Application of derivatives Questions and Answers

A 10 foot ladder is leaning against a wall If the top of the ladder slides down the wall at a rate of 2 feet per second how fast is the bottom moving along the ground when the bottom of the ladder is 1 feet from the wall

A tank contains 70 kg of salt and 2000 L of water Pure water enters a tank at the rate 12 L min The solution is mixed and drains from the tank at the rate 6 L min a What is the amount of salt in the tank initially kg amount b Find the amount of salt in the tank after 2 5 hours kg amount c Find the concentration of salt in the solution in the tank as time approaches infinity Assume your tank is large enough to hold all the solution

Find the horizontal H A asymptote and the vertical s asymptotes V A of the function 2x 2 f x x x 2

Find the asymptotes of the function f x 2 x 4 x x 2

Let f x 2 x 3 Find 1

Find the vertical asymptotes VA of f x x 5x 6 x 9 a Lines x 2 and x 3 b Lines x 2 and x 3 c Line x 2 d Lines y 2 and y 3 e Lines y 1 Are they any holes Where

If a function is one to one then could points 4 2 and 4 2 lie on the graph of the function and why If a function is even then could points 4 2 and 4 2 lie on the graph of the function and why If a function is even then could points 4 2 and 4 2 lie on the graph of the function and why If a function is odd then could points 4 2 and 4 2 lie on the graph of the function and why If a function is odd then could points 4 2 and 4 2 lie on the graph of the function and why

3 x Solve the inequality 3 Express the solution set in interval notation x 2

x 0 x4 43 lim x sin m x x18

2 20 pts Consider the quadratic function g x 2x 16x 29 a What is the y intercept of the function for yintercept put O y 2x0 16 Y0 29 29 b What are the x intercepts of the function If the function has no x intercepts say so but show your work c Use completing the square to write the function in standard form d List a sequence of transformations to get from the graph of y x to the graph of g Sketch a graph of y g x Does the graph open upward or downward Where does the global maximum or minimum value of the function occur What is this global maximum or minimum value Where is the function increasing and decreasing same 0772 mo m T fo Re

a Use definition 1 p 133 to find the slope of the line tangent to the graph off at P b Determine an equation of the tangent line at P c Plot the graph of f and the tangent line at P 15 f x x 5 P 3 4 16 f x 3x 5x 1 P 1 7 1 17 f x P 1 1 X 4 18 f x 2 P 1 4 19 f x 3x 3 P 2 3 720 f x P 4 1

23 28 a Find the intervals on which f is increasing or decreasing b Find the local maximum and minimum values of f c Find the intervals of concavity and the inflection points 23 f x x 2x 3 24 f x X x 1

Find the consumer surplus for the demand curve p 100 3q when q 5 units are sold at the equilibrium price The consumer surplus is 187 5 X

23 f x 25 f x A B Cex X X X 24 f x 26 f x 1 xe x ex ax b CX d

1 Find the derivative of y x 1 x 1 in two ways by using the Product Rule and by performing the multiplication first Do your answers agree 2 Find the derivative of the function F x x 3x x x in two ways by using the Quotient Rule and by simplifying first Show that your answers are equivalent Which method do you prefer

What is p x when p x log4 Select the correct answer below O p x O p x O p x O p x 12x 5 6x 5x 2 x 4ln 4 6x 5x 2 6 I 1 6x 5x 2 ln 4 12x 5 6x 5x 2 ln 4 1 6 zln 4 6 ln 4 6x 5x 2 x64 1 1

Consider the following rational expression Step 2 of 2 Determine what value s if any of the variable must be excluded Answer Select the number of values that must be excluded 27x 64 3x 4 Selecting an option will display any text boxes needed to complete your answer 00 1 02 03

x 1 What is the first derivative of g x ln 1 Select the correct answer below 3x Og x x 1 3x x 1 Og x Og x Og x x 1 3x 3x x 1 473 x 1 4x 2 1 x 1 4x 47 1

27x 6 3x 4

Graph the function f x x 6x by starting with the graph of y x and using transformations shifting stretching compressing and or reflecting Use the graphing tool to graph the function Click to enlarge graph Ay SHEEPSK 30 20

The monthly revenue R achieved by selling x wristwatches is figured to be R x 71x 0 2x The monthly cost C of selling x wristwatches is C x 30x 1750 a How many wristwatches must the firm sell to maximize revenue What is the maximum revenue b Profit is given as P x R x C x What is the profit function c How many wristwatches must the firm sell to maximize profit What is the maximum profit d Provide a reasonable explanation as to why the answers found in parts a and c differ Explain why a quadratic function is a reasonable model for reventa a The firm must sell wristwatches to maximize revenue Round to the nearest integer as needed The maximum revenue is Round to two decimal places as needed

Graph the function f x x 2 4 by starting with the graph of y x and using transformations shifting stretching compressing and or reflecting Use the graphing tool to graph the function Click to enlarge graph 10 6 4 10 8

What is t x when t x ln In 2 3 2 2 Select the correct answer below O t x 2 O t x 2 O t x O t x 2x 4x 3 2x 3x 2 4x 3 2x 3x 2 2r 3x 2 4x 3 4x 3 2x 3x 2

Graph the following function by starting with the graph of y x and using transformations shifting compressing stretching and or reflection 52 f x 4x Lan the graphing tool to graph the function

9 16 Find the intervals on which f is increasing or decreasing and find the local maximum and minimum values of f 9 f x 2x 15x 24x 5 10 f x x 6x 135x 11 f x 6x4 16x 1 x 24 x 5 13 f x 12 f x x x 3 4 14 f x x 2 X

