the graph Visual Pattern How are the shapes changing in each step Graph How do you see the pattern reflected in the graph Figure 1 y 10x Linear Figure 2 Figure 3 Figure 1 y 3 Exponential Figure 2 Figure 3

graph of f and g p x f x g x 9 x f x g x Lo 2 a Find p 5 p 5 10 b Find q 1 q 1 8 6 2 2 b 2 S 8 9 10

Determine the point at which the graph of the function below has a horizontal tangent line If the function has no horizontal tangent line enter NONE 9 y x y

Determine the points at which the graph of the function has a horizontal tangent line x X 7 f x x y x y smaller x value larger x value

Use the function and its derivative to determine any points on the graph off at which the tangen line is horizontal over the interval 0 2 f x x 2sinx f x 1 2cosx

Determine the points in the interval 0 2x at which the graph of the function below has a horizontal tangent line f x 10 cos x 5 sin 2x x y x y x y smallest x value largest x value

The value V of a machine t years after it is purchased is inversely proportional to the square root of t 1 The initial value of the machine is 10 a Write V as a function of t V b Find the rate of depreciation when t 1 Round your answer to two decimal places dollars year c Find the rate of depreciation when t 3 dollars year

Identify a function f that has the given characteristics Then sketch the function f 3 14 f x 4 x f x S No Solution 10 9 8 7 6 5 4 3 2 1 20 18 16 14 12 10 8 6 4 2 2 4 6 2 3 4 5 6 7 8 10 9 10 71 Fill

Sketch the graph of an example of a function that satisfies all of the given conditions RO 0 lim f x 0 lim x 00 fis odd x 6 x480 10 10 y 10 10 10 10 X 10 10 y 10 10 X 10 10 10 10 o D 10

a Use a graph of PX 1 1 f x to estimate the value of lim f x correct to two decimal places X b Complete the following table of values of f x to estimate the limit to four decimal places Round your answers to six decimal places 10 000 100 000 1 000 000 f x Estimate the limit to four decimal place

Determine whether f x approaches or as x approaches 2 from the left and from the right TX 4 f x sec lim sec X 2 lim sec X 2 11 11 F 6 H 2 2 6

Consider the following function and graph F x 6 24 10 8 Determine whether f x approaches or as x approaches 2 from the left and from the lim f x a x 2 b lim f x 10

population is modeled by the differential equation N 1 3N 1 4400 dN dt where N t is the number of individuals at time t measured in days a For what values of N is the population increasing Enter your answer using interval notation b For what values of N is the population decreasing Enter your answer using interval notation c What are the equilibrium solutions Enter your answers as a comma separated list N

relative extrema Also give a possible formula for the graph below

1 point The graph below shows the fuel consumption in miles per gallon mpg of a car driving at various speeds in miles pe mph 2 npg 30 HH 20 a How much gas is used on a 500 mile trip at 40 mph amount of gas v nph 10 20 90 40 58 60 78 80 gallons click on image to enlarge 90 100 b How much gas is saved by traveling 60 mph instead of 70 mph on a 300 mile trip saved gas gallons c According to this graph what is the most fuel efficient speed to travel most fuel efficient speed mph

Find the complete solution of the difference equation by using the method of undetermined coefficients x n 10x n 1 25x n 2 8 5 n where x 0 6 x 1 10

The real solutions to 3x6x4 12x 4 0 are 3 3 2 3 O 2 1 2 O 1 1 2 5 3 2