A roast turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room where the temperature is 75 F Round your answers to the nearest whole number a If the temperature of the turkey is 150 F after half an hour what is the temperature after 60 minutes T 60 F b When will the turkey have cooled to 90 t min

u cos2x dv 4x 4x cos2x dx du V Seurax ife U 4x du 4 dx 2sin 2x elix 4 du Scos2x Seas 4x e4x 47 e4x cos 2x 4 cos 2x 4 cos 2x cos 2x LX e ex sin 2x 8 4X U 2sin 2x du 4052 dv e4xs 4 V e x 16 4x 23 in 23 e4x fett 7 16 16 4cos2x 2 sin 2x sin 2x S 1 4x cos2x dx 4x e4 cos 2x 4

1 2 Use the given graph of f to find the following a The open intervals on which f is increasing b The open intervals on which f is decreasing c The open intervals on which f is concave upward d The open intervals on which f is concave downward e The coordinates of the points of inflection 1 y 2 y 1 0 1 x 4 a State the First Derivative Test b 5 yt 1 0 3 Suppose you are given a formula for a function f a How do you determine where f is increasing or decreasing b How do you determine where the graph of f is concave upward or concave downward c How do you locate inflection points 1 State the Second Derivative Test Under what circum stances is it inconclusive What do you do if it fails y f x 5 6 The graph of the derivative f of a function f is shown a On what intervals is f increasing Decreasing b At what values of x does f have a local maximum Local minimum X 6 YA 2 YA 0 y f x 7 In each part state the x coordinates of the inflection points of f Give reasons for your answers a The curve is the graph of f b The curve is the graph of f c The curve is the graph of f 2 4 4 6 X m 6 8 8 The graph of the first derivative f of a function f is shown a On what intervals is f increasing Explain b At what values of x does f have a local maximum or minimum Explain y f x X c On what intervals is f concave upward or concave downward Explain d What are the x coordinates of the inflection points of f Why

The square root of a number is greater than the square of the number For what set of numbers is this true 0 1 O 0 1 O 1 00 O 1 00

A vitamin degrades in the body according to the function represented in the table where x is the time in hours and f x is the dosage in milligrams X 0 4 8 12 24 500 00 158 20 50 06 15 8 0 50 What was the original dose O 0 50 milligrams O 8 milligrams O24 milligrams 500 milligrams

The inequality log 2x 7 2 3x 2 can be broken into two related inequalities Review the graphs of the two related inequalities 5 4 3 2 1 5 4 3 2 2 3 5 y 1 2 3 5 X Based on the graph which set shows the solution to log 2x 7 2 x 2 O 00 00 O 3 5 1 5 O 3 5 00 O 1 5 00

An investor has purchased 1 000 shares of stock at a price of 10 per share The stock pays no dividends and historically has a growth rate of 12 5 per year compounded continuously Which function models this scenario O A t 1 000e12 5t O A t 1 000e 0 125 OA 10 000e12 5t A t 10 000e 0 125t

If f x n x find f 1 f 1 9 Submit Answer

Differentiate f x the function f x sin 7 In x

a b The derivative can be used to find the O a point of intersection b common factors d Both b and c are correct What value is the following limit calculating O ld c instantaneous rate of change lim 4th 4 h 0 0 for f x x 4 fi 4 for f x x ft 16 for f x x fi 4 for f x x lim fiz h f x h lim h e h ho IC lim h f a h 0 h A notation for derivative a f x d dy da The derivative of a function f in terms of a limit is defined by which of the following lim f x f a a 20 b lim c fd x

Derivatives of an inverse function Use the values of a one to one d ferentiable function fin Table 3 7 to compute the indicated derivatives or state that t derivative cannot be determined a 5 Table 3 7 X f x x 1 2 1 2 b 2 0 3 2 1 5 3 2 C 2 6 3 7 2 3

If x 9 find x O f x 4x 948 1 O f x 4 ln 9 94 Of x 9In 4 94 Of x 4 94

Let h x f x g x If f x 4x 3x and g x In 3x 1 what is h x Select the correct answer below Oh x 3 8x In 3x 1 Oh x 8x 3 3x 1 Oh x 8x 3 In 3x 1 Oh x 8x 3 In 3x 1 Oh x 8x 3 ln 3x 1 12x 9x 1 3x 12x 9x 3x 1 4x 3x 3x 1 12x 9x 3x 1 12x 9x 3x 1

Let h x f x g x If f x x 1 and g x x 4x 2 what is h 2 Do not include h 2 in your answer For example if you found h 2 7 you would enter 7

Let z z 13 2x 6 Find an equation for the tangent line to the graph of fat z 4 Tangent line y

Find the derivative of g x 87 U 26

Find the average value of f x x on the interval 2 6 3

Let f x 2 sin cos

If f x then x 0 t 4t 7 dt

Local Maximum or Minimum on an Interval f x 2x 9x 4 x 6x 3 x 2x 16 x 2 3x 4x Sketch 36x 20 Interval 1 x 5 4 x 8 0 x 14 3 x 4 1 J 4 Endpoints 0 14 3 4 5

At the start of a trip the odometer on a car reads 20809 miles At the end of the trip 15 hours later the odometer reads 21370 miles What is the average speed the car traveled during this trip miles hour

Find the intervals where the function y f x whose graph is given below is increasing and where it is decreasing and find the value of the absolute minimum Increasing Decreasing 1 0 Click on graph to enlarge Absolute minimum y help intervals help intervals help numbers

Find the average rate of change for f t 4t 9 over the interval 2 2 h help formulas

Find the average rate of change for f x 5x 3x over the interval 4 5 Be sure to simplify the answe help numbers

a The graph of y f x is shown Draw the graph of y f x Ex